Finite Difference Methods for the Solution of FractionalDiffusion Equations

Orlando Miguel Reis e Ribeiro Santos

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisors: Prof. José Carlos Fernandes PereiraProf. José Manuel da Silva Chaves Ribeiro Pereira

Examination Committee

Chairperson: Prof. Filipe Szolnoky Ramos Pinto CunhaSupervisor: Prof. José Carlos Fernandes Pereira

Member of the Committee: Prof. Duarte Pedro Mata de Oliveira Valério

November 2016

ii

Dedicated to my family.

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Acknowledgments

I would like to start by expressing my deepest gratitude to my supervisors, Prof. Jose Carlos Pereira

and Prof. Jose Chaves Pereira, for the all the knowledge, feedback and constant support that always

kept me motivated during the development of this thesis.

A big thank you to my colleagues at LASEF, for their help and for always making me feel welcome.

I am also grateful to all my friends, for all the shared laughs and moments that made this journey

much more joyful.

I would like to end by remembering all the support and encouragement I received from my family and

my girlfriend. Each in their own way, they have helped me become who I am and without them I would

not be writing this thesis.

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Resumo

O calculo fraccional e uma disciplina matematica que lida com integrais e derivadas de ordem ar-

bitraria que tem vindo a encontrar aplicacoes na fısica, processamento de sinais, engenharia, biociencias

e financas. A difusao anomala tem recebido bastante atencao por parte do calculo fraccional. Nas

equacoes fraccionais de difusao as derivadas normais sao substituıdas por derivadas de ordem frac-

cional, dando origem a equacoes fraccionais no tempo, espaco e tempo-espaco. Dado que a solucao

analıtica de equacoes de difusao fraccionarias e difıcil de obter, os metodos de diferencas finitas

tornaram-se bastante populares havendo um grande numero de esquemas recentemente publicado.

Foram seleccionados tres esquemas com ordens crescentes para cada um dos subtipos de equacao

de difusao fraccional. A construcao de cada um dos esquemas e sumarizada e cada um e implemen-

tado de modo a permitir a sua validacao e comparacao com os restantes.

Apesar do seu sucesso, as equacoes de difusao faccionais de ordem constante mostraram dificul-

dades na modelacao de fenomenos mais complexos. Para as ultrapassar, foram propostas derivadas

de ordem variavel, funcao do tempo e/ou espaco,sendo importante entender claramente como e que

a ordem variavel afecta o comportamento de um sistema difusivo. Uma equacao de difusao de ordem

variavel no tempo e resolvida atraves de um esquema de diferencas finitas e a sua forma matricial

e apresentada. A implementacao e validada e usada para estudar a ordem variavel como funcao do

espaco, tempo ou ate da solucao da equacao que sao comparadas com a ordem constante.

Palavras-chave: Calculo Fraccional, Derivada Fraccional, Equacao da Difusao Fraccional,

Difusao Anomala, Metodos de Differencas Finitas, Ordem Variavel

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Abstract

Fractional calculus is a mathematical field dealing with integrals and derivatives of arbitrary order. In

recent times fractional calculus has found applications in physics, signal-processing, engineering, bio-

science, and finance. Anomalous diffusion has received particular interest in the framework of fractional

calculus applications. In fractional diffusion equations, standard derivatives are replaced by fractional

order counterparts, originating time, space and time-space fractional diffusion equations. Since the an-

alytical solution of fractional differential equations is hard to obtain, finite difference methods in particular

became very popular and a large number of schemes has been published very recently. Three different

schemes with increasing order of accuracy were selected for time, space and time-space fractional diffu-

sion equations. To fulfil the first objective of this work, the construction of these schemes is summarized

and then with numerical examples, solved through self-written code, a comparison is made in terms of

accuracy and computational cost.

Despite their success, constant fractional order differential equations showed difficulties in modelling

complex phenomena. To overcome these difficulties, variable order fractional derivatives, whose order is

function of time and/or space have been proposed, becoming important to understand how the variable

order behaviour affects a diffusive system. A variable order time fractional diffusion equation is solved

via a finite difference scheme and its matrix form is presented in detail. A self-written implementation is

validated and then used to study the influence of the variable order time derivative, function of time and

space, in the solution of variable order time fractional diffusion equations.

Keywords: Fractional Calculus, Anomalous Diffusion, Fractional Diffusion Equation, Finite Dif-

ference Methods, Fractional Derivative, Variable-Order.

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Topic Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Finite Difference Solution of Fractional Diffusion Equations 9

2.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Finite difference approximations for fractional derivatives . . . . . . . . . . . . . . . . . . . 11

2.2.1 Approximations for time fractional derivatives . . . . . . . . . . . . . . . . . . . . . 11

2.2.1.1 The Grunwald-Letnikov Approximation . . . . . . . . . . . . . . . . . . . 11

2.2.1.2 L1 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1.3 Third order weighted and shifted Grunwald difference approximation . . . 12

2.2.2 Approximations for space fractional derivatives . . . . . . . . . . . . . . . . . . . . 14

2.2.2.1 The shifted Grunwald approximation . . . . . . . . . . . . . . . . . . . . . 14

2.2.2.2 Second order approximation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2.3 Fourth order compact finite difference approximation . . . . . . . . . . . . 16

2.3 Finite difference approximations of integer order derivatives . . . . . . . . . . . . . . . . . 18

2.4 Time fractional diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.2 First order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.3 Second order implicit finite difference scheme . . . . . . . . . . . . . . . . . . . . . 21

2.4.4 Third order compact finite difference scheme . . . . . . . . . . . . . . . . . . . . . 23

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2.4.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Space fractional diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.2 First order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.3 Second order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.4 Fourth order quasi-compact finite difference scheme . . . . . . . . . . . . . . . . . 31

2.5.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 Time-space fractional diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6.2 First order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6.3 Second order finite difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6.4 Fourth order finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Influence of variable-order operators in the behaviour of sub-diffusive systems 47

3.1 Numerical solution of variable order time fractional diffusion equations . . . . . . . . . . . 47

3.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Influence of variable order differential operators in anomalous diffusion . . . . . . . . . . . 53

3.2.1 Constant fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 Time dependent fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.3 Space dependent fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.4 Solution dependent fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Conclusions 65

References 69

A Schemes for fractional diffusion equations in matrix form A.1

A.1 Time Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1

A.1.1 First Order Weighted Average Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.1

A.1.2 Second Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.1

A.1.3 Third Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . A.2

A.2 Space Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3

A.2.1 First Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . A.3

A.2.2 Second Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.3

A.2.3 Fourth Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.4

A.3 Time-space Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5

A.3.1 First Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . A.5

A.3.2 Second Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.5

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A.3.3 Fourth Order Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . A.6

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List of Tables

2.4.1L∞ error and its order of convergence with decrease of the temporal step size, for the

presented schemes for the time fractional diffusion equation. The results were taken with

h = 1/2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.2Time of computation for each scheme for each of the presented schemes. The results

correspond to a constant space step h = 1/2000. . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.1L∞h,τ errors and their order of convergence with the refinement of the space step for equa-

tion (2.5.19)-(2.5.21) with µ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.2L∞h,τ errors and their order of convergence with the refinement of the space step for equa-

tion (2.5.19)-(2.5.21) with µ = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.3L∞h,τ errors and their order of convergence with the refinement of the space step for equa-

tion (2.5.19)-(2.5.21) with µ = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.4Time of computation in the solution of problem (2.5.19)-(2.5.21) for the three schemes

presented with 1/τ = 20000 and 1/h = 128 for α = 1.5. . . . . . . . . . . . . . . . . . . . . 36

2.6.1L∞h,τ error and the respective order of convergence with space step refinement for the first

order in space scheme with a constant τ = 1/8000 . . . . . . . . . . . . . . . . . . . . . . 43

2.6.2L∞h,τ error and the respective order of convergence with time step refinement for the first

order in space scheme, h ≈ τ (2−γ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.3L∞h,τ error and the respective order of convergence with space step refinement for the

second order in space scheme, h = 1/1000. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6.4L∞h,τ error and the respective order of convergence with time step refinement for the sec-

ond order in space scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6.5L∞h,τ error and the respective order of convergence with space step refinement for the

fourth order in space scheme, for t ∈ [0, 0.1] and τ = 1/100000. . . . . . . . . . . . . . . . 45

2.6.6L∞h,τ error and the respective order of convergence with time step refinement for the fourth

order in space scheme, h = 1/1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6.7Computing times with refinement of the time interval with the first, second and fourth order

in space schemes and h = 1/2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1Error behaviour with decreasing temporal gridsize, h = 1/500. . . . . . . . . . . . . . . . . 52

3.1.2Error behaviour with decreasing temporal gridsize, t = h2. . . . . . . . . . . . . . . . . . . 53

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List of Figures

2.1 Exact solution of problem (2.4.47)-(2.4.49) with γ = 0.5. . . . . . . . . . . . . . . . . . . . 26

2.2 Absolute errors for each numerical scheme for the solution of time fractional diffusion

equations. All the the plots shown correspond to τ = 1/2000, h = 1/512 and γ = 0.5. . . . 27

2.3 Analytical solutions of problem (2.5.19)-(2.5.21) for µ = 2, 3, 4. . . . . . . . . . . . . . . . . 33

2.4 Absolute error in the numerical solution of the problem (2.5.19)-(2.5.21) with µ = 4 for the

three schemes presented. The errors correspond to h = 1/128 and τ = 1/20000. . . . . . 34

2.5 Exact solution of problem (2.6.27)-(2.6.29) with γ = 0.5 . . . . . . . . . . . . . . . . . . . 41

3.1 Exact solution of problem (3.1.20)-(3.1.22). . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Absolute error in the solution of (3.1.20)-(3.1.22), h = 1/250 and τ = 1/500. . . . . . . . . 52

3.3 Solution of the standard diffusion equation. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Solution versus time plot at x = 5 with different constant fractional orders. . . . . . . . . . 56

3.5 Solution at t = 0.2 with different constant fractional orders. . . . . . . . . . . . . . . . . . . 56

3.6 Solution at t = 5 with different constant fractional orders. . . . . . . . . . . . . . . . . . . . 57

3.7 Time evolution in x = 5 with different time dependent fractional orders. . . . . . . . . . . . 58

3.8 The three space dependent fractional orders considered. . . . . . . . . . . . . . . . . . . 59

3.9 Time evolution at x = 5 modelled with different space dependent fractional orders. . . . . 60

3.10 Solution at t = 10 modelled with different space dependent fractional orders. . . . . . . . 61

3.11 Solution dependent fractional order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.12 Time evolution in x = 5 with a solution dependent variable order model. . . . . . . . . . . 63

3.13 Solution t = 10 with a solution dependent variable order model. . . . . . . . . . . . . . . . 64

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Nomenclature

α Space fractional derivative order.

δtun+1i Backward operator for the first order time derivative.

δtun+ 1

2i Central difference operator for the first order time derivative.

δ2xuni Central differences operator for the second order space derivative.

γ Time fractional derivative order.

CaD

αt Left Caputo fractional derivative.

CtD

αb Right Caputo fractional derivative.

GLδγt Grunwald-Letnikov difference operator for the Riemann-Liouville time fractional derivative.

L1δγt un L1 operator for the Caputo time fractional derivative.

L1δγt un L1 operator for the Riemann-Liouville time fractional derivative.

RLaD−αt Left Riemann-Liouville fractional derivative.

RLaD−αt Left Riemann-Liouville fractional integral.

RLtD−αb Right Riemann-Liouville fractional derivative.

RLtD−αb Right Riemann-Liouville fractional integral.

RZDαt Riesz fractional derivative.

WS2δαx,+ Second order weighted and shifted Grunwald difference operator for the left Riemann-Liouville

space derivative.

WS2δαx,− Second order weighted and shifted Grunwald difference operator for the right Riemann-Liouville

space derivative.

WS3δγt u(t) Weighted and shifted difference operator for the Riemann-Liouville time derivative.

WS4δαx,+ Weighted and shifted Grunwald difference operator for the fourth order approximation to the

left Riemann-Liouville space derivative .

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WS4δαx,− Weighted and shifted Grunwald difference operator for the fourth order approximation tos the

right Riemann-Liouville space derivative.

pδαx,+ Shifted Grunwald-Letnikov operator for the left Riemann-Liouville space derivative.

pδαx,− Shifted Grunwald-Letnikov operator for the right Riemann-Liouville space derivative.

τ Time interval size.

0Dα(x,t)t Coimbra variable-order time derivative.

h Space interval size.

K Diffusivity coefficient.

L∞h,τ Maximum error.

M Space mesh size.

N Time mesh size.

Rni Truncation error.

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Glossary

EOC Error order of convergence.

TOC Time of computation.

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Chapter 1

Introduction

1.1 Motivation

Fractional Calculus is a mathematical field dealing with integrals and derivatives of arbitrary order.

Even though the concept dates back to 1695 [1], it was only on the last century that the most impressive

achievements were made. Particularly in the last three decades fractional calculus has found applica-

tions in physics, signal-processing, engineering, bio-science, and finance [2, 3, 4, 5, 6, 7].

The field of Aerospace Engineering has been an early adopter of fractional calculus and one may

find its application regarding viscoelasticity and modelling of unsteady aerodynamic forces in AIAA con-

ferences and papers since the 1980’s, see e.g. [8, 9, 10, 11, 12, 13, 14]. The topic has also captured

both the attention of NASA and ESA in the solution of viscoelastic[15] and astrophysical problems [16].

Aerospace engineering beeing such a multidisciplinary field, can find applications of fractional calculus

in a vast number of areas including acoustics [17], fracture mechanics [18], composite materials [19]

and control theory [20, 21]. Recent developments have been also made in the fields of heat conduction

[22, 23] and flow in porous media [24, 25]. Interest in fractional calculus is currently experiencing an un-

precedented growth and there is no doubt aerospace engineering will benefit from future technologies

potentiated by this mind opening mathematical theory.

Anomalous diffusion has received particular interest in the framework of fractional calculus applica-

tions, see e.g. [3, 26, 27, 28, 29, 30, 31, 32]. So far, constant order fractional differential equations have

been the most used in anomalous diffusion modelling. Even if successful, constant order models have

failed to describe more complex phenomena whose behaviour is dependent of time, space and system

properties. Variable order differential operators, on the other hand allow the order of the derivative to

be a function of time, space or even the function itself, providing the flexibility to solve many of these

phenomena. A significant portion of this research effort is however concentrated in the mathematical

theory or numerical solution of these schemes. On the other hand, since the analytical solutions of frac-

tional differential equations are difficult to obtain, numerical methods for the solution of these equations

become extremely important. Finite difference methods in particular became very popular and a large

number of schemes has been published very recently. Consequently it becomes important to under-

1

stand how they compare in terms of accuracy, stability and computing times. The majority of the works

related to the numeric solution of fractional differential equations provides numerical examples based on

manufactured solutions, through which is not possible to gain the much needed intuitive understanding

of how the order of variable order derivatives affects the system behaviour.

1.2 Topic Overview

In fractional diffusion equations, integer order derivatives are replaced by fractional order counter-

parts, originating what may be considered as three different types of equations: i) time fractional, ii)

space fractional and iii) space-time fractional equations. Enjoying non-local properties, fractional inte-

grals and derivatives may describe more accurately anomalous diffusion processes. For instance it has

been suggested that the probability density function u(x, t) that describes anomalous subdiffusion par-

ticles follows the time fractional subdiffusion equation [3, 33, 34, 35]. Naturally, each type of fractional

diffusion equation has attracted in its own right a considerable number of works regarding its solution.

Loking at time- fractional diffusion equations, implicit shemes are more favorable than their explicit

counterparts, see e.g. [36, 37, 38, 39, 40]. Compact schemes have also attracted many researchers

because of the advantadge of keeping the tridiagonal nature, see e.g. [41]. Gao and Sun [42] applied

the L1 approximation for the time-fractional derivative and developed a compact finite difference scheme

for the fractional sub-diffusion equation. Most of these methods focus on the improvement of the space

order accuracy. It is however remarked that other numerical methods have been sought to solve time

fractional diffusion equations namely finite element [43] and spectral methods [44, 45]. High order in

space and time was sought by Ji and Sun [46] have proposed a high order compact difference scheme

able to solve the time fractional diffusion equation with third order accuracy in time. Most recently, Hu and

Zhang [47] have also proposed a second order implicit finite difference method in time for the fractional

diffusion equation.

Concerning the space fractional diffusion equation, Meerschaert and Tadjeran [48] proposed a shifted

Grunwald formula to approximate the space fractional derivative, overcoming the instability of the stan-

dard formula and applied it in the construction of schemes for the space fractional diffusion equation

[49, 50]. Tadjeran et al. [51] have also presented the Taylor expansion of the error of the shifted Grunwald

formula that enabled the construction of many high order schemes. Tian et al. [52], constructed a class

of second-order finite difference approximations with weighted and shifted Grunwald difference approx-

imations for Riemann-Liouville derivatives. Combining these approximations with a compact technique,

Zhou et al. [53] then suggested a third order scheme. Following this work, Chen et al. [54] further pro-

posed a class of second, third and fourth order difference approximations to solve the space fractional

diffusion equations. Hao et al. [55] have also proposed a fourth-order difference approximation for the

Riemann-Liouville space derivatives combining the weighted average of the shifted Grunwald operators

with a compact technique. Noticing that the matrices for the solution of space fractional diffusion equa-

tions have a structure of Toeplitz type, a fast finite difference solver has been proposed, reducing storage

and computing cost [56].

2

Schemes for time-space fractional diffusion equations have recently received considerable attention.

Liu et al. [57] proposed an implicit finite difference approximation to the time–space fractional diffusion

equation with first order accuracy in time and space. Yang et al. [45] derived a novel numerical method

based on the matrix transfer technique in space and finite difference scheme (or Laplace transform) in

time to deal with the time–space fractional diffusion equations in two dimensions. Chen et al. [58] applied

the L1 approximation to the time fractional derivative and second-order finite difference discretizations

to the space fractional derivative for solving the two-dimensional time–space Caputo-Riesz fractional

diffusion equation with variable coefficients in a finite domain. Ding [59] recently presented a numerical

method for the space–time Caputo-Riesz fractional diffusion equation, discretizing the Riesz derivative

by a fourth- order fractional-compact difference scheme and changing the space–time fractional diffu-

sion equation into a fractional ordinary differential equation system. Sun et al. [60] proposed several

difference schemes for one-dimensional and two-dimensional space and time fractional Bloch-Torrey

equations with second and fourth order in space. Wang et al. [61] proposed an alternating direction im-

plicit scheme with second-order accuracy in both time and space that is also considered the time–space

fractional subdiffusion equation. Pang and Sun have also proposed a fourth order accurate compact

difference scheme in space, using the L1 approximation for the time fractional derivative.

From the brief survey above, there are a multitude of finite difference methods for the approximation of

fractional derivatives and their application on schemes for time, space and space-time fractional diffusion

equations. The non-local properties of fractional derivatives translate into approximations that have

much longer computing times than integer order derivatives. For this reason, the stability criteria of

explicit schemes for fractional diffusion equations may lead to prohibitively high computational costs and

difficulties in the analysis of the orders of accuracy. As such, this work focuses on implicit schemes that

are unconditionally stable. Three different schemes with increasing order of accuracy were selected for

time, space and time-space fractional diffusion equations. The objective is the comparison of these finite

difference schemes in terms of accuracy and computational cost.

For time fractional diffusion equations the compared schemes are: i) the weigted average scheme

developed by Yuste [40],with the classic Grunwald-Letnikov Approximation with first order accuracy in

time, ii) the recent scheme proposed by Hu and Zhang [47] with second order accuracy and iii) the

third-order in time compact finite difference scheme developed by Ji and Sun [46].

The comparison of space fractional diffusion schemes is made with the following works: i) the first

order in space scheme developed by Meerschaert and Tadjeran, using the shifted Grunwald difference

formula for space fractional derivatives ii) the second order in space scheme developed by Tian et

al. [62], that used a weighted combination of the shifted Grunwald difference operator of the previous

scheme and iii) the fourth order in space scheme that uses weighted and shifted Grunwald difference

operators together with a compact technique. These works are representative of the evolution that has

been seen on space fractional diffusion equations since going from the first suggestion of the shifted

Grunwald difference formula to the more recent trend of compact difference schemes for fractional diffu-

sion equations that includes the use of weighted and shifted Grunwald difference operators.

Regarding the schemes for time-space fractional diffusion equations the L1 method stands as the

3

most common approximation for time fractional derivatives while the search continues for high-order

finite difference schemes both in time and space. Serving as a needed common ground, a comparison

is made with schemes that use the L1 approximation for the Caputo time fractional derivative with time

accuracy (2 − γ). In the schemes that were chosen, the discretization of space fractional derivatives is

made with the three approximations that were used in space fractional diffusion equations. The selected

schemes are: i) the first order in space scheme proposed by Li et al. [57] ii)a second order in space

scheme combining the L1 method and the second order approximation for space fractional derivatives

iii) the fourth order in space scheme proposed by Pang et al. [63].

Even if successful, constant order models have failed to describe more complex phenomena whose

behaviour is dependent of time, space and system properties. Variable order models, on the other hand,

have received far less attention than constant fractional order ones. To tackle this problem, several

authors have proposed different definitions of variable order operators [64] and distributed order [65].

Random order fractional differential equations have also been considered by Sun et al. [66, 67, 68]

that concluded that each type of fractional differential operator has distinct advantages and potential

applications for the modelling of diffusion processes. Distributed order models are the best at describing

multi-scale diffusion processes while the variable order models suit the description of diffusion processes

whose diffusion pattern changes with time evolution or space variation. To describe diffusion processes

subjected to an oscillating field or unstable system parameters, the random order model may be more

adequate.

Important works involving variable order calculus have been reviewd by Samko [69] that has also

proposed the concept of variable order differential operator, investigating the properties of variable order

Riemann-Liouville integrals and derivatives [70]. Lorenzo and Hartley [71] studied some mathematical

properties of candidate variable-order operators . Coimbra et al. [64] has investigated the dynamics

and control of nonlinear viscoelasticity oscillator with variable order operators . With a time dependent

variable order operator, Ingman et al. [72, 73] modeled the viscoelastic deformation process. Pedro et

al. [74] modelled the motion of particles suspended in a viscous fluid where the drag force is calculated

recurring to variable order calculus. Kobolev et al. [75] studied the statistical physics of dynamic systems

with variable memory. Chechkin et al. [76] studied the evolution of a composite system consisting of two

separate regions with the time-fractional diffusion equation with a space variable fractional order time

derivative.

This work focuses on variable order fractional diffusion equations that present an important tool in

the study of complex anomalous diffusion phenomena. Going beyond the constant fractional order

exponent allows modelling of situations where the diffusive behaviour may change with time evolution,

space location, the concentration of the diffusing species as well as system parameters. Very often

these issues are mainly addressed in the mathematical framework without discussion of effect of the

variable order on the solution. Attending to the vast number of possibilities, attention on this work will

be focused for simplicity on the variable order time fractional diffusion equation, capable of depicting

sub-diffusive processes in which temporal fractional derivatives is solution of i)time, ii) space and iii)

the dependent solution itself. The results of the different variable order functions are compared and the

4

results discussed. The diffusion equation is considered on the form

0Dγ(x,t)t u(x, t) = K

∂2u(x, t)

∂x2+ f(x, t), x ∈ [0, L]t ∈ [0, T ] (1.2.1)

where K > 0 is a generalized diffusion coefficient, the variable u(x, t) is a physical quantity of interest

such as temperature, concentration or survival propability of a particle and 0Dγ(x,t)t denotes the Coimbra

variable order derivative [64].

Variable time dependent fractional order diffusion equations are useful in the modelling of processes

for which the diffusive behaviour changes with time. There are situations for instance, where processes

tend to Fickian diffusion with the evolution of time [77, 78, 79]. This behaviour can found in biology,

plasma physics and economy. The opposite behaviour can also be observed, with diffusion rates de-

creasing with time evolution. Conventional solutions to these problems are often found through inte-

ger order differential equations using time dependent diffusion exponents [80, 81]. Good data fitting is

sometimes provided by such methods, but they cannot achieve a general formulation for time dependent

diffusion processes because they do not capture the origin of these problems [68].

The space dependent variable order time derivative may be thought of as the memory rate depending

on the space location in the diffusive system. While the constant order diffusion models seem to be ade-

quate to model homogeneous media, that is not the case for inhomogeneous and isotropic situations. In

this case, the diffusive behaviour changes with spacial location making the case for a space dependent

variable order. In recent years, anomalous diffusion in complex media has captured the attention of

many scholars in fields such as geophysics, environmental science, hydrology and biology [82]. Frac-

tional diffusion equations have here helped to model heat conduction and fluid flow in porous media

seismic waves and protein dynamics. Modelling of these problems is often made through nonlinear dy-

namics, statistical mechanics and memory formalisms [83]. With the significance of these problems, the

necessity for the investigation of diffusive behaviour in porous systems becomes apparent.

Solution dependent variable order diffusion equation allows the memory rate to vary along with the

solution the diffusive system, capturing information that would otherwise be coded in a complex expres-

sion of the diffusion coefficient. There are situations in physics, chemistry and biology where concentra-

tion plays the key role in diffusive behaviour [84]. Examples of this behaviour are the diffusive transport of

macromolecules in biological tissue and diffusion processes associated with chemical reactions where

the concentration of reactant will determine the characteristics of the chemical diffusion process. The

most common approaches to deal with these situations involve nonlinear or variable coefficient partial

differential equations [85, 86]. In these cases the expression adopted for the diffusion coefficient often

presents parameters that are difficult to physically analyse or obtain experimentally [87].

Finite difference methods stand as the most popular solution methods for fractional calculus. Nonethe-

less other methods have also been sought for the solution of variable order fractional differential equa-

tions, namely spectral methods [88]. Coimbra et al. [64] proposed a first order accurate approximation

for variable order differential equations. Soon et al. [89] have employed a second-order Runge-Kutta

method consisting of an explicit Euler predictor step followed by an implicit Euler corrector step to nu-

5

merically integrate the variable order differential equation. Zhuang et al. [90] constructed explicit and

implicit Euler approximations for the variable-order fractional advection-diffusion equation with a nonlin-

ear source term and several other approximations have been proposed.

Lin et al. [91] constructed an explicit finite difference scheme for spatial VO fractional differential

equation with a generalized Riesz fractional derivative of variable order α(x, t), (1 < α(x, t) ≤ 2) with

linear convergence on both time and space. Chen et al. [92] proposed a numerical scheme with first

order temporal accuracy and fourth order spatial accuracy for the fractional sub-diffusion equation. In

[93], the variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid with

a fourth order accurate numerical scheme is studied. The variable-order nonlinear reaction-subdiffusion

equation was considered by [94]. In [95] an implicit scheme for the variable order space fractional

diffusion equation in two dimensions was proposed and in [96] an alternating direction implicit method

for new two-dimensional variable-order fractional percolation equation with variable coefficients. Sun et

al. [97] developed three finite difference schemes for the variable-order fractional sub-diffusion equation

and suggested [98] a finite difference scheme with first order accuracy in time and second order in

space for the fractional subdiffusion equation. Shen et al. [99] developed a numerical scheme for the

variable order advection-diffusion equation with a nonlinear source term with first order accuracy. Zhang

et al. [100] proposed an implicit difference method, first order accurate in time and space, for the time

fractional variable order mobile-immobile-advection-dispersion equation.

The finite difference schemes that have been mentioned for variable order fractional diffusion equa-

tions exhibit first order convergence in time. Zhao et al. [101] derives two second- order approximation

formulas for the variable-order fractional time derivatives involved in anomalous diffusion and wave prop-

agation. It should be noted that not all the schemes here mentioned adopt the same definition of variable

order fractional derivative. The Coimbra [102] definition of fractional derivative is adopted in this work.

Several other authors adopt this definition, see e.g. [98, 101, 103].

1.3 Objectives

The scientific community’s interest in fractional calculus is undergoing exponential growth, many

applications have been found so far but the majority is yet to be unravelled. For fractional calculus to

succeed in engineering applications, a proper understanding of the underlying mathematical theory and

the tools to deal with it have to be acquired. As such this work represents and effort in the building of a

bridge between the mathematical and engineering standpoints, laying the ground for future applications.

This purpose is fulfilled with two objectives. Firstly, through the implementation and comparison of

several finite difference schemes for fractional diffusion equations, a good grasp on particularities of this

numerical solution technique, applied to fractional calculus, is intended. Secondly, an intuitive insight

on how constant and variable order fractional differential operators affect the solution of system is to be

gained that will certainly prove valuable in the development of an engineering application.

6

1.4 Thesis Outline

This document is organized as follows. Chapter 2 will concern the comparison of finite difference

schemes for fractional diffusion equations. Section 2.1 begins with a brief overview of the most important

mathematical definitions of fractional integrals and derivatives.

In section 2.2 the finite difference approximations that will be used to in the construction of schemes

for fractional diffusion equations will be provided. For time fractional derivatives the Grunwald-Letnikov,

L1 and a third order weighted and shifted Grunwald approximation aproximation will be presented. For

space fractional derivatives the first order shifted Grunwald difference approximation, the second order

weighted and shifted Grunwald difference approximation and a fourth order compace difference approx-

imation are presented. In section 2.3 finite difference approximations for the first order time derivative

and second order space derivative are given.

Section 2.4 studies the behaviour of three schemes with increasing order of accuracy for the time

fractional diffusion equation. The initial-boundary value problem is stated in section 2.4.1. In section

2.4.2 a first order weighted average finite difference method is given, section 2.4.3 presents a second

order scheme and section 2.4.4 will deal with a third order in time compact difference scheme. In

section 2.4.5 a numerical example is solved with the three schemes, with different time fractional orders

and time interval refinement, serving both for code validation and to compare the schemes in terms of

convergence order and computing time.

With a similar structure to section 2.4, section 2.5 studies the behaviour of three schemes with in-

creasing order of accuracy for the space fractional diffusion equation. The initial-boundary value problem

is stated in section 2.5.1. In section 2.5.2 a first order in space finite difference method is given, section

2.5.3 refers to a second order scheme and section 2.5.4 will deal with a fourth order in space compact

difference scheme. In section 2.5.5 a numerical example is solved with the three schemes, with different

space fractional orders and space interval refinement, serving both for code validation and to compare

the schemes in terms of convergence order and computing time.

Section 2.6 will deal with schemes for the time-space fractional diffusion equation under a common

fractional time derivative approximation. As in the two previous sections, the initial-boundary value

problem is stated in section 2.6.1. In section 2.6.2 a first order in space finite difference method is given,

section 2.6.3 refers to a second order scheme and section 2.6.4 will deal with a fourth order in space

compact difference scheme. In section 2.6.5 a numerical example is solved with the three schemes

for time-space, with different space and time fractional orders and refinement of the space and time

intervals, serving both for code validation and to compare the schemes in terms of convergence order

and computing time.

In chapter 3 an investigation is made into the effects of variable order differentiation in the behaviour

of sub-diffusive systems. Section 3.1 introduces a scheme able to solve variable order time-fractional

sub-diffusion equations. The initial-boundary value problem is stated in section 3.1.1. In section 3.1.2 a

difference scheme able to solve time fractional diffusion equations with variable coefficients dependent

on time and space [98] is implemented and provided to the reader in matrix form. The convergence

7

of the scheme is then validated with a numerical example against the analytic solution in section 3.1.3.

In section 3.2 the effects of order dependence on time, space and the solution itself will be analysed

through a numerical example. Departure is made in section 3.2.1 from the comparison of the standard

diffusion equation with constant order fractional diffusion which is then taken as reference for the analysis

of the behaviour of anomalously diffusive systems with variable order. Sections 3.2.2, 3.2.3 and 3.2.4,

study of the behaviour of a sub-diffusive with variable orders dependent of time, space and the system

solution, respectively.

In chapter 4 important conclusions are made regarding both the schemes for the different types

of fractional diffusion equations and the behaviour of the solution in the case of a variable order time

fractional diffusion equations. The main achievements are stated and suggestions for future works are

made.

8

Chapter 2

Finite Difference Solution of

Fractional Diffusion Equations

This purpose of this chapter is to compare finite difference schemes for fractional diffusion equations.

Section 2.1 begins with a brief overview of the most important mathematical definitions of fractional

integrals and derivatives.

In section 2.2 the finite difference approximations that will be used to in the construction of schemes

for fractional diffusion equations will be provided. For time fractional derivatives the Grunwald-Letnikov,

L1 and a third order weighted and shifted Grunwald approximation aproximation will be presented. For

space fractional derivatives the first order shifted Grunwald difference approximation, the second order

weighted and shifted Grunwald difference approximation and a fourth order compace difference approx-

imation are presented. In section 2.3 finite difference approximations for the first order time derivative

and second order space derivative are given.

Section 2.4 studies the behaviour of three schemes with increasing order of accuracy for the time

fractional diffusion equation. The initial-boundary value problem is stated in section 2.4.1. In section

2.4.2 a first order weighted average finite difference method is given, section 2.4.3 presents a second

order scheme and section 2.4.4 will deal with a third order in time compact difference scheme. In

section 2.4.5 a numerical example is solved with the three schemes, with different time fractional orders

and time interval refinement, serving both for code validation and to compare the schemes in terms of

convergence order and computing time.

With a similar structure to section 2.4, section 2.5 studies the behaviour of three schemes with in-

creasing order of accuracy for the space fractional diffusion equation. The initial-boundary value problem

is stated in section 2.5.1. In section 2.5.2 a first order in space finite difference method is given, section

2.5.3 refers to a second order scheme and section 2.5.4 will deal with a fourth order in space compact

difference scheme. In section 2.5.5 a numerical example is solved with the three schemes, with different

space fractional orders and space interval refinement, serving both for code validation and to compare

the schemes in terms of convergence order and computing time.

Section 2.6 will deal with schemes for the time-space fractional diffusion equation under a common

9

fractional time derivative approximation. As in the two previous sections, the initial-boundary value

problem is stated in section 2.6.1. In section 2.6.2 a first order in space finite difference method is given,

section 2.6.3 refers to a second order scheme and section 2.6.4 will deal with a fourth order in space

compact difference scheme. In section 2.6.5 a numerical example is solved with the three schemes

for time-space, with different space and time fractional orders and refinement of the space and time

intervals, serving both for code validation and to compare the schemes in terms of convergence order

and computing time.

2.1 Mathematical Preliminaries

In this chapter, the mathematical definitions of fractional integrals and derivatives used throughout

this paper are introduced [32].

Definition 2.1. The left and right fractional Riemann-Liouville integrals of order α > 0 of a given function

f(t), t ∈ (a, b) are defined as

RLaD−αt f(t) =

1

Γ(α)

∫ t

a

(t− s)α−1f(s)ds (2.1.1)

RLtD−αb f(t) =

1

Γ(α)

∫ b

t

(s− t)α−1f(s)ds (2.1.2)

respectively, where Γ(·) denotes Euler’s gamma function.

Definition 2.2. The left and right Riemann-Liouville derivatives with order α > 0 of the function f(t),

t ∈ (a, b) are defined as

RLaD

αt f(t) =

dm

dtm[D−(m−α)a,t f(t)] =

1

Γ(m− α)

dm

dtm

∫ t

a

(t− s)m−α−1f(s)ds (2.1.3)

RLtD

αb f(t) = (−1)m

dm

dtm[D−(m−α)t,b f(t)] =

(−1)m

Γ(m− α)

dm

dtm

∫ b

t

(s− t)m−α−1f(s)ds (2.1.4)

respectively, where m is a positive integer satisfying m− 1 ≤ α < m and Γ(·) is Euler’s gamma function.

Definition 2.3. The left and right Caputo derivatives with order α > 0 of the function f(t), t ∈ (a, b) are

defined as

CaD

αt f(t) = RL

aD−(m−α)t [f (m)(t)] =

1

Γ(m− α)

∫ t

a

(t− s)m−α−1f (m)(s)ds (2.1.5)

CtD

αb f(t) =

(−1)m

Γ(m− α)

∫ b

t

(s− t)m−α−1f (m)(s)ds (2.1.6)

respectively, where m is a positive integer satisfying m− 1 ≤ α < m and Γ(·) is Euler’s gamma function.

Although the definitions of the Riemann-Liouville and of the Caputo derivatives cannot be assumed

equal, they do have the following relationship

10

RLaD

αt f(t) = C

aDα

t f(t) +

m−1∑k=0

f (k)(a)(t− a)k−α

Γ(k + 1− α)(2.1.7)

where m − 1 < α < m is a positive integer and fm is integrable on [a, t]. On the special case where

fk(0) = 0 with k = 0, 1, 2, · · · ,m− 1,m− 1 < α < m the Riemann-Liouville and Caputo derivatives are

equivalent.

Furthermore, if the definitions of fractional integral and derivative are compared, it can be seen that

the Caputo derivative of order α is equivalent to the fractional integral of order (m − α) of f (m)(t), with

m− 1 < α < m [104].

Definition 2.4. The Riesz derivative of order α > 0 for a given function f(t), t ∈ (a, b) is defined as

RZDαt f(t) = cα(RLaD

αt f(t) + RL

tDαb f(t)) (2.1.8)

where cα = − 12 cos(απ/2) and α 6= 2k + 1, k = 0, 1, · · ·.

Definition 2.5. The Coimbra variable order time derivative of order α(x, t) ∈ [0, 1] for a given function

f(x, t), t ∈ (a, b) is defined as

0Dα(x,t)t f(x, t) =

1

Γ(1− α(x, t))

∫ t

0

(t− σ)−α(x,t)∂f(x, σ)

∂σdσ +

(f(x, 0+)− f(x, 0−))t−α(x,t)

Γ(1− α(x, t)(2.1.9)

2.2 Finite difference approximations for fractional derivatives

2.2.1 Approximations for time fractional derivatives

2.2.1.1 The Grunwald-Letnikov Approximation

The Grunwald-Letnikov approximation [40] is one of the most used for time fractional derivatives. Let

t ∈ [0, T ], τ = T/N so that tn = nτ , Ωτ = tn|0 ≤ n ≤ N and u(tn) = un. If u(t) is suitably smooth, the

left Riemann-Liouville derivative can be approximated with first order accuracy by

[RL0 Dγ

t u(t)]t=tn

= GLδγt un +O(τ) (2.2.1)

where the left Grunwald-Letnikov difference operator is given by

GLδγt un =1

τγ

n∑k=0

ω(γ)j un−k (2.2.2)

The Grunwald-Letnikov wheights ω(γ)k = (−1)k

(γk

), with k ≥ 0, are the coefficients of the power

series of the generating function (1− z)γ =∑∞k=0 ω

(γ)k zk. These weights satisfy the recursive formula

ω(γ)k =

(1− γ + 1

k

)ω(γ)k−1, w

(γ)0 = 1. (2.2.3)

11

However, other formulas for the calculation of these weights exit, leading to higher order approximations

[105, 32].

2.2.1.2 L1 Approximation

The L1 method [36] is another popular choice for the approximation of time fractional derivatives.

This approximation is found in many unconditionally stable schemes and is suitable for (0 < γ < 1).

Nevertheless, similar methods exist for 1 < γ < 2.

Let t ∈ [0, T ], τ = T/N so that tn = nτ and Ωτ = tn|0 ≤ n ≤ N. Additionally, let u(tn) = un. The

left Riemann-Liouville derivative can be discretized with (2− γ)th order accuracy by

[RL

0Dγ

t u(t)]t=tn

= L1δγt un +O(τ2−γ) (2.2.4)

where the L1δγt un operator is defined as

L1δγt un =τ−γ

Γ(2− γ)

n−1∑k=0

bn−k−1 [uk+1 − uk] +u0t−γn

Γ(1− γ)(2.2.5)

with

bk =[(k + 1)1−γ − k1−γ

](2.2.6)

If, on the other hand the Caputo definition of time fractional derivative is considered then the scheme

can be approximated with (2− γ)th order accuracy by

[C0D

γ

t u(t)]t=tn

= L1Cδ

γt un +O(τ2−γ) (2.2.7)

where the L1Cδ

γt un operator is defined as

L1Cδ

γt un =

τ−γ

Γ(2− γ)

n−1∑k=0

bn−k−1 [uk+1 − uk] (2.2.8)

2.2.1.3 Third order weighted and shifted Grunwald difference approximation

In [46] Ji and Sun developed a third order accurate weighted and shifted Grunwald difference operator

for the Riemann-Liouville derivative, that they used to construct in a compact difference scheme for the

time fractional diffusion equation. The construction of this approximation is now summarized.

Let t ∈ [0, T ], τ = T/N so that tn = nτ and Ωτ = tn|0 ≤ n ≤ N. Additionally, let u(tn) = un.

Supposing that u ∈ L1(R) ∩ Cγ+1(R), the Riemann-Liouville derivative (2.1.3) evaluated from negative

infinity (a = −∞) can be approximated with first order accuracy by

[RL−∞D

γt u(t)

]t=tn

= pδ(γ)t un +O(τ) (2.2.9)

12

where pδ(γ)t is the shifted Grunwald difference operator [48], defined as

pδ(γ)t un =

1

τγ

∞∑k=0

ω(γ)k un−(k−p) (2.2.10)

uniformly for t ∈ R as τ → 0. The integer p corresponds to the number of shifts and as in the Grunwald-

Letnikov approximation, the weights ω(γ)k are the coefficients of the power series of the generating func-

tion (1− z)γ , given in equation (2.2.3).

Moreover, if u(t) ∈ L1(R), −∞Dγ+3t u(t) and its Fourier transform belong to L1(R), then the operator

in (2.2.10) can be used to construct a third order approximation for RL−∞D

γt u(t)

[RL−∞D

γ

tu(t)

]t=tn

= p,q,rδ(γ)t un +O(τ3) (2.2.11)

where p,q,rδ(γ)t is a weighted and shifted Grunwald difference operator defined by

p,q,rδ(γ)t un = ρ1 pδ

(γ)t un + ρ2 qδ

(γ)t un + ρ3 rδ

(γ)t un (2.2.12)

with shifts p,q and r are defined in [46] as (p, q, r) = (0,−1,−2) and

ρ1 =12qr − (6q + 6r + 1)γ + 3γ2

12(qr − pq − pr + p2), ρ2 =

12pr − (6p+ 6r + 1)γ + 3γ2

12(pr − pq − qr + q2),

ρ3 =12pq − (6p+ 6q + 1)γ + 3γ2

12(pq − pr − qr + r2)

Introducing now,

u(t) =

u(t), t ∈ [0, T ]

0, t ∈ [−∞, 0]

(2.2.13)

it naturally occurs that RL0Dγ

t u(t) = RL−∞D

γ

tu(t) and therefore RL

0Dγ

t u(t) can be approximated with third

order accuracy by [RL

0Dγ

t u(t)]t=tn

= WS3δγt un + +O(τ3) (2.2.14)

where the weighted and shifted difference operator WS3δγt u(t) is defined as

WS3δγt un =1

τγ

[ρ1

n∑k=0

ω(γ)k un−k + ρ2

n−1∑k=0

ω(γ)k un−(k+1) + ρ3

n−2∑k=0

ω(γ)k un−(k+2)

](2.2.15)

For simplicity, the WD3δγt operator can be written as

WD3δγt un =1

τγ

n∑k=0

g(γ)k un−k, n = 2, 3, ..., N (2.2.16)

13

where g(γ)0 = ρ1ω

(γ)0

g(γ)1 = ρ1ω

(γ)1 + ρ2ω

(γ)0

g(γ)k = ρ1ω

(γ)k + ρ2ω

(γ)k−1 + ρ3ω

(γ)k−2, k ≥ 2

(2.2.17)

2.2.2 Approximations for space fractional derivatives

2.2.2.1 The shifted Grunwald approximation

The first approximation considered for space fractional derivatives will be the shifted Grunwald ap-

proximation, used by Meerschaert and Tadjeran [49] to construct a first order in space scheme for the

solution of space fractional diffusion equations. For (1 < α < 2), the standard approximation leads to

unstable numerical schemes, this problem is solved if the shifted approximation is chosen. Moreover,

the shifted approximation can be employed in the construction of higher order approximations through

the weighted combination of different shifts, as will later be demonstrated.

Let x ∈ [a, b], h = (b − a)/M so that xi = ih and Ωh = xi|0 ≤ i ≤ M. Additionally, let u(xi) = ui.

Similarly to the shifted Grunwald approximation introduced in section 2.2.1.3, let u(x) ∈ L1(R)∩Cγ+1(R).

The left and right Riemann-Liouville derivatives (2.1.3) evaluated with (a = −∞) and (b = +∞) can be

approximated with first order accuracy by

[RL−∞D

α

xu(x)

]x=xi

= Gpδαx,+ui +O(h) (2.2.18)[

RLxD

α

∞u(x)]x=xi

= Gpδαx,−ui = +O(h) (2.2.19)

where the shifted left and right Grunwald operators are defined as

Gpδαx,+ui =

1

hα

∞∑k=0

ω(α)k ui−k+p (2.2.20)

Gpδαx,−ui =

1

hα

∞∑k=0

ω(α)k ui+k−p (2.2.21)

respectively, as h→ 0 and where p ∈ Z is the number of shifts. It was found that optimum performance

comes from p = 1 when 1 < γ ≤ 2 [49].

If a zero extension of the function is made,

u(x) =

u(x), t ∈ [a, b]

0, x ∈ [−∞, a] ∪ [b,+∞]

(2.2.22)

it occurs that on the interval x ∈ [a, b], RLaDαxu(x) = RL

−∞Dγ

xu(x) and RL

xDαb u(x) = RL

xDγ

∞u(x). Hence,

the use of (2.2.30) and (2.2.31) to approximate RLaD

αxu(x) and RL

xDαb u(x) results in

[RLaD

α

xu(x)]x=xi

= pδαx,+ui +O(h) (2.2.23)

14

[RLxD

α

b u(x)]x=xi

= pδαx,−ui = +O(h) (2.2.24)

where the shifted left and right Grunwald-Letnikov operators are defined as

pδαx,+ui =

1

hα

i+p∑k=0

ω(α)k ui−k+p (2.2.25)

pδαx,−ui =

1

hα

M−i+p∑k=0

ω(α)k ui+k−p (2.2.26)

respectively. These approximations are also referred to as the Grunwald-Letnikov approximations. One

drawback of these is that they lack first order accuracy when the values at the boundaries are not zero

[106]. For instance, considering the left sided derivative, if u(a) 6= 0, first order accuracy can be achieved

with

[RLaD

α

xu(x)]x=xi

=[RLaD

α

x [u(x)− u(a)]]x=xi

+u(a)x−αiΓ(1− α)

=1

hα

i+p∑k=0

ω(α)k (ui−k+p − u(a)) +

u(a)x−αiΓ(1− α)

+O(h)

(2.2.27)

2.2.2.2 Second order approximation

Zhou, Tian and Deng [62] develop second order approximations for left and right Riemann-Liouville

derivatives and use them on the same paper for the construction of schemes for the solution of the

space fractional diffusion equation. These approximations are made with weighted and shifted Grunwald

difference operators, inspired in the shifted Grunwald difference operator introduced in the previous

section. A summary of the construction of such operators will be made, departing from the shifted

operators introduced in the previous section.

As before, let x ∈ [a, b], h = (b − a)/M so that xi = ih and Ωh = xi|0 ≤ i ≤ M. Additionally, let

u(xi) = ui. Assuming that u ∈ L1(R), RL−∞Dα+2

xu and its Fourier transform belong to L1(R) the weighted

and shifted Grunwald difference operators can be defined as

Gp,qδ

α

x,+ui =

α− 2q

2(p− q)Gpδα

x,+ui +

2p− α2(p− q)

Gqδα

x,+ui (2.2.28)

Gp,qδ

α

x,−ui =α− 2q

2(p− q)Gpδα

x,−ui +2p− α

2(p− q)Gqδα

x,−ui (2.2.29)

allowing the left and right Riemann-Liouville derivatives (2.1.3) and (2.1.4) to be evaluated for (a = −∞)

and (b = +∞) with second order accuracy

[RL−∞D

α

xu(x)

]x=xi

= Gp,qδ

α

x,+ui +O(h2) (2.2.30)

[RLxD

α

∞u(x)]x=xi

= Gp,qδ

α

x,−ui +O(h2) (2.2.31)

15

uniformly for x ∈ R, where p and q are integers with p 6= q.

Considering that the function u is well defined in the interval [a, b], if u(a) = 0 = u(b) = 0, a zero

extension of u(x) can be made for (x < a ∪ x > b). In this manner the Riemann-Liouville derivatives of

order α of u(x) can be approximated by (2.2.28) and (2.2.29) with second order accuracy, resulting in

[RLaD

α

xu(x)]x=xi

=µ1

hα

[ x−ah ]+p∑k=0

ω(α)k ui−(k−p) +

µ2

hα

[ x−ah ]+q∑k=0

ω(α)k ui−(k−q) +O(h2)

=µ1

hα pδαx,+ui +

µ2

hα qδαx,+ui +O(h2)

(2.2.32)

[RLxD

α

b u(x)]x=xi

=µ1

hα

[ b−xh ]+p∑k=0

ω(α)k ui+(k−p) +

µ2

hα

[ b−xh ]+q∑k=0

ω(α)k ui+(k−q) +O(h2)

=µ1

hα pδαx,−ui +

µ2

hα qδαx,−ui + o(h2)

(2.2.33)

where µ1 = α−2q2(p−q) , µ2 = 2p−α

2(p−q) and pδαx,+ and pδ

αx,− are defined in equations (2.2.25) and (2.2.26).

The choice of p and q in (2.2.32) and (2.2.33) must satisfy |p| ≤ 1 and |q| ≤ 1, ensuring that

the nodes at which the values of u needed in (2.2.32) and (2.2.33) are within the bounded interval,

when employing the finite difference method with weighted and shifted Grunwald difference formulas

for numerically solving non-periodic fractional differential equations on bounded intervals. Otherwise,

an alternative discretization method is necessary when x is near a boundary. Having the authors of

the approximation already concluded that (p, q) = (0,−1) is unstable for time dependent problems,

only (p, q) = (1, 0) and (p, q) = (1,−1) remain for for the construction of the quasi-compact difference

approximations. Further conclusions on these coefficients will be provided upon the derivation of the

difference scheme for space fractional diffusion equations employing the weighted and shifted Grunwald

operators in (2.2.34) and (2.2.35). Equations (2.2.32) and (2.2.33) can be simplified to yield

[RLaD

α

xu(x)]x=xi

=1

hα

i+1∑k=0

g(α)k ui−k+1 +O(h2) = WS2δαx,+ui +O(h2) (2.2.34)

[RLxD

α

b u(x)]x=xi

=1

hα

N−i+1∑k=0

g(α)k ui+k−1 +O(h2) = WS2δαx,−ui +O(h2) (2.2.35)

where(p, q) = (1, 0), g

(α)0 =

α

2ω(α)0 , g

(α)k =

α

2ω(α)k +

2− α2

ω(α)k−1, k ≥ 1

(p, q) = (1,−1), g(α)0 =

2 + α

4ω(α)0 , g

(α)1 =

2 + α

4ω(α)1 , g

(α)k =

2 + α

4ω(α)k +

2− α4

ω(α)k−2, k ≥ 2

(2.2.36)

2.2.2.3 Fourth order compact finite difference approximation

Recently, Hao, Sun and Cao [55] developed a fourth-order approximation for Riemann-Liouville frac-

tional derivatives. Yet again, this approximation departs from the use of a weighted average of shifted

16

Grunwald operators with different shifts combining the compact technique. The main idea is to vanish

the low order leading terms in asymptotic expansions for the truncation errors by means of a weighted

average.

Tadjerran et al. [107] provided the Taylor expansion for the error in the shifted Grunwald finite differ-

ence formula, fundamental to many high order schemes. If 1 < α < 2 and u ∈ Cn+3(R) such that all

derivatives of u up to order n+ 3 belong to L1(R), it can be obtained for any integer r ≥ 0 that

Gpδα

x,+u(x) = RL

−∞Dα

xu(x) +

n−1∑l=1

cα,rlRL−∞D

α+l

xu(x)hl +O(hn) (2.2.37)

uniformly for x ∈ R, where Gpδα

x,+was defined in (2.2.25) and cα,rl are the coefficients of the power series

expansion of function Wr(z) = ( 1−e−zz )αerz. The condition that u ∈ Cn+3(R) can however be weakened

to u ∈ Cn+α(R) .

As usual, let x ∈ [a, b], h = (b − a)/M so that xi = ih and Ωh = xi|0 ≤ i ≤ M. Additionally, let

u(xi) = ui, u ∈ L1(R) and f ∈ C4+α(R). For current case, the weighted and shifted Grunwald difference

operators for 1 < α ≤ 2,are defined by

WSG4δα

x,+ui = λ1G1 δ

α

x,+ui + λ0G0 δ

α

x,+ui + λ−1G−1δ

α

x,+ui, (2.2.38)

WSG4δα

x,−ui = λ1G1δα

x,−ui + λ0G0δα

x,−ui + λ−1G−1δ

α

x,−ui (2.2.39)

respectively, where the shifted Grunwald difference operators for Riemann-Liouville fractional derivatives

are given by (2.2.20) and (2.2.21) and

λ1 =α2 + 3α+ 2

12, λ0 =

4− α2

6, λ−1 =

α2 − 3α+ 2

12(2.2.40)

In [55], it was showed that the operators in (2.2.38) and (2.2.39) have second order accuracy for

approximating Riemann-Liouville fractional derivatives. Considering the second order central difference

operator in (2.3.3), the following difference operator is defined

Aαui = (1 + cαh2δ2x)ui with cα =

−α2 + α+ 4

24, (2.2.41)

Applying Aα to RL−∞D

α

xu(x) and RL

x Dα

+∞u(x) Hao et al. reach fourth-order approximations , this

operator will naturally have to be applied to the remaining of th equation when deriving the scheme.

Letting u(x) ∈ L1(R) and u(x) ∈ C4+α(R), one obtains

Aα(RL−∞Dα

xu(x)) = δαx,+u(x) +O(h4) (2.2.42)

Aα(RLx Dα

+∞u(x)) = δαx,−u(x) +O(h4) (2.2.43)

Combining (2.2.38) and (2.2.39) with (2.2.20) and (2.2.21), the weighted and shifted difference oper-

ators can be written in an abbreviated form

17

WSG4δα

x,+ui =1

hα

+∞∑k=0

g(α)k ui−(k−1) (2.2.44)

WSG4δα

x,−ui =1

hα

+∞∑k=0

g(α)k ui+(k−1) (2.2.45)

where g(α)0 = λ1ω

(α)0

g(α)1 = λ1ω

(α)1 + λ0ω

(α)0

g(α)k = λ1ω

(α)k + λ0ω

(α)k−1 + λ−1ω

(α)k−2, k ≥ 2

(2.2.46)

and the weights ω(α)k are given by equation (2.2.3).

If u(x) ∈ C[a, b] with u(a) = u(b) = 0, a zero extension of u can be made. Supposing u(x) ∈ C4+α(R),

equations (2.2.42) and (2.2.43) lead to

Aα(RLaDα

xui) =1

hα

i∑k=0

g(α)k ui−(k−1) +O(h4) = WS4δ

α

x,+ui +O(h4) (2.2.47)

Aα(RLxDα

b ui) =1

hα

M−i∑k=0

g(α)k ui+(k−1) +O(h4) = WS4δ

α

x,−ui +O(h4) (2.2.48)

2.3 Finite difference approximations of integer order derivatives

In addition to fractional derivatives, integer order derivative will also need to be approximated through-

out the coming sections. First order time derivatives will appear in space fractional diffusion equations

and are approximated either by central or backward difference operators. On the other hand, second

order difference operators are used to approximate second order derivatives in time fractional diffusion

equations and in the construction of high-order schemes. Hence, this section introduces the operators

used to approximate integer order derivatives, common to several of the coming schemes.

Take two positive integers M, N and let h = (b − a)/M and τ = T/N . Define xi = ih(0 ≤ i ≤ M),

tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and Ωτ = tn|0 ≤ n ≤ N. The computational domain

[a, b]× [0, T ] is then covered by Ωτh = Ωh × Ωτ . Moreover, let uni = u(xi, tn).

At time level n+ 1/2 it holds that

∂u

∂t

∣∣∣∣xi,tn+1

2

= δtun+ 1

2i +O(τ2), where δtu

n+ 12

i =un+1i − uni

τ(2.3.1)

which holds a similar result, apart from the truncation error, to the approximation of the first time deriva-

tive at time t = (n+ 1)τ with backward differences

∂u

∂t

∣∣∣∣xi,tn+1

= δtun+1i +O(τ), where δtu

n+1i =

un+1i − uni

τ(2.3.2)

18

The second order space derivative can be approximated at x = ih with

∂2u

∂x2

∣∣∣∣xi,tn

= δ2xuni +O(h2), where δ2xu

ni =

uni−1 − 2uni + uni+1

h2(2.3.3)

2.4 Time fractional diffusion equations

2.4.1 Problem statement

Attention will now be devoted to the development of schemes for time fractional diffusion equations.

To this purpose, the two most common forms of these equations are presented. Throughout this section

take two positive integers M, N and let h = (b − a)/M and τ = T/N . Define xi = ih(0 ≤ i ≤ M),

tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and Ωτ = tn|0 ≤ n ≤ N. The computational domain

[a, b]× [0, T ] is then covered by Ωτh = Ωh × Ωτ . Moreover, let uni = u(xi, tn).

Equations (2.4.1)-(2.4.3) give the first form of time fractional diffusion equations, where the time

fractional derivative is of the Riemann-Liouville type. This form of the time fractional diffusion equation

is used in the schemes within sections 2.4.2 and 2.4.3.

∂u

∂t= RL

0D1−γt

[Kγ

∂2u

∂x2

]+ f(x, t), x ∈ [a, b], t ∈ [0, T ] (2.4.1)

u(0, t) = φ(t), u(L, t) = Φ(t), t ∈ [0, T ] (2.4.2)

u(x, 0) = 0, x ∈ [a, b] (2.4.3)

where Kγ is the diffusion coefficient and RL0D

1−γt is the Riemann-Liouville derivative of order (1 − γ) of

function u as defined in section 2.1.

Alternatively, the Caputo fractional derivative can be used, in which case the initial-boundary value

problem of the form (2.4.4)-(2.4.6). Time fractional derivatives of the Caputo type will be used in the

scheme of section 2.4.4.

C0D

γt u(x, t) = Kγ

∂2u(x, t)

∂x2+ F (x, t), x ∈ [0, L], t ∈ [0, T ] (2.4.4)

u(0, t) = φ(t), u(L, t) = Φ(t), t ∈ [0, T ] (2.4.5)

u(x, 0) = 0, x ∈ [a, b] (2.4.6)

once again, Kγ is the diffusion coefficient and C0 D

γt is the Caputo derivative with order γ of the function

as defined in section 2.1.

The conversion between these two forms can be made in a straightforward manner when u(x, t =

0) = 0. In this case, if a Riemann-Liouville integration of order (1− γ) is made on both sides of equation

(2.4.1), the Caputo derivative naturally comes up on the left side because the Caputo derivative of order

γ is equivalent to the fractional integral of order (1 − γ) of u(1)(t), with (0 < γ < 1) (see section 2.1).

In this situation, the Riemann-Liouville fractional derivative on the left hand side of (2.4.1) vanishes and

19

the source term in equation (2.4.4) is given by the Riemann-Liouville fractional integral of order (1 − γ)

of f(x, t), denoted F (x, t).

2.4.2 First order finite difference scheme

The first scheme here shown for the time fractional diffusion equation was developed by Yuste in

[40], to which a source term was added. This scheme can be thought of as an extension of the weighted

average schemes for integer order differential equations.

Let us consider, equation (2.4.1) at the off-lattice point (xi, tn+ 12)

∂

∂tun+1/2i = Kγ

RL0D

(1−γ)t

(∂2

∂x2un+1/2i

)+ f

n+1/2i = 0 (2.4.7)

The integer order time and space derivatives in this equation are now replaced by the three-point centred

operator (2.3.1), for the first order time derivative and a weighted average of the three-point centred

finite difference operator in (2.3.3), evaluated at times tn and tn+1. Furthermore, the Riemman-Liouville

derivative is substituted by the Grunwald-Letnikov difference operator defined in (2.2.2).

δtun+1/2i −

[θKγδ

1−γt δ2xu

ni + (1− θ)Kγδ

1−γt δ2xu

n+1i + θfni + (1− θ)fn+1

i

]= T

n+1/2j (2.4.8)

Neglecting the truncation error and expanding the difference operators using equations (2.3.1),

(2.3.3) and (2.2.2) a computable finite difference scheme is achieved

− Sun+1j−1 + (1 + 2S)un+1

j − Sun+1j = R, 1 ≤ i ≤M − 1, 0 ≤ n ≤ N − 1 (2.4.9)

U0i = 0, 1 ≤ i ≤M − 1 (2.4.10)

Un0 = φ(tn), unM = Φ(tn), 0 ≤ n ≤ N (2.4.11)

where

S = (1− θ)ω(1−γ)0 S, S = Kγ

(τ)γ

(h)2(2.4.12)

and

R = unj + S

n∑k=0

[(1− θ)ω(1−γ)

k+1 + θω(1−γ)k

] [un−kj−1 − 2un−kj + un−kj+1

]+ τγ

[θfni + (1− θfn+1

i

]. (2.4.13)

Though the scheme is in general implicit, some particular cases are to be pointed out. If θ = 1 the

scheme is fully explicit while for θ = 0 the scheme is fully explicit. For θ = 1/2, a Crank-Nicholson type

scheme is achieved.

In [40], Yuste concluded that the truncation error Tn+1/2j in equation 2.4.8 is O(h2 + τ q), with q = 1 if

θ 6= 12 and q = 2 if θ = 1

2 and a second order discretization scheme is used for the fractional derivative.

This means that if the scheme is used, as given in [40] there is no significant improvement between the

20

semi-implicit (θ = 1/2) and fully implicit (θ = 1).

On the same paper, it was proven through a von Neumann stability analysis that the stability of

the method is strongly dependent on the chosen θ, being unconditionally stable for 0 ≤ θ ≤ 12 and

conditionally stable for 12 ≤ θ ≤ 1. This criterion is summarized in equation (2.4.14).

1

S≥ 1

S×≡ 2(2θ − 1)W (−1, 1− γ) (2.4.14)

where W (z, γ) is the generating function of the coefficients, in this caseW (z, γ) = (1− z)γ .

2.4.3 Second order implicit finite difference scheme

In [47] Hu and Zhang develop, through an integration method, a second order difference scheme for

the time fractional diffusion equation. In the derivation of the scheme they make use of the following

lemma.

Lemma 2.4.1. [92] If u(x, t) is sufficiently smooth, then we have

u(xi, t)−(tn+1 − t)u(xi, tn)− (t− tn)u(xi, tn+1)

τ=

1

2

∂2u(xi, tn)

∂t2(tn − t)(tn+1 − t) ≤ C1τ

2,

C1 = max

∣∣∣∣12 ∂2u(x, t)

∂t2

∣∣∣∣ (2.4.15)

Integrating equation 2.4.1, one gets

u(xi, tn+1)− u(xi, tn) =Kγ

Γ(γ)

∫ tn+1

0

uxx(xi, ξ)

(tn+1 − ξ)1−γdξ − Kγ

Γ(γ)

∫ tn

0

uxx(xi, ξ)

(tn − ξ)1−γdξ +

∫ tn+1

tn

f(xi, ξ)dξ

= I1 − I2 + I3

(2.4.16)

Applying Lemma 2.4.1 and the central difference formula for the second order space derivative,

I1,I2,I3 are discretized

I1 =Kγ

Γ(γ)

n∑k=0

∫ tk+1

tk

[(tk+1 − ξ)uxx(xi, tk) + (ξ − tk)uxx(xi, tk+1)

τ

]1

(tn+1 − ξ)1−γdξ +Rn+1

1i

= r

n∑k=0

[ω(γ)k δ2xu(xi, tn−k) + υ

(γ)k δ2xu(xi, tn−k+1)

]+Rn+1

1i +Rn+12i

(2.4.17)

I2 = r

n−1∑k=0

[ω(γ)k δ2xu(xi, tn−k−1) + υ

(γ)k δ2xu(xi, tn−k)

]+Rn1i +Rn2i (2.4.18)

I3 =τ

2[f(xi, tn) + f(xi, tn+1] +Rn+1

3i (2.4.19)

where

r =Kγτ

γ

Γ(1 + γ)(2.4.20)

21

ω(γ)i =

1

1 + γ

[(i+ 1)1+γ − i1+γ

](2.4.21)

υ(γ)i =

1

1 + γ

[(i+ 1)1+γ − i1+γ

]− iγ (2.4.22)

and

Rn1i =Kγ

Γ(γ)

n−1∑k=0

∫ tk+1

tk

[uxx(xi, ξ)−

(tk+1 − ξ)uxx(xi, tk) + (ξ − tk)Lu(xi, tk+1)

τ

]1

(tn − ξ)1−γdξ

(2.4.23)

Rn2i = r

n−1∑k=0

[ω(γ)k (uxx(xi, tn−k−1)− δ2xu(xi, tn−k−1)) + υ

(γ)k (uxx(xi, tn−k)− δ2xu(xi, tn−k))

](2.4.24)

After some handling of the equation, defining ω(γ)−1 = 0 and replacing uni by its numerical approxima-

tion Uni , it is possible to obtain

Un+1i − rυ(γ)0 δ2xU

n+1i = Uni + r

n−1∑k=0

[(ω(γ)k − ω(γ)

k−1 + υ(γ)k+1 − υ

(γ)k

)δ2xU

n−ki

]+ r

(ωγ)n − ω

γ)n−1

)δ2xu

0i + τf

n+ 12

i , 1 ≤ i ≤M − 1, 1 ≤ n ≤ N

(2.4.25)

Un0 = φ(tn), unM = ψ(tn), 1 ≤ n ≤ N (2.4.26)

U0i = 0, 0 ≤ i ≤M (2.4.27)

where the omitted truncation error Rn+1i is equal to

Rn+1i = Rn+1

1i −Rn1i +Rn+12i −Rn2i +Rn+1

3i (2.4.28)

Hu and Zhang have also estimated that |Rn+1i | ≤ Cr(τ2 + h2)(ω

(γ)n + υ(γ)). In the same paper, the

stability of the scheme was proven and summarized in the following theorem.

Theorem 2.4.1. When 0 < γ ≤ log2 3−1 the scheme is stable to the initial data and the inhomogeneous

term in the L∞ norm, defined as

||u||∞ = max1≤i≤M−1

|ui| (2.4.29)

The convergence of the scheme was also proved and the following theorem holds

Theorem 2.4.2. Let u(x, t) ∈ C4,3x,t ([0, L] × [0, T ]) be the solution of the problem (2.4.1)-(2.4.3) and

Uni |0 ≤ i ≤M, 0 ≤ n ≤ N be the solution of the scheme (2.4.25)-(2.4.27). Denote eni = u(xi, tn)−Uni ,

0 ≤ i ≤M , 0 ≤ n ≤ N . Then for nτ ≤ T and 0 < γ ≤ log2 3− 1, there exists a positive constant C, such

that ||en||∞ ≤ C(τ2 + h2).

Though the stability and convergence of this scheme is only proven for γ ∈ [0, log2 3 − 1], Hu and

Zhang point out that numerical experiments show evidence of unconditional stability and convergence,

leaving this proof an open problem.

22

2.4.4 Third order compact finite difference scheme

The final scheme here presented for time fractional diffusion equations was developed by Ji and Sun

in [46]. This high-order compact difference scheme uses the third order accurate time weighted and

shifted Grunwald difference operator for time discretization defined in (2.2.15). For the spacial direction,

a compact technique is employed. Even though Ji and Sun derive an approximation for the Riemann-

Liouville derivatives, they then focus on the particular cases where there is equivalence between the

Riemann-Liouville and Caputo forms of the time fractional diffusion problem, developing a scheme for

the Caputo form of the time fractional initial-boundary value problem in equations (2.4.4)-(2.4.6). Before

proceeding further with the discretization there are two lemmas in [46] which are fundamental to the

development of the scheme that will now be stated.

Lemma 2.4.2. If u(0)=0, then it holds that 0D−γt (C0 D

γt u(t)) = u(t) for 0 < γ < 1.

Lemma 2.4.3. Define θ(s) = (1− s)3 = [5− 3(1− s)2]. if g(x) ∈ C6[a, b],h = (b− a)/M , xi = a+ ih(0 ≤

i ≤M) it holds that

1

12[g′′(xi−1) + 10g′′(xi) + g′′(xi+1)]

=g(xi−1 − 2g(xi) + g(xi+1)

h2+

h4

360

∫ 1

0

[g(6)(xi − sh) + g(6)(xi + sh)]θ(s)ds, 1 ≤ i ≤M − 1

(2.4.30)

In addition, let an average operator be defined as

Auni =

112 (uni−1 + 10uni + uni+1) = (I + h2

12 δ2x)uni , 1 ≤ i ≤M − 1

uni , i = 0 or M

(2.4.31)

Looking at the structure of the time weighted and shifted Grunwald difference operator in (2.2.15),

it can be seen that the discretization of the first time level for equation (2.4.4) needs to be considered

separately from the second to the Nth time levels. It will be further assumed that u(x, t) ∈ C6,5x,t ([a, b] ×

[0, T ]) and ∂ku(x,0)∂tk

= 0 for k = 0, 1, ..., 5, which allows for C0Dγt u(xi, tn) = RL

0Dγt u(xi, tn)).

The discretization for time levels with 2 ≤ n ≤ N will be first considered. At grid point (xi, tn) equation

2.4.4 givesC0D

γt u

ni = Kγ

∂2uni∂x2

+ fni , 0 ≤ i ≤M, 2 ≤ n ≤ N (2.4.32)

If the weighted and shifted Grunwald difference operator is chosen to approximate C0D

γt u(xi, tn) ,

followed by the application of the average operator A to both sides of the equation and using Lemma

2.4.3 will lead to

1

τγ

n∑k=0

g(γ)k Au

n−ki = Kγδ

2xu

ni +Afni +Rni , 1 ≤ i ≤M − 1, 2 ≤ n ≤ N (2.4.33)

where

|Rni | ≤ C1(τ3 + h4), 1 ≤ i ≤M − 1, 2 ≤ n ≤ N (2.4.34)

To obtain the discretization at the first time step, the Riemann-Liouville integral operator RL0D−γt is

23

applied on both sides of equation (2.4.4). Making use of lemma 2.4.2 one obtains

u(x, t1) =Kγ

Γ(γ)

∫ t1

0

uxx(x, ξ)

(t1 − ξ)1−γdξ + F (x, t1) (2.4.35)

using uxx(x, 0), uxxt(x, 0) and uxx(x, t1) to make an Hermite interpolation of uxx(x, ξ) on the interval

[0, t1], it follows that

P (x, ξ) = uxx(x, 0) + uxxt(x, 0)(ξ − 0) +uxx(x, t1)− uxx(x, 0)− τuxxt(x, 0)

τ2(ξ − 0)2 (2.4.36)

If u(x, 0) = 0 and ut(x, 0) = 0 one obtains

u(x, t1) ≈ u(x, t1) =Kγ

Γ(γ)

∫ t1

0

P (x, ξ)

(t1 − ξ)1−γdξ + F (x, t1) =

2Kγ

Γ(γ + 3)τγuxx(x, t1) + F (x, t1) (2.4.37)

where F (x, t) =RL0 D−γf(x, t)

Once again, applying the space average operator A and using Lemma 2.4.3 gives

1

τγAu1i =

2Kγ

Γ(γ + 3)δ2xu

1i +

1

τγAF (xi, t1) +R1

i , 1 ≤ i ≤M − 1 (2.4.38)

where

|R1i | ≤ C3(τ3 + h4), 1 ≤ i ≤M − 1 (2.4.39)

Finally, omitting the error terms Rni and replacing uni with the numerical approximation Uni the final

scheme is1

τγ

n∑k=0

g(γ)k AU

n−ki = Kγδ

2xU

ni +Afni , 1 ≤ i ≤M − 1, 2 ≤ n ≤ N (2.4.40)

1

τγAU1

i =2Kγ

Γ(γ + 3)δ2xU

1i +

1

τγAF (xi, t1), 1 ≤ i ≤M − 1 (2.4.41)

U0i = 0, 1 ≤ i ≤M − 1 (2.4.42)

Un0 = φ(tn), unM = Φ(tn), 0 ≤ n ≤ N (2.4.43)

Equations (2.4.40) and (2.4.41) are systems of linear diagonally dominant equations, having a unique

solution and easily solved. Having discretized the scheme and provided an error estimation Ji and Sun

proceed with the stability and convergence analysis of the scheme resulting in

Theorem 2.4.3. The difference scheme (2.4.40)-(2.4.43) is unconditionally stable to the right hand therm

and initial value for all γ ∈ [0, γ∗], with γ∗ = 0.9569347.

Theorem 2.4.4. Assume that u(x, t) ∈ C6,5x,t ([a, b]×[0, T ]) is the solution of problem (2.4.4) to (2.4.6), and

uni |0 ≤ i ≤M, 0 ≤ n ≤ N is the solution of the finite difference scheme (2.4.40) to (2.4.43). Suppose

∂ku(x, 0)

δtk= 0, k = 0, 1, ..., 5 (2.4.44)

24

Denote

eni = u(xi, tn)− uni , 0 ≤ i ≤M, 0 ≤ n ≤ N (2.4.45)

Then when Nτ ≤ T it holds that

τ

N∑m=1

‖em‖∞ ≤=b− a

22

√C4T (c21T + C2

3 )(τ3 + h4). (2.4.46)

2.4.5 Numerical examples

In this section numerical, the behaviour of the selected schemes for the time fractional diffusion equa-

tions is studied. A comparison of the schemes is made, confronting the solutions of the same problem

given by different schemes, in terms of convergence order, error and computational cost. Consider the

following time fractional diffusion equation of the form (2.4.1)-(2.4.3)

∂u

∂t= RL

0D1−γt

[K∂2u

∂x2

]+ f(x, t), x ∈ [0, L], t ∈ [0, T ] (2.4.47)

u(x, t = 0) = 0, x ∈ [0, L] (2.4.48)

u(x = 0, t) = 0, U(x = L, t) = t4−γ sin(1), t ∈ [0, t] (2.4.49)

where K = 1, L = 1, T = 1 and source term f(x, t) given by

f(x, t) = sin(x)

[(4− γ)t3−γ

Γ(5− γ)t3

6

](2.4.50)

The exact solution of the problem is u(x, t) = t4−γ sin(x), depicted on Figure 2.1 for γ = 0.5. Since

u(x, t = 0) = 0, equation (2.4.47) can easily be converted to the form of equation (2.4.4) form by means

of the procedure described in section 2.4.1.

Figure 2.2, shows the absolute errors for γ = 0.5, for each of the schemes. As expected the highest

errors are present on the first order scheme followed by the second order (thousands of times smaller)

and the compact third order scheme which shows a maximum error in the order of 10−10.

Table 2.4.1 lists the results of the time convergence analysis of the three schemes that were anal-

ysed. For each scheme three different fractional orders (γ = 0.2, γ = 0.5 and γ = 0.8) were considered

and the L∞h,τ error was registered with the refinement of the time interval, allowing the computation of

the convergence order. The L∞h,τ error is defined as follows

L∞h,τ = max |Uni − uni | , 0 ≤ i ≤M, 0 ≤ n ≤ N (2.4.51)

Where uni is the exact and Uni the numerical solution of problem (2.4.47)-(2.4.49), with the mesh

step-sizes h and τ at the grid point (xi, tn). If h τa/b the order of convergence EOC in time of an error

E(h, τ) may be calculated by

25

Figure 2.1: Exact solution of problem (2.4.47)-(2.4.49) with γ = 0.5.

EOC = log τ1τ2

(E(h, τ1)

E(h, τ2)

)(2.4.52)

The time step was kept at value h = 1/2000, guaranteeing that in every test case the contribution

of the space truncation error to the solution is minimal when compared with the time contribution. The

first, second and third order schemes were shown, in the previous sections, to have errors O(τ + h2),

O(τ2 + h2) and O(τ3 + h4), respectively and any h ≤ τ will result for the three cases in a smaller

contribution of the space error.

The results for the first order scheme, listed in the table correspond to θ = 0, the fully implicit situation.

Other values of θ were tested namely θ = 1/2, but use of first order weights in the computation of the

time fractional derivative prevents that higher than first order convergence is achieved. Consequently the

results with other θ values display similar results. The first order scheme clearly shows the expected first

order of convergence,with the halving of the maximum error with the halving of the time step. Moreover,

a slight error decrease can be seen with the increase in γ, all within the same order of magnitude.

The second order scheme follows also the theoretical second order predictions for error convergence.

This scheme naturally allows a significant reduction of the maximum error when compared with the first

order scheme. A slight error decrease with the increase of γ is also observed in this case.

The third order scheme, even if it can be said that it behaves according to the theoretical results,

shows a need for smaller time steps to reach asymptotic convergence in the case of γ = 0.8. On the

other hand, the results for γ = 0.2 and γ = 0.5 show third order convergence even with the coarse

grids. Contrary to the two previous schemes, a slight increase in the error is observed with the increase

γ. Naturally, the maximum errors observed are orders of magnitude smaller than with the two previous

26

(a) Absolute error in the numerical solution of the problem (2.4.47)-(2.4.49) with the first order implicit scheme.

(b) Absolute error in the numerical solution of the problem (2.4.47)-(2.4.49) with the second order implicit scheme.

(c) Absolute error in the numerical solution of the problem (2.4.47)-(2.4.49) with the third order compact scheme.

Figure 2.2: Absolute errors for each numerical scheme for the solution of time fractional diffusion equa-tions. All the the plots shown correspond to τ = 1/2000, h = 1/512 and γ = 0.5.

27

schemes.

Table 2.4.2 lists the computing times of each of the solutions used to produce the results of Table

2.4.1 with (γ = 0.5) and no significant differences were observed for other γ values. The results on Table

2.4.1 show that the time of computation is decreasing with the order of the scheme partly because the

computation of time fractional derivatives involves all previous time steps, regardless of the order of the

scheme the number of time steps involved will be the same. It was seen that this decrease is mainly due

to the computing times of the right side matrix, when building the scheme. The matrix associated with

the implicit time step showed similar spectral radius for every scheme and the times for the solution of

the system of equations at each time step also revealed to be identical. The first order scheme requires

the computation of the weighted average of the time fractional derivative of the space derivative in two

time steps, while the second order scheme evaluates the fractional derivative at only one time step, thus

explaining the observed drop in computing time. The third order scheme also reveals a considerable

decrease of computing times with respect to the second order schemes that may be related with the

solution of the fractional diffusion equation in the Caputo form.

These results indicate that high order schemes for the solution of the time fractional diffusion equation

are the best choice and the ability of reaching the same order of magnitude with less time steps is an

enormous advantage, with significant reduction of computing times.

Table 2.4.1: L∞ error and its order of convergence with decrease of the temporal step size, for thepresented schemes for the time fractional diffusion equation. The results were taken with h = 1/2000.

γ = 0.2 γ = 0.5 γ = 0.8

1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

1stOrder 8 2.696E-02 - 2.438E-02 - 2.162E-02 -16 1.327E-02 1.02 1.223E-02 1.00 1.100E-02 0.9732 6.581E-03 1.01 6.123E-03 1.00 5.550E-03 0.9964 3.277E-03 1.01 3.064E-03 1.00 2.787E-03 0.99128 1.635E-03 1.00 1.533E-03 1.00 1.397E-03 1.00256 8.168E-04 1.00 7.664E-04 1.00 6.991E-04 1.00512 4.082E-04 1.00 3.832E-04 1.00 3.497E-04 1.00

2ndOrder 8 1.223E-03 - 8.840E-04 - 8.613E-04 -16 2.939E-04 2.06 2.156E-04 2.04 2.101E-04 2.0432 7.097E-05 2.05 5.301E-05 2.02 5.164E-05 2.0264 1.721E-05 2.04 1.310E-05 2.02 1.276E-05 2.02128 4.187E-06 2.04 3.248E-06 2.01 3.164E-06 2.01256 1.022E-06 2.03 8.079E-07 2.01 7.871E-07 2.01512 2.510E-07 2.03 2.019E-07 2.00 1.967E-07 2.00

3rdOrder 8 5.792E-05 - 1.164E-04 - 2.053E-04 -16 7.518E-06 2.95 1.461E-05 2.99 3.357E-05 2.6132 9.568E-07 2.97 1.829E-06 3.00 4.982E-06 2.7564 1.207E-07 2.99 2.288E-07 3.00 7.362E-07 2.76128 1.517E-08 2.99 2.863E-08 3.00 1.007E-07 2.87256 1.926E-09 2.98 3.608E-09 2.99 1.337E-08 2.91512 2.691E-10 2.84 4.915E-10 2.88 1.772E-09 2.92

28

Table 2.4.2: Time of computation for each scheme for each of the presented schemes. The resultscorrespond to a constant space step h = 1/2000.

Time Of Computation (s)

1/τ 1stOrder 2ndOrder 3rdOrder

8 0.095 0.090 0.33216 0.105 0.093 0.36332 0.149 0.116 0.36764 0.288 0.225 0.400

128 0.949 0.672 0.636256 3.390 2.331 1.287512 14.125 9.124 4.209

2.5 Space fractional diffusion equations

2.5.1 Problem statement

On this section focus will be given to space fractional diffusion equations. The space fractional initial-

boundary value problem is considered as follows

∂u(x, t)

∂t= K1

RLaD

αxu(x, t) +K2

RLxD

αb u(x, t) + f(x, t), (x, t) ∈ [a, b]× [0, T ] (2.5.1)

u(x, 0) = ξ(x), x ∈ [a, b] (2.5.2)

u(a, t) = 0, u(b, t) = 0, t ∈ [0, T ] (2.5.3)

where RLaD

αxu and RL

xDαb u are the left and right Riemann-Liouville derivatives defined in equations

(2.1.3) and (2.1.4) with 1 < α < 2. The diffusion coefficients K1 and K2 are nonnegative constants with

K21 +K2

2 6= 0. If K1 6= 0 then ψ(t) = 0 and if K2 6= 0 then φ(t) = 0.

Throughout the section let xi = ih(0 ≤ i ≤ M), tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and

Ωτ = tn|0 ≤ n ≤ N. The computational domain [a, b] × [0, T ] is then covered by Ωτh = Ωh × Ωτ .

Moreover, consider uni = u(xi, tn).

2.5.2 First order finite difference scheme

The first scheme considered for space fractional diffusion equations, first order in time and space,

was introduced by Meerschaert and Tadjeran [49]. Here the left and right Riemann-Liouville fractional

derivatives are approximated with the shifted Grunwald difference operators defined in (2.2.25) and

(2.2.26), respectively.

Equation (2.5.1) at gridpoint (xi, tn+1) gives

∂tun+1i = K1

RLa Dα

xun+1i +K2

RLx Dα

b un+1i + fn+1

i (2.5.4)

If first order backward differences are used for time and the the fractional derivatives are approxi-

29

mated with operators (2.2.25) and (2.2.26) with p=1, the optimal value, it is obtained that

un+1i − uni

τ=

1

hα

[K1

i+1∑k=0

ω(α)k un+1

i−k+1 +K2

N−i+1∑k=0

ω(α)k un+1

i+k−1

]+ fn+1

i +O(h+ τ) (2.5.5)

Solving for u at time level tn + 1, omitting the truncation error and denoting by Uni the numerical

approximation of uni yields,

Un+1i − τ

hα

[K1

i+1∑k=0

ω(α)k Un+1

i−k+1 +K2

N−i+1∑k=0

ω(α)k Un+1

i+k−1

]= Uni + τfn+1

i (2.5.6)

Meerschaert and Tadjeran proved in [49], that this implicit method is unconditionally stable for 1 ≤

α ≤ 2. As an addition, it is noted that if this method was used to solve the equation explicitly, the method

would be stable if τ/hα ≤ 1/[α(K1 +K2)]

2.5.3 Second order finite difference scheme

This section describes the construction of the scheme developed by Tian et al. in [62]. The scheme

is second order accurate in time and space, using the shifted Grunwald difference operators developed

in section 2.2.2.2. For this scheme it will be assumed that u ∈ L1(R) and u ∈ C2+α(R).

If the Crank-Nicholson technique is used for time discretization, it is obtained

δun+1/2i − 1

2

(K1(RLaD

αxu)ni +K1(RLaD

αxu)n+1

i +K2(RLxDαb u)ni +K2(RLxD

αb u)n+1

i

)= f

n+1/2i +O(τ2)

(2.5.7)

For space, the approximation of the left and right Riemann-Liouville derivatives with the weighted

and shifted operators WS2δα

x,+ and WS2δα

x,− with (p, q) = (1, 0)or(1,−1) defined in section 2.2.2.2 leads

to

δun+1/2i − 1

2

(K1

WS2δα

x,+uni +K1

WS2δα

x,+un+1i +K2

WS2δα

x,−uni +K2

WS2δα

x,−un+1i

)= f

n+1/2i +O(τ2 + h2)

(2.5.8)

Substitution of the fractional difference operator with equations (2.2.34) and (2.2.35), separation of

time layers, omission of the truncation error and denoting by Uni the numerical approximation of uni gives

Un+1i − K1τ

2hα

i+1∑k=0

g(α)k Un+1

i−k+1 −K2τ

2hα

N−i+1∑k=0

g(α)k Un+1

i+k−1

= Uni +K1τ

2hα

i+1∑k=0

g(α)k Uni−k+1 +

K2τ

2hα

N−i+1∑k=0

g(α)k Uni+k−1 +

τ

2(fn+1i + fni )

(2.5.9)

Following the construction of the finite difference scheme Tian et al. proved that the scheme in

30

unconditionally stable and convergent with order O(h2 + τ2).

2.5.4 Fourth order quasi-compact finite difference scheme

The construction of the high order scheme made by Hao et al. in [55] is summarized in this section.

This scheme employs the fourth-order approximations for left and right Riemann-Liouville derivatives

introduced in section 2.2.2.3. For this scheme it will be assumed that u ∈ L1(R) and u ∈ C4+α(R).

For convenience,let

Dαx = (K1

RLaD

αx +K2

RLxD

αb ), δαx = (K1

WS4δα

x,+ +K2WS4δ

α

x,−) (2.5.10)

where the weighted difference operators WS4δα

x,+ and WS4δα

x,− are defined by (2.2.47) and (2.2.48),

respectively. Considering equation (2.5.1) at point (xi, t) gives

ut(xi, t) = Dαxu(xi, t) + f(xi, t), 0 ≤ i ≤M (2.5.11)

Applying now the average difference operator in equation (2.2.41) to equation (2.5.11)

Aut(xi, t) = ADαxu(xi, t) +Af(xi, t), 1 ≤ i ≤M − 1 (2.5.12)

Equations (2.2.47) and (2.2.48) allow to write this equation as

Aut(xi, t) = δαxu(xi, t) +Af(xi, t) +O(h4), 1 ≤ i ≤M − 1 (2.5.13)

The above equation is now averaged at time levels t = tn and t = tn+1, yielding

Aδtun+1/2i − δαxu

n+1/2i = Afn+1/2

i +Rni , 1 ≤ i ≤M − 1, 0 ≤ n ≤ N − 1 (2.5.14)

where Rni is the truncation error with |Rni | ≤ c1(τ2 + h4)

Omitting the truncation error and replacing uni by its numerical approximation Uni the following finite

difference scheme is obtained

AδtUn+1/2i − δαxU

n+1/2i = Afn+1/2

i , 1 ≤ i ≤M − 1, 0 ≤ n ≤ N − 1 (2.5.15)

Un0 = φa(x0), unM = φb(xM ), 1 ≤ n ≤ N (2.5.16)

U0i = u0(xi), 0 ≤ i ≤M (2.5.17)

Hao, Sun and Cao proved in [55] that the difference scheme (2.5.15)-(2.5.17) is unconditionally stable

to the initial value u0 and right hand term f for all 1 < α ≤ 2. They further proved that the scheme is

convergent and the estimate

||un − Un|| ≤ 3c1√b− aT (τ2 + h4), 1 ≤ n ≤ N (2.5.18)

31

holds for all 1 < α ≤ 2 where uni is the exact solution of the problem (2.5.1)-(2.5.3) and Uni is the

corresponding solution of the scheme (2.5.15)-(2.5.17).

2.5.5 Numerical examples

The following space fractional initial-boundary value problem, of type (2.5.1)-(2.5.3) is considered.

∂u(x, t)

∂t= RL

0Dα

xu(x, t) + RLxD

α

1u(x, t) + f(x, t), (x, t) ∈ [0, 1]× [0, T ] (2.5.19)

u(x, t = 0) = xµ(1− x)µ, x ∈ [a, b] (2.5.20)

u(x = 0, t) = u(x = 1, t) = 0, t ∈ [0, T ] (2.5.21)

with source term

f(x, t) = e−t(u(x, 0)− RL0D

αxu(x, t)− RL

xDα1u(x, t)) (2.5.22)

and exact solution u(x, t) = e−txµ(1− x)µ.

Considering µ = 2, 3, 4 will allow some conclusion regarding the smoothness requirements for the

schemes to behave according to the theoretical predictions. In figure 2.3 the plots of the analytical

solutions for µ = 2, 3, 4 are shown. The comparison of these three plots clearly shows the increase of

the smoothness in the boundaries with increasing µ.

Figure 2.4 depicts the absolute error in the solution of the three schemes for µ = 4, with the same

grid (τ = 1/20000 and h = 1/128). Oscillations in the absolute errors can be seen in the space direction,

whose frequency increases with the order of the schemes. Moreover, it is seen that with the increase in

the order of the scheme there is a local maximum of the error approaching the boundaries.

If a numerical scheme has a truncation error O(τa + hb) then optimal step sizes will yield τa ≈ hb. If

τ hb/a, the contribution of time discretization of the overall error will be minimal when compared with

the space discretization contribution. Bearing this conclusion in mind, a fixed time step τ = 1/20000 will

be considered to analyse the space convergence.

In each of the three following Tables 2.5.1 , 2.5.2 and 2.5.3, a convergence analysis in the space

direction is made for the three schemes under different fractional orders, with each table corresponding

to a different value of µ. Tables 2.5.1, 2.5.2 and 2.5.3 correspond to µ = 2, µ = 3 and µ = 4.

Regarding the first order scheme, it can be seen that asymptotic convergence requires smaller space

intervals than the remaining schemes and that the theoretical first order prediction is achieved. As α

increases from α = 1.2 to α = 1.8 there is also a clear reduction in the maximum error which may reach

two orders of magnitude and the increased smoothness leads to smaller errors.

Focusing now on the second order scheme, the tables show that second order convergence is

achieved for every µ. The second order scheme has two possibilities, (p, q) = (1, 0) and (p, q) = (1,−1),

that lead to unconditional stability being the former case the one listed in the tables. Only a slightly better

convergence rate was observed than for the latter. In this case, asymptotic convergence reached with

courser space grids. As in the first order scheme, an increase of µ leads to smaller maximum errors and

32

(a) µ = 2

(b) µ = 3

(c) µ = 4

Figure 2.3: Analytical solutions of problem (2.5.19)-(2.5.21) for µ = 2, 3, 4.

33

(a) Absolute error in the numerical computation with the first order scheme.

(b) Absolute error in the numerical computation with the second order scheme.

(c) Absolute error in the numerical computation with the fourth order scheme.

Figure 2.4: Absolute error in the numerical solution of the problem (2.5.19)-(2.5.21) with µ = 4 for thethree schemes presented. The errors correspond to h = 1/128 and τ = 1/20000.

34

no significant changes in the maximum error with the variation of the fractional order α.

Looking now at the fourth order scheme, the first thing that meets the eye is the change in behaviour

with µ. It is seen that with µ = 2 second order of convergence is achieved, third order with µ = 3 and

fourth order with µ = 4. This undoubtedly proves that the importance of the smoothness criteria in the

numerical solution, particularly for high-order schemes. Looking for instance at Table 2.5.3, with µ = 4, it

can be proved that the integer order space derivatives of u(x, t) are continuous up to the third derivative

at the boundaries (x ∈ [0, 1]), therefore the function u ∈ C3 in space. Numerical results then suggest

that the continuity requirements may be eased past the C4+α(R) that was assumed in the theoretical

analysis. As the value of µ decreases, so does the convergence rate of the fourth order scheme. It may

then be said that the fourth order scheme is convergent with order µ for µ ≤ 4. This means that for the

fourth order scheme the maximum error greatly increases with the decrease of µ ≤ 4 without significant

variations across different values of α.

Table 2.5.1: L∞h,τ errors and their order of convergence with the refinement of the space step for equation(2.5.19)-(2.5.21) with µ = 2

α = 1.2 α = 1.5 α = 1.8

1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

First Order Scheme 8 1.3921E-02 - 4.0300E-03 - 1.5458E-03 -16 8.4348E-03 0.723 2.6013E-03 0.632 1.8557E-04 3.05832 4.6923E-03 0.846 1.4432E-03 0.850 1.0165E-04 0.86864 2.4808E-03 0.919 7.5459E-04 0.936 8.8970E-05 0.192

128 1.2747E-03 0.961 3.8462E-04 0.972 5.3671E-05 0.729256 6.4533E-04 0.982 1.9388E-04 0.988 2.9041E-05 0.886512 3.2439E-04 0.992 9.7260E-05 0.995 1.5049E-05 0.948

1024 1.6254E-04 0.997 4.8693E-05 0.998 7.6515E-06 0.976

Second Order Scheme 8 1.8813E-03 - 2.9161E-03 - 3.3964E-03 -16 4.6022E-04 2.031 7.0247E-04 2.054 8.3020E-04 2.03232 1.5325E-04 1.586 1.6817E-04 2.062 2.0174E-04 2.04164 4.3826E-05 1.806 4.0196E-05 2.065 4.8889E-05 2.045

128 1.1775E-05 1.896 9.6206E-06 2.063 1.1836E-05 2.046

Fourth Order Scheme 8 2.0239E-04 - 4.5958E-04 - 8.3249E-04 -16 5.9058E-05 1.777 1.1855E-04 1.955 2.0970E-04 1.98932 1.6822E-05 1.812 3.1574E-05 1.909 5.4195E-05 1.95264 4.6011E-06 1.870 8.2943E-06 1.929 1.4035E-05 1.949

128 1.2216E-06 1.913 2.1446E-06 1.951 3.5974E-06 1.964

Table 2.5.4 lists the times of computation for the numerical solutions with three different values of µ

for α = 1.5 , 1/τ = 20000 and 1/h = 128. The fourth order scheme to gives slightly better results for the

less smooth µ function than the second order scheme. The homogeneity in computing times is however

to be expected, since the matrices used to compute the left and right space fractional derivatives are

full. In other words, since the value of the space fractional derivatives depends on all the space values

at a given instant, the higher order scheme has no significant increase in computational cost. If a large

space domain is to be analysed with small errors, the high order scheme becomes even more important

in fractional diffusion equations since it will allow the same error magnitude with a courser space grid.

The fourth order finite difference scheme, for the same absolute error magnitude presents a smaller

computing cost than the first and second order schemes because less points may be considered.

35

Table 2.5.2: L∞h,τ errors and their order of convergence with the refinement of the space step for equation(2.5.19)-(2.5.21) with µ = 3

α = 1.2 α = 1.5 α = 1.8

1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

First Order Scheme 8 3.6888E-03 - 1.1387E-03 - 2.6918E-04 -16 2.2923E-03 0.686 7.1007E-04 0.681 5.3609E-05 2.32832 1.3136E-03 0.803 3.9075E-04 0.862 3.9861E-05 0.42864 7.1174E-04 0.884 2.0443E-04 0.935 2.6149E-05 0.608

128 3.7183E-04 0.937 1.0447E-04 0.969 1.4615E-05 0.839256 1.9020E-04 0.967 5.2786E-05 0.985 7.6915E-06 0.926512 9.6201E-05 0.983 2.6528E-05 0.993 3.9415E-06 0.965

1024 4.8377E-05 0.992 1.3297E-05 0.996 1.9946E-06 0.983

Second Order Scheme 8 3.1605E-04 - 3.3891E-04 - 2.8112E-04 -16 8.4564E-05 1.902 8.7296E-05 1.957 7.0401E-05 1.99832 2.2056E-05 1.939 2.2411E-05 1.962 1.7812E-05 1.98364 5.6565E-06 1.963 5.7032E-06 1.974 4.4948E-06 1.987

128 1.4344E-06 1.979 1.4405E-06 1.985 1.1300E-06 1.992

Fourth Order Scheme 8 7.7491E-04 - 4.9583E-04 - 6.1105E-04 -16 1.0862E-04 2.835 6.0868E-05 3.026 1.0479E-04 2.54432 1.3956E-05 2.960 5.4423E-06 3.483 1.3285E-05 2.98064 1.8088E-06 2.948 8.2337E-07 2.725 1.7370E-06 2.935

128 2.3531E-07 2.942 1.1825E-07 2.800 2.0022E-07 3.117

Table 2.5.3: L∞h,τ errors and their order of convergence with the refinement of the space step for equation(2.5.19)-(2.5.21) with µ = 4

α = 1.2 α = 1.5 α = 1.8

1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

First Order Scheme 8 9.6878E-04 - 2.9516E-04 - 8.2201E-05 -16 6.2081E-04 0.642 1.8834E-04 0.648 1.6002E-05 2.36132 3.6606E-04 0.762 1.0585E-04 0.831 9.3134E-06 0.78164 2.0230E-04 0.856 5.6071E-05 0.917 6.6894E-06 0.477

128 1.0696E-04 0.919 2.8850E-05 0.959 3.8588E-06 0.794256 5.5078E-05 0.957 1.4629E-05 0.980 2.0570E-06 0.908512 2.7953E-05 0.978 7.3626E-06 0.991 1.0589E-06 0.958

1024 1.4076E-05 0.990 3.6905E-06 0.996 5.3542E-07 0.984

Second Order Scheme 8 2.1098E-04 - 2.4372E-04 - 2.0179E-04 -16 5.4244E-05 1.960 6.1119E-05 1.996 5.0377E-05 2.00232 1.3887E-05 1.966 1.5467E-05 1.982 1.2660E-05 1.99364 3.5249E-06 1.978 3.9019E-06 1.987 3.1748E-06 1.995

128 8.8876E-07 1.988 9.8070E-07 1.992 7.9501E-07 1.998

Fourth Order Scheme 8 9.4148E-06 - 1.8968E-05 - 2.6585E-05 -16 8.3720E-07 3.491 1.5792E-06 3.586 2.0566E-06 3.69232 5.9107E-08 3.824 1.0861E-07 3.862 1.3693E-07 3.90964 3.8954E-09 3.924 6.9461E-09 3.967 8.6772E-09 3.980

128 2.8474E-10 3.774 4.3124E-10 4.010 5.3745E-10 4.013

Table 2.5.4: Time of computation in the solution of problem (2.5.19)-(2.5.21) for the three schemespresented with 1/τ = 20000 and 1/h = 128 for α = 1.5.

1st Order Scheme 2nd Order Scheme 4th Order Scheme

TOC (s) L∞h,τ TOC (s) L∞h,τ TOC (s) L∞h,τ

µ = 2 3.49 3.8462E-04 3.58 9.6206E-06 4.42 2.1446E-06µ = 3 3.47 1.0447E-04 3.27 1.4405E-06 4.23 1.1825E-07µ = 4 3.61 2.8850E-05 3.25 9.8070E-07 4.32 4.3124E-10

36

2.6 Time-space fractional diffusion equations

2.6.1 Problem statement

On this section focus will be given to space fractional diffusion equations. The space fractional initial-

boundary value problem is considered as follows

C0D

γ

t u(x, t) = K1RLaD

α

xu(x, t) +K2RLxD

α

b u(x, t) + f(x, t), (x, t) ∈ [a, b]× [0, T ] (2.6.1)

u(x, 0) = ξ(x), x ∈ [a, b] (2.6.2)

u(a, t) = 0, u(b, t) = 0, t ∈ [0, T ] (2.6.3)

where RLa Dα

xu and RLx Dα

b u are the left and right Riemann-Liouville derivatives defined in equations

(2.1.3) and (2.1.4) with 1 < α < 2. The time fractional derivative C0D

γ

t u(x, t), is considered to be of the

Caputo type as defined in (2.1.5) with 0 ≤ γ ≤ 1. The diffusion coefficients K1 and K2 are nonnegative

constants with K21 +K2

2 6= 0. If K1 6= 0 then ψ(t) = 0 and if K2 6= 0 then φ(t) = 0.

Throughout the section let xi = ih(0 ≤ i ≤ M), tn = nτ(0 ≤ n ≤ N), Ωh = xi|0 ≤ i ≤M and

Ωτ = tn|0 ≤ n ≤ N. The computational domain [a, b] × [0, T ] is then covered by Ωτh = Ωh × Ωτ .

Moreover, consider uni = u(xi, tn).

An extensive research effort revealed that the L1 method stands as the most common approximation

for time fractional derivatives in time-space fractional diffusion equations. The search for high-order finite

difference schemes both in time and space, with stability and convergence analysis failed to produce any

results. Bearing these facts in mind and serving as a needed common ground for the comparison of the

schemes all the finite difference schemes now introduced will use the L1 approximation for the Caputo

time fractional derivative found in section 2.2.1.2. For the discretization of space fractional derivatives

the three approximations that were previously given for Riemann-Liouville space fractional derivatives

will be used. Consequently, three schemes will be available with first, second and fourth order in space

and (2 − γ)th order in time and since they will all use the same approximation for the time derivative,

they will be named after their space convergence order.

2.6.2 First order finite difference scheme

The first finite difference scheme for time-space fractional diffusion equations that will be studied was

developed by Liu et al. [57], to which the right time fractional derivative and the source term were added.

This scheme will make use of the L1 method for the discretization of the Caputo time derivative and

the Riemann-Liouville derivatives will be approximated through the shifted Grunwald operators, already

used in the first order scheme for space fractional diffusion equations. Consider equation (2.6.1), at

(xi, tn) it becomes

C0D

γ

t uni = K1

RLaD

α

xuni +K2

RLxD

α

b uni + fni , (x, t) ∈ [a, b]× [0, T ] (2.6.4)

37

Approximating the Caputo time derivative with the L1 operator as in equation (2.2.7) and the Riemann-

Liouville space derivatives with the shifted Grunwald operators defined in (2.2.23) and (2.2.24), with

p = 1, leads to

L1Cδ

γ

t uni = K11δ

αx,+u

ni +K21δ

αx,−u

ni + fni (2.6.5)

Substitution of the fractional difference operators with equations (2.2.8), (2.2.25) and (2.2.26) yields

τ−γ

Γ(2− γ)

n−1∑k=0

bn−k−1(uk+1i − uki ) =

K1

hα

i+1∑k=0

ωαk uni−k+1 +

K2

hα

M−i+1∑k=0

ωαk uni+k−1 + fni (2.6.6)

Development of the left side of the equation will result in

τ−γ

Γ(2− γ)

[uni −

n−1∑k=1

(bn−k−1 − bn−k)uki − bn−1u0i

]

=K1

hα

i+1∑k=0

ωαk uni−k+1 +

K2

hα

M−i+1∑k=0

ωαk uni+k−1 + fni

(2.6.7)

Rearrangement and substitution of uni by its numerical approximation Uni gives

Uni − µ1

i+1∑k=0

ωαkUni−k+1 − µ2

M−i+1∑k=0

ωαkUni+k−1

=

n−1∑k=1

(bn−k−1 − bn−k)Uki + bn−1U0i + µff

ni , 1 ≤ i ≤M − 1, 1 ≤ n ≤ N − 1

(2.6.8)

Un0 = 0, unM = 0, 1 ≤ n ≤ N (2.6.9)

U0i = u0(xi), 0 ≤ i ≤M (2.6.10)

where µ1 =K1τ

γΓ(2− γ)

hα, µ2 =

K2τγΓ(2− γ)

hαand µf = τγΓ(2− γ).

After constructing the scheme, Liu et al. have made a stability analysis, concluding that it is uncon-

ditionally stable . They have also determined that the scheme is convergent, existing a positive constant

C such that |uni − Uni | ≤ C(τ + h).

2.6.3 Second order finite difference

A second order in space scheme for time-space fractional diffusion equations will now be developed.

Similarly to the first order scheme, this scheme will use the L1 method for the discretization of the Caputo

time derivative and the Riemann-Liouville derivatives will be approximated through the weighted shifted

Grunwald operators defined in section 2.2.2.2, used in the second order scheme for space fractional

diffusion equations. Considering equation (2.6.1) at (xi, tn) gives

C0D

γ

t uni = K1

RLaD

α

xuni +K2

RLxD

α

b uni + fni (2.6.11)

Approximating the Caputo time derivative with the L1 operator as in equation (2.2.7) and the Riemann-

38

Liouville space derivatives with the wheighted and shifted Grunwald operators defined in (2.2.34) and

(2.2.35), with (p, q) = (1, 0), leads to

L1Cδ

γ

t uni = K1

WS2δα

x,+uni +K2

WS2δα

x,−uni + fni , (x, t) ∈ [a, b]× [0, T ] (2.6.12)

Substitution of the fractional difference operators with equations (2.2.8), (2.2.25) and (2.2.26) yields

τ−γ

Γ(2− γ)

n−1∑k=0

bn−k−1(uk+1i − uki ) =

K1

hα

i+1∑k=0

gαk uni−k+1 +

K2

hα

M−i+1∑k=0

gαk uni+k−1 + fni (2.6.13)

Development of the left side of the equation will result in

τ−γ

Γ(2− γ)

[uni −

n−1∑k=1

(bn−k−1 − bn−k)uki − bn−1u0i

]

=K1

hα

i+1∑k=0

gαk uni−k+1 +

K2

hα

M−i+1∑k=0

gαk uni+k−1 + fni

(2.6.14)

Rearrangement and substitution of uni by its numerical approximation Uni gives

Uni − µ1

i+1∑k=0

gαkUni−k+1 − µ2

M−i+1∑k=0

gαkUni+k−1

=

n−1∑k=1

(bn−k−1 − bn−k)Uki + bn−1U0i + µff

ni , 1 ≤ i ≤M − 1, 1 ≤ n ≤ N − 1

(2.6.15)

Un0 = 0, unM = 0, 1 ≤ n ≤ N (2.6.16)

U0i = u0(xi), 0 ≤ i ≤M (2.6.17)

where µ1 =K1τ

γΓ(2− γ)

hα, µ2 =

K2τγΓ(2− γ)

hαand µf = τγΓ(2− γ). As it can be seen, equation

(2.6.15) is very similar to equation (2.6.8), with the only difference being the replacement of the first

order weights in space fractional derivatives by second order ones.

2.6.4 Fourth order finite difference scheme

In this section the difference scheme recently developed by Pang et al. [63] is visited. This scheme

combines the L1 method for the approximation of the temporal derivative with the compact difference

scheme developed by Hao et al. [55] for space fractional derivatives, seen on section 2.5.4. The tech-

nique to reach fourth-order approximations for left and right Riemann-Liouville derivatives was intro-

duced in section 2.2.2.3.

As in section 2.5.4, conveniently let

Dαx = (K1

RLaD

αx +K2

RLxD

αb ), δαx = (K1

WS4δα

x,+ +K2WS4δ

α

x,−) (2.6.18)

where the weighted difference operators WS4δα

x,+ and WS4δα

x,− are defined by (2.2.47) and (2.2.48),

respectively.

39

Considering equation (2.6.1) at point (xi, tn) gives

C0D

γ

t uni = Dα

xuni + fni , 0 ≤ i ≤M, 1 ≤ n ≤ N (2.6.19)

The average difference operator in (2.2.41) is now applied to equation (2.6.19) to give

A(C0Dγ

t uni ) = A(Dα

xuni ) +A(fni ), 1 ≤ i ≤M − 1, 1 ≤ n ≤ N (2.6.20)

Equations (2.2.47) and (2.2.48) allow to write this equation as

A(C0Dγ

t uni ) = δαxu

ni +A(fni ) +O(h4) (2.6.21)

Approximating the Caputo time derivative with the L1 operator as in equation (2.2.7) yields

A(L1Cδγ

t uni ) = δαxu

ni +A(fni ) +O(h4 + τ2−γ) (2.6.22)

Substitution of the L1 operator with (2.2.8) leads to

τ−γ

Γ(2− γ)A

[uni −

n−1∑k=1

(bn−k−1 − bn−k)uki − bn−1u0i

]= δαxu

ni +A(fni ) +O(h4 + τ2−γ) (2.6.23)

Omitting the truncation error, replacing uni by its numerical approximation Uni , and denoting µ =

τγΓ(2− γ) the following finite difference scheme is obtained

A

[Uni −

n−1∑k=1

(bn−k−1 − bn−k)Uki − bn−1U0i

]= µδαxU

ni + µA(fni ), 1 ≤ i ≤M − 1, 1 ≤ n ≤ N − 1

(2.6.24)

Un0 = 0, unM = 0, 1 ≤ n ≤ N (2.6.25)

U0i = u0(xi), 0 ≤ i ≤M (2.6.26)

Pang et al., after studying the stability and convergence of the scheme have concluded that for all

0 < γ < 1 and 1 < α < 2 the scheme is unconditionally stable that ||eni || = ||uni − uni || ≤ C(τ2−γ + h4).

2.6.5 Numerical Examples

The following time-space fractional initial-boundary value problem of type (2.6.1)-(2.6.3) is consid-

ered.

C0D

γt u(x, t) = RL

0Dαxu(x, t) + RL

xDα1u(x, t) + f(x, t), (x, t) ∈ [0, 1]× [0, T ] (2.6.27)

u(x, t = 0) = (x4(1− x)4), x ∈ [0, 1] (2.6.28)

40

u(x = 0, t) = u(x = 1, t) = 0, t ∈ [0, T ] (2.6.29)

with source term

f(x, t) =

(Γ(3 + γ)

Γ(3)t2 +

Γ(2)

Γ(2− γ)

)x4(1− x)4 −

(t2−γ + t+ 2

) [ Γ(9)

Γ(9− α)x8−α

− 4Γ(8)

Γ(8− α)x7−α +

6Γ(7)

Γ(7− α)x6−α − 4Γ(6)

Γ(6− α)x5−α +

Γ(5)

Γ(5− α)x4−α

]−(t2−γ + t+ 2

) [ Γ(9)

Γ(9− α)(1− x)8−α − 4Γ(8)

Γ(8− α)(1− x)7−α +

6Γ(7)

Γ(7− α)(1− x)6−α

− 4Γ(6)

Γ(6− α)(1− x)5−α +

Γ(5)

Γ(5− α)(1− x)4−α

](2.6.30)

The exact solution of the problem is given by u(x, t) = (t2+γ + t+ 2)x4(1− x)4 and it is depicted on

Figure 2.5 for the case γ = 0.5

Figure 2.5: Exact solution of problem (2.6.27)-(2.6.29) with γ = 0.5

To validate and compare the implemented schemes, the L∞h,τ error and its order of convergence,

already defined in (2.4.51) and (2.4.52), were calculated for each of the schemes with the refinement of

the space and time intervals.

Tables 2.6.1 and 2.6.2 list the L∞h,τ error along with the respective order of convergence with the

refinement of the space and time intervals, respectively, for the first order in space scheme. The results

listed on Table 2.6.1 were taken with a constant value τ = 1/8000, guaranteeing a smaller contribution

of the space truncation error when compared with the time contribution. It can be seen that in this case,

the error and its convergence order are insensitive to variations in γ. Although first order convergence

can be observed, the higher value of α seems to require smaller space steps to reach asymptotic

41

convergence than the lower values of α. In Table 2.6.2 the results of the time convergence analysis for

the first order in space scheme are listed. This analysis was made optimizing the space interval with

h ≈ τ2−γ because the attempt to verify first order convergence with a fixed h τ2−γ failed to provide a

satisfactory outcome. The results show that in this manner the (2−γ) order of accuracy is achieved, with

higher values of γ naturally showing larger errors. At the same time the error magnitude is decreasing

with the increase of α. This increase is due to the higher orders of convergence in coarser grids seen

with high α, before asymptotic convergence is achieved with time step refinement.

Tables 2.6.3 and 2.6.4 list the L∞h,τ error along with the respective order of convergence with the re-

finement of the space and time intervals, respectively, for the second order in space scheme. The results

of Table 2.6.3 were taken with a constant τ = 1/80000 and reveal a perfect second order convergence

with space step refinement, while remaining relatively in sensitive to variations in both α and γ.

Table 2.6.3 lists the maximum error and its order of convergence with time step refinement for h =

1/1000. It is seen that for γ = 0.5 and γ = 0.8, the orders of convergence are very close to (2 − γ). On

the other hand, it is seen that for γ = 0.2, the results seem to show a significantly lower convergence

order than what was expected. The error however is still decreasing with the increase of γ.

Tables 2.6.5 and 2.6.6 list the L∞h,τ error along with the respective order of convergence with the

refinement of the space and time intervals, respectively, for the fourth order in space scheme. In Table

2.6.5 the space convergence of the L∞h,τ scheme had to be carried out with t ∈ [0, 0.1] for reasons of

computational cost. Since the scheme is of order O(τ2−γ , h4) the computational cost of keeping the time

contribution to the error lower than the space contribution became too high to carry such an extensive

analysis. It can bee seen that fourth order convergence space is achieved, although a slight increase in

error is seen with increasing α. Furthermore, it is seen that the effect of γ in the error order of magnitude

is very small. Table 2.6.6, was built using a constant value of h = 1/1000 and the results exhibit (2− γ)

convergence order. A slight increase on convergence order is also seen with increasing α, that explains

the decreasing errors observed.

Table 2.6.7 lists the errors and computing times that were verified with the refinement of the time

interval for a fixed h = 1/2000. Here the error for the first order scheme shows to be insensitive to the

refinement of the space interval and order γ. Leading to smaller errors than the first order scheme, the

second and third order schemes exhibit errors of the same order of magnitude. Due to the similarity in

the construction and solution of the schemes the first and second order in time schemes are exhibiting

similar computing times. On the other hand, the fourth order scheme is exhibiting computing times more

than ten times larger than the second order scheme. This huge increase in the computing time for the

high order scheme is here due to the application of the averaging operator of the compact scheme to the

time fractional derivative, greatly increasing the computational cost. These conclusions indicate that the

most efficient choice for time-space fractional diffusion equation is the second order scheme. To unlock

the potential of a high order approximation in space, a high order approximation should also be used for

the time fractional derivative, enabling the use a coarser time grid to get the same error magnitude.

42

Table 2.6.1: L∞h,τ error and the respective order of convergence with space step refinement for the firstorder in space scheme with a constant τ = 1/8000

γ = 0.2 γ = 0.5 γ = 0.8

α 1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

1.2 4 6.9409E-03 - 6.9470E-03 - 6.9741E-03 -8 5.1888E-03 0.420 5.1900E-03 0.421 5.2069E-03 0.42216 3.3637E-03 0.625 3.3625E-03 0.626 3.3699E-03 0.62832 1.9769E-03 0.767 1.9755E-03 0.767 1.9782E-03 0.76864 1.0860E-03 0.864 1.0851E-03 0.864 1.0861E-03 0.865128 5.7173E-04 0.926 5.7121E-04 0.926 5.7157E-04 0.926

1.5 4 1.4609E-03 - 1.4628E-03 - 1.4677E-03 -8 1.4461E-03 0.015 1.4476E-03 0.015 1.4510E-03 0.01712 9.2856E-04 0.639 9.2946E-04 0.639 9.3140E-04 0.64016 5.2280E-04 0.829 5.2329E-04 0.829 5.2433E-04 0.82964 2.7707E-04 0.916 2.7732E-04 0.916 2.7785E-04 0.916128 1.4258E-04 0.958 1.4271E-04 0.958 1.4297E-04 0.959

1.8 4 1.9696E-03 - 1.9704E-03 - 1.9713E-03 -8 3.5892E-04 2.456 3.5904E-04 2.456 3.5914E-04 2.45712 7.3973E-05 2.279 7.4015E-05 2.278 7.4061E-05 2.27816 4.3100E-05 0.779 4.3138E-05 0.779 4.3174E-05 0.77964 3.1069E-05 0.472 3.1091E-05 0.472 3.1104E-05 0.473128 1.7931E-05 0.793 1.7943E-05 0.793 1.7942E-05 0.794

Table 2.6.2: L∞h,τ error and the respective order of convergence with time step refinement for the firstorder in space scheme, h ≈ τ (2−γ).

α = 1.2 α = 1.5 α = 1.8

γ 1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

0.2 4 4.05E-03 - 1.13E-03 - 1.44E-04 -8 1.56E-03 1.374 4.06E-04 1.477 3.87E-05 1.89016 4.99E-04 1.646 1.24E-04 1.713 1.56E-05 1.31632 1.48E-04 1.751 3.62E-05 1.775 4.84E-06 1.68664 4.30E-05 1.786 1.05E-05 1.793 1.42E-06 1.770128 1.24E-05 1.796 3.01E-06 1.798 4.09E-07 1.793

0.5 4 5.12E-03 - 1.42E-03 - 3.53E-04 -8 2.58E-03 0.985 6.98E-04 1.020 3.31E-05 3.41716 1.07E-03 1.269 2.72E-04 1.358 2.86E-05 0.21132 4.05E-04 1.403 9.99E-05 1.447 1.23E-05 1.22064 1.47E-04 1.463 3.58E-05 1.481 4.60E-06 1.413128 5.24E-05 1.487 1.27E-05 1.494 1.66E-06 1.471

0.8 4 5.72E-03 - 1.40E-03 - 1.13E-03 -8 3.97E-03 0.529 1.09E-03 0.362 1.32E-04 3.09516 2.18E-03 0.861 5.77E-04 0.917 3.17E-05 2.05832 1.06E-03 1.044 2.67E-04 1.114 2.52E-05 0.33264 4.89E-04 1.116 1.20E-04 1.153 1.33E-05 0.922128 2.19E-04 1.160 5.30E-05 1.178 6.25E-06 1.091

43

Table 2.6.3: L∞h,τ error and the respective order of convergence with space step refinement for thesecond order in space scheme, h = 1/1000.

γ = 0.2 γ = 0.5 γ = 0.8

α 1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

1.2 4 4.1593E-03 - 4.1602E-03 - 4.1767E-03 -8 1.0548E-03 1.979 1.0554E-03 1.979 1.0584E-03 1.98116 2.7021E-04 1.965 2.7036E-04 1.965 2.7105E-04 1.96532 6.9060E-05 1.968 6.9101E-05 1.968 6.9332E-05 1.96764 1.7509E-05 1.980 1.7522E-05 1.980 1.7645E-05 1.974128 4.4120E-06 1.989 4.4180E-06 1.988 4.5129E-06 1.967

1.5 4 4.7018E-03 - 4.7060E-03 - 4.7122E-03 -8 1.1158E-03 2.075 1.1166E-03 2.075 1.1178E-03 2.07612 2.8216E-04 1.984 2.8235E-04 1.984 2.8267E-04 1.98316 7.1518E-05 1.980 7.1565E-05 1.980 7.1673E-05 1.98064 1.8036E-05 1.987 1.8049E-05 1.987 1.8102E-05 1.985128 4.5310E-06 1.993 4.5352E-06 1.993 4.5742E-06 1.985

1.8 4 3.7413E-03 - 3.7426E-03 - 3.7437E-03 -8 8.8624E-04 2.078 8.8654E-04 2.078 8.8686E-04 2.07812 2.2559E-04 1.974 2.2568E-04 1.974 2.2579E-04 1.97416 5.6921E-05 1.987 5.6943E-05 1.987 5.6984E-05 1.98664 1.4285E-05 1.994 1.4292E-05 1.994 1.4315E-05 1.993128 3.5775E-06 1.998 3.5795E-06 1.997 3.5987E-06 1.992

Table 2.6.4: L∞h,τ error and the respective order of convergence with time step refinement for the secondorder in space scheme.

α = 1.2 α = 1.5 α = 1.8

γ 1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

0.2 4 1.6593E-05 - 6.2447E-06 - 3.1952E-06 -8 5.3092E-06 1.644 2.0004E-06 1.642 1.0261E-06 1.63916 1.6680E-06 1.670 6.3548E-07 1.654 3.2931E-07 1.64032 5.2425E-07 1.670 2.0763E-07 1.614 1.1103E-07 1.56864 1.7156E-07 1.612 7.5866E-08 1.452 4.3841E-08 1.341

0.5 4 9.2840E-05 - 3.5656E-05 - 1.8281E-05 -8 3.5368E-05 1.392 1.3511E-05 1.400 6.9145E-06 1.40312 1.3116E-05 1.431 4.9999E-06 1.434 2.5588E-06 1.43416 4.7930E-06 1.452 1.8305E-06 1.450 9.3936E-07 1.44664 1.7413E-06 1.461 6.7158E-07 1.447 3.4775E-07 1.434

0.8 4 3.1483E-04 - 1.2645E-04 - 6.5451E-05 -8 1.4469E-04 1.122 5.8155E-05 1.121 3.0021E-05 1.12412 6.4925E-05 1.156 2.6065E-05 1.158 1.3436E-05 1.16016 2.8749E-05 1.175 1.1533E-05 1.176 5.9425E-06 1.17764 1.2641E-05 1.185 5.0730E-06 1.185 2.6155E-06 1.184

44

Table 2.6.5: L∞h,τ error and the respective order of convergence with space step refinement for the fourthorder in space scheme, for t ∈ [0, 0.1] and τ = 1/100000.

γ = 0.2 γ = 0.5 γ = 0.8

α 1/h L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

1.2 4 1.6450E-04 - 1.1525E-04 - 4.4287E-05 -8 2.0260E-05 3.021 1.2400E-05 3.216 4.4206E-06 3.32516 1.8460E-06 3.456 1.2015E-06 3.367 4.8230E-07 3.19632 1.3077E-07 3.819 8.7423E-08 3.781 5.2136E-08 3.21064 8.6077E-09 3.925 6.9323E-09 3.657 6.0357E-09 3.111

1.5 4 3.6896E-04 - 2.9775E-04 - 2.0223E-04 -8 4.2480E-05 3.119 3.5738E-05 3.059 2.2893E-05 3.14312 3.5837E-06 3.567 3.0039E-06 3.573 1.9867E-06 3.52616 2.4483E-07 3.872 2.0874E-07 3.847 1.4237E-07 3.80364 1.5664E-08 3.966 1.3370E-08 3.965 9.2127E-09 3.950

1.8 4 5.6067E-04 - 4.9140E-04 - 3.6127E-04 -8 5.9297E-05 3.241 5.4091E-05 3.183 4.4163E-05 3.03212 4.6038E-06 3.687 4.1167E-06 3.716 3.2129E-06 3.78116 3.0638E-07 3.909 2.7752E-07 3.891 2.2345E-07 3.84664 1.9423E-08 3.979 1.7562E-08 3.982 1.4168E-08 3.979

Table 2.6.6: L∞h,τ error and the respective order of convergence with time step refinement for the fourthorder in space scheme, h = 1/1000.

α = 1.2 α = 1.5 α = 1.8

γ 1/τ L∞h,τ EOC L∞h,τ EOC L∞h,τ EOC

0.2 4 1.6575E-05 - 6.2261E-06 - 3.1805E-06 -8 5.2911E-06 1.647 1.9817E-06 1.652 1.0114E-06 1.65316 1.6498E-06 1.681 6.1683E-07 1.684 3.1463E-07 1.68532 5.0604E-07 1.705 1.8898E-07 1.707 9.6357E-08 1.70764 1.5335E-07 1.722 5.7218E-08 1.724 2.9166E-08 1.724

0.5 4 9.2823E-05 - 3.5637E-05 - 1.8266E-05 -8 3.5350E-05 1.393 1.3493E-05 1.401 6.8998E-06 1.40512 1.3098E-05 1.432 4.9812E-06 1.438 2.5441E-06 1.43916 4.7748E-06 1.456 1.8118E-06 1.459 9.2469E-07 1.46064 1.7231E-06 1.470 6.5292E-07 1.472 3.3307E-07 1.473

0.8 4 3.1481E-04 - 1.2643E-04 - 6.5437E-05 -8 1.4467E-04 1.122 5.8137E-05 1.121 3.0006E-05 1.12512 6.4907E-05 1.156 2.6047E-05 1.158 1.3421E-05 1.16116 2.8731E-05 1.176 1.1514E-05 1.178 5.9278E-06 1.17964 1.2623E-05 1.187 5.0544E-06 1.188 2.6008E-06 1.189

45

Table 2.6.7: Computing times with refinement of the time interval with the first, second and fourth orderin space schemes and h = 1/2000.

1st Order 2nd Order 4th Orderγ 1/τ L∞h,τ TOC(s) L∞h,τ TOC(s) L∞h,τ TOC(s)

0.2 16 8.753E-06 1.345E+00 6.355E-07 1.356E+00 6.168E-07 9.160E+0132 9.181E-06 2.414E+00 2.076E-07 2.453E+00 1.890E-07 9.227E+0164 9.313E-06 4.746E+00 7.587E-08 4.726E+00 5.722E-08 9.539E+01

128 9.353E-06 9.162E+00 3.582E-08 9.518E+00 1.717E-08 1.001E+02

0.5 16 4.781E-06 1.275E+00 5.000E-06 1.346E+00 4.981E-06 9.090E+0132 7.568E-06 2.376E+00 1.830E-06 2.446E+00 1.812E-06 9.207E+0164 8.726E-06 4.594E+00 6.716E-07 4.799E+00 6.529E-07 9.448E+01

128 9.145E-06 8.949E+00 2.525E-07 9.451E+00 2.339E-07 9.943E+01

0.8 16 1.768E-05 1.296E+00 2.607E-05 1.302E+00 2.605E-05 9.223E+0132 8.106E-06 2.289E+00 1.153E-05 2.532E+00 1.151E-05 9.059E+0164 4.801E-06 4.544E+00 5.073E-06 4.760E+00 5.054E-06 9.079E+01

128 7.190E-06 8.993E+00 2.229E-06 9.397E+00 2.210E-06 9.557E+01

46

Chapter 3

Influence of variable-order operators

in the behaviour of sub-diffusive

systems

In this chapter an investigation is made into the effects of variable order differentiation in the be-

haviour of sub-diffusive systems. Section 3.1 introduces a scheme able to solve variable order time-

fractional sub-diffusion equations. The initial-boundary value problem is stated in section 3.1.1. In

section 3.1.2 a difference scheme able to solve time fractional diffusion equations with variable coeffi-

cients dependent on time and space [98] is implemented and provided to the reader in matrix form. The

convergence of the scheme is then validated with a numerical example against the analytic solution in

section 3.1.3. In section 3.2 the effects of order dependence on time, space and the solution itself will

be analysed through numerical examples. Departure is made in section 3.2.1 from the comparison of

the standard diffusion equation with constant order fractional diffusion which is then taken as reference

for the analysis of the behaviour of anomalously diffusive systems with variable order. Sections 3.2.2,

3.2.3 and 3.2.4, study of the behaviour of a sub-diffusive with variable orders dependent of time, space

and the system solution, respectively.

3.1 Numerical solution of variable order time fractional diffusion

equations

3.1.1 Problem Statement

The following variable order time fractional diffusion equation will be considered [98]

0Dγ(x,t)t u(x, t) = K

∂2u(x, t)

∂x2+ f(x, t), x ∈ [0, L]t ∈ [0, T ] (3.1.1)

47

with the following initial and boundary conditions

u(x, 0) = ξ(x), x ∈ [0, L] (3.1.2)

u(0, t) = ψ(t); u(L, t) = φ(t), t ∈ [0, T ] (3.1.3)

In equation (3.1.1), K > 0 is a generalized diffusion coefficient, the variable u(x, t) is a physical

quantity of interest such as the concentration and 0Dγ(x,t)t denotes the Coimbra [64] variable order

fractional derivative

0Dγ(x,t)t u(x, t) =

1

Γ(1− γ(x, t))

∫ t

0

(t− σ)−γ(x,t)∂u(x, σ)

∂σdσ +

(u(x, 0+)− u(x, 0−))t−γ(x,t)

Γ(1− γ(x, t))(3.1.4)

where the order γ(x, t) ∈ [0, 1] may be a function of time, space or both.

If u(x, 0+) = u(x, 0−) then the Coimbra definition is equivalent to the Caputo type definition [97]

C0 D

γ(x,t)t =

1

Γ(1− γ(x, t))

∫ t

0

(t− σ)−γ(x,t)∂u(x, σ)

∂σdσ (3.1.5)

However, it is pointed out that there are more definitions for the variable order fractional differential

operator, namely definitions that account for the memory of the history of the derivative [108, 109].

3.1.2 Numerical Scheme

In this section, the main steps regarding the scheme developed by Shen et al. in [98] will be stated.

The matrix form of the scheme is then provided to expedite future implementations.

Let the solution u(x, t) of problem (3.1.1)-(3.1.3) be an adequately smooth function. In addition let,

the domain be represented by an equally spaced mesh with M + 1 points in the spacial domain and

N + 1 points in the temporal domain. That is, xi = ih, i = 0, 1, ...,M and tn = nτ, n = 0, 1, ..., N , where

the spacial and temporal grid sizes are h = L/M and τ = T/N , respectively.

The second order space derivative in equation (3.1.1) will be approximated by following central finite

difference formula, denoting by δ2x the second order difference operator

∂2u(xi, tn)

∂x2= δ2xu(xi, tn) +O(h)2,

δ2xu(xi, tn) =u(xi−1, tn)− 2u(xi, tn) + u(xi+1, tn)

h2

(3.1.6)

Following the scheme given in [98], the variable order time fractional derivative can be approximated

by

0Dγ(xi,tn)t u(xi, tn) =

τ−γ(xi,tn)

Γ(2− γ(xi, tn))

n−1∑j=0

di,j,n[u(xi, tj+1)− u(xi, tj)] + rni (3.1.7)

where

di,j,n = (n− j)1−γ(xi,tn) − (n− j − 1)1−γ(xi,tn), j = 0, 1, ..., n− 1 (3.1.8)

48

and rni is the truncation error

rni =C · τ2−γ(xi,tn)

Γ(2− γ(xi, tn))n1−γ(xi,tn) ≤ C · T 1−γ(xi,tn)

Γ(2− γ(xi, tn))· τ ≤ C · τ (3.1.9)

Substituting the derivatives in equation (3.1.1) by their respective numerical approximations and de-

noting by Uni the numerical approximation of u(xi, tn), f(xi, tn) by fni and γ(xi, tn) by γni results in

τ−γni

Γ(2− γni )

n−1∑j=0

di,j,n

(U j+1i − U ji

)= K

Uni−1 − 2Uni + Uni+1

h2+ fni (3.1.10)

Leading to

Uni −KΓ(2− γni )τγ

ni

h2(Uni−1 − 2Uni + Uni+1) =

Un−1i −n−2∑j=0

di,j,n(U j+1i − U ji ) + Γ(2− γni )τγ

ni fni

i = 1, 2, ...,M − 1, n = 1, 2, .., N

(3.1.11)

Discretization of the initial and boundary conditions completes the scheme

U0i = ξ(xi), i = 0, 1, ...,M (3.1.12)

Un0 = ψ(tn), UnM = φ(tn), n = 0, 1...N (3.1.13)

Equation (3.1.11) can be written in matrix for as

(E − K

h2GnS

)Un = Un−1 −

n−2∑j=0

Dnj

(U j+1 − U j

)+Gnfn +

K

h2XGnbU

nb (3.1.14)

where Un = [Un1 , Un2 , ..., U

nM−1]T , Unb = [Un0 , U

nM ]T , fn = [fn1 , f

n2 , ..., f

nM−1]T . S, Gn and Dn

j are diagonal

matrices of size (M − 1)× (M − 1)

S =

−2 1 0 0 · · · 0 0

1 −2 1 0 · · · 0 0

0 1 −2 1 · · · 0 0...

......

.... . .

......

0 0 0 0 · · · −2 1

0 0 0 0 · · · 1 −2

(M−1)×(M1)

(3.1.15)

49

Gn =

Γ(2− γn1 )τγ

n1 0 · · · 0

0 Γ(2− γn2 )τγn2 · · · 0

......

. . ....

0 0 · · · Γ(2− γnM−1)τγnM−1

(M−1)×(M1)

(3.1.16)

Dnj =

d1,j,n 0 · · · 0

0 d2,j,n · · · 0...

.... . .

...

0 0 · · · dM−1,j,n

(M−1)×(M1)

(3.1.17)

Furthermore, the boundary terms can be included through the following matrices

X =

1 0...

...

0 1

(M−1)×2

(3.1.18)

Gnb =

Γ(2− γn1 )τγn1 0

0 Γ(2− γnM−1)τγnM−1

2×2

(3.1.19)

Unconditional stability of the scheme (3.1.11)-(3.1.13) was proved in [98] through Fourier analysis.

The scheme also proved to be both convergent with order O(τ + h2) and uniquely solvable.

3.1.3 Numerical Example

In order to carry out a numerical test of the convergence order of scheme (3.1.11)-(3.1.13), consider

the following initial and boundary value problem of the type (3.1.1)-(3.1.3) [98]

0Dγ(x,t)t u(x, t) = K

∂2u(x, t)

∂x2+ f(x, t), x ∈ [0, L], t ∈ [0, T ] (3.1.20)

with the following initial and boundary conditions

u(x, 0) = 10x2(1− x), x ∈ [0, 1] (3.1.21)

u(0, t) = u(1, t) = 0, t ∈ [0, 1] (3.1.22)

with order γ(x, t) =2 + sin(xt)

4and source term f(x, t) given by

f(x, t) = 20x2(1− x)

[t2−γ(x,t)

Γ(3− γ(x, t))+

t1−γ(x,t)

Γ(2− γ(x, t))

]− 20(t+ 1)2(1− 3x) (3.1.23)

50

The exact solution, depicted in Figure 3.1, is

u(x, t) = 10x2(1− x)(t+ 1)2 (3.1.24)

Figure 3.1: Exact solution of problem (3.1.20)-(3.1.22).

Let, u(xi, tn) and Uni be the exact and numeric solutions, respectively. Then the L2h,τ and L∞h,τ errors

are defined as

L2h,τ =

√√√√ 1

P

M∑i=0

N∑n=0

[u(xi, tn)− Uni ]2 (3.1.25)

L∞h,τ = max |u(xi, tn)− Uni | (3.1.26)

where P is the total number of mesh points domain.

Furthermore, let the order of convergence of an error e, EOC, be given by the following expression

EOC = log2

eτeτ/2

(3.1.27)

In figure 3.2 the absolute error in each point of the domain can be observed. Although the plot

shown corresponds to the situation where h = 1/250 and τ = 1/500, the morphology remained constant

throughout all numerical test with an increasing in error with time and towards the centre of the space

interval.

51

Figure 3.2: Absolute error in the solution of (3.1.20)-(3.1.22), h = 1/250 and τ = 1/500.

To numerically verify the order of convergence of the scheme, Tables 3.1.1 and 3.1.2 were produced,

where the L2 and L∞ errors can be found along with the respective orders of convergence. The L2 error

presents an average of the error in the entirety of the domain, while the L∞ error depicts the error in the

most critical situation.

Table 3.1.1: Error behaviour with decreasing temporal gridsize, h = 1/500.

1/τ L2 EOC(L2) L∞ EOC(L∞)

8 2.0911× 10−3 - 4.1270× 10−3 -

16 8.1127× 10−4 1.37 1.6213× 10−3 1.35

32 3.1097× 10−4 1.38 6.3430× 10−4 1.35

36 1.1838× 10−4 1.39 2.4759× 10−4 1.36

128 4.4901× 10−5 1.40 9.6527× 10−5 1.36

256 1.7002× 10−5 1.40 3.7614× 10−5 1.36

The error behaviour with the reduction of the temporal grid size is shown on Table 3.1.1. The nu-

merical tests were carried out with h = 1/500, a value that guarantees a negligible contribution of the

spacial discretization to the error because the scheme has second order in space. The L2 error order

of convergence is just slightly superior to the L∞ with both errors having the same order of magnitude.

This similarity suggests a homogeneous error behaviour, a conclusion supported by additional numerical

tests at different time steps and positions where the variable fractional order did not appear to change

the rate of convergence. The EOC exhibits an order in between 1.35 and 1.40 , nonetheless, consider-

ably above first order the theoretical prediction. In fact, the observed EOC is much closer to 2 − γavg,

52

with γavg being the average value of the order γ across the computational domain, thus resembling the

order observed in the L1 scheme for contant order fractional diffusion equations [63].

Table 3.1.2: Error behaviour with decreasing temporal gridsize, t = h2.

1/h L2 EOC(L2) L∞ EOC(L∞)

4 7.5920× 10−4 - 1.6252× 10−3 -

8 1.1306× 10−4 2.75 2.4936× 10−4 2.70

16 1.6561× 10−5 2.77 3.7585× 10−5 2.73

32 2.4031× 10−6 2.78 5.7123× 10−6 2.72

In Table 3.1.2 lists the L2 and L∞ errors and their rate of convergence with spacial grid size reduction.

As in the previous case, the L2 and L∞ errors exhibit the same magnitude. The order of convergence

ranges between 2.70 and 2.78, also above the second order theoretical calculations.

Attending to the results of this numerical simulation, a successful implementation of a numerical

scheme able to solve variable order fractional diffusion equations was made. Furthermore, the scheme

is able to deal with a fractional order function dependent of time and space. This scheme can be used

to analyse the impact that different dependences of the fractional order of differentiation have in the

behaviour of the solution of the diffusion equation.

3.2 Influence of variable order differential operators in anomalous

diffusion

Let’s consider the one-dimensional diffusion equation in variable fractional order with the initial and

boundary value problem of the type (3.1.1)-(3.1.3). The initial distribution is given and so are the bound-

ary conditions, maintained at a zero constant value.

0Dγ(x,t)t u(x, t) =

∂2u(x, t)

∂x2, x ∈ [0, 10], t ∈ [0, T ] (3.2.1)

with the following initial and boundary conditions

u(x, 0) = sin2(xπ

10

), x ∈ [0, 10] (3.2.2)

u(0, t) = u(10, t) = 0, t ∈ [0, T ] (3.2.3)

where u may represent for instance the temperature or concentration and the diffusion coefficient was

set to K = 1. An analysis of the effect of the variable order will only require the adjustment of the order

γ for each of the intended cases. The initial and boundary conditions will remain the same for the time,

space and solution dependent cases. The following results were taken with a time and space steps

equal to τ = 1/25 and h = 1/25, respectively.

53

3.2.1 Constant fractional order

In order to investigate the effects of different order dependences of time derivatives in anomalous

diffusion modelling it is important to understand the behaviour of the solution under different constant

fractional orders and how this fractional behaviour differs from standard diffusion (γ = 1). The behaviour

of the constant order solution may then be taken as reference for the analysis of variable order modelling.

For the numerical solution of constant order fractional diffusion equations, the previously introduced

scheme can be used with a constant order γ and equation (3.2.1) will become

0Dγt u(x, t) = K

∂2u(x, t)

∂x2, x ∈ [0, 10], t ∈ [0, T ] (3.2.4)

Figure 3.3: Solution of the standard diffusion equation.

The scheme proved able to also solve the standard diffusion with γ = 1, the corresponding plot is

shown on Figure 3.3 with first order accuracy in time. Figure 3.4 shows the time evolution of the solution

of (3.2.4) for x = 5 with different fractional orders, along with the solution of the standard diffusion

equation (γ = 1) . As the order of the time fractional derivative increases so does the diffusion rate

and the solution approaches the standard diffusion equation. Figure 3.6 portrays the solution in space

for t = 5 for the same orders, where the results with the different fractional orders behave as expected,

again it is seen that with each increase in order the solution get closer to the standard diffusion equation.

However, one intriguing feature does stand out, in the initial time steps a decrease in the order of the

derivative corresponds to an increase in the diffusion rate, this behaviour can also be observed in Figure

54

3.5 which depicts the solution at t = 0.2. This behaviour is counter-intuitive when one is looking for

sub-diffusion, and some attempts to modify this initial behaviour were made. The inclusion of a constant

value history before the initial time step was considered and so was the reduction of to a zero initial value

problem, with persistence of this initial super-diffusive behaviour. After unfruitful checks to the initial

conditions and to the well-posedness of the problem, a dimensional analysis was made. From the last,

some conclusions are worth remarking. If a physical problem is considered, for instance the evolution

of the concentration given an initial distribution and boundary conditions, it is known that [u] = ML−3,

[∂u/∂t] = ML−3T−1 , [∂2u/∂x2] = ML−5 and thus [K] = L2T−1. Now, if the intent is to model diffusion

in a porous media, through the substitution of the integer order time derivative by the fractional one

and with the same diffusivity coefficient as would be used in the standard diffusion equation then the

dimensions of the right and left sides of equation (3.2.4) do not match. Indeed, if [Dγ0,tu] = ML−3T−γ

then the dimensions of the diffusivity coefficient should be [K] = L2T−γ . To account for this change, a

Riemann-Liouville integration [32] of order 1 − γ was made to the diffusion coefficient so that it would

have the desired dimensions, as in the following equation

0D−(1−γ)t K = Kγ(t) =

1

Γ(1− γ)

∫ t

0

(t− σ)−γKdσ =Kt1−γ

(1− γ)Γ(1− γ)(3.2.5)

Hence, the fractional time fractional derivative may require a temporal adjustment of the diffusivity

coefficient. As the order of the fractional derivative tends to the unity, the dependence on time disappears

and the diffusion coefficient tends to one, retrieving standard diffusion. On the other hand, as the order

of the fractional derivative tends to zero, Kγ approaches Kγ = Kt. In figure 3.4, the dashed lines

correspond to the computation of equation (3.2.4) after substitution of the standard diffusion coefficient

by the time varying coefficient. It can be seen that the super-diffusive behaviour disappears and that

as the order of the fractional derivative increases from 0 to 1, the diffusion rate increases towards the

standard diffusion solution as expected. It is noted that the overall solution decay with time is higher

than with a constant diffusion coefficient when comparing derivatives of the same order because the

time dependent diffusion coefficient increases with time. Despite this increase , it was verified that the

solution remains higher than the standard diffusion solution.

3.2.2 Time dependent fractional order

To evaluate the impact of a time varying order, equation (3.2.1) will be solved with a time only depen-

dent order becoming

0Dγ(t)t u(x, t) = K

∂2u(x, t)

∂x2, x ∈ [0, 10]t ∈ [0, T ] (3.2.6)

To analyse the behaviour of the solution in this case, the following three order functions will be

considered

γ1(t) = 0.5 + 0.25

(t

T

)(3.2.7)

γ2(t) = 0.75− 0.25

(t

T

)(3.2.8)

55

0 1 2 3 4 5 6 7 8 9 10

0.4

0.5

0.6

0.7

0.8

0.9

1

t

u(5,

t)

γ=1γ=0.25γ=0.5γ=0.75γ=0.25 with K

γ(t)

γ=0.5 with Kγ(t)

γ=0.75 with Kγ(t)

Figure 3.4: Solution versus time plot at x = 5 with different constant fractional orders.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u(x,

0.2)

γ=1γ=0.25γ=0.5γ=0.75u(x,t=0)

Figure 3.5: Solution at t = 0.2 with different constant fractional orders.

56

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

u(x,

5)

γ=1γ=0.25γ=0.5γ=0.75γ=0.25 with K

γ(t)

γ=0.5 with Kγ(t)

γ=0.75 with Kγ(t)

Figure 3.6: Solution at t = 5 with different constant fractional orders.

57

γ3(t) = 0.5 + 0.25 sin

(πt

T

)(3.2.9)

where the maximum time considered is T = 10.

0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1

t

u(5,

t)

γ=0.5(t)γ=0.75γ1(t)

γ2(t)

γ3(t)

Figure 3.7: Time evolution in x = 5 with different time dependent fractional orders.

Figure 3.7 shows the evolution of the solution at x = 5. Focusing on γ1(t), the plot clearly demon-

strates a diffusion rate that follows the constant order solution with γ = 0.5 at initial time steps, again

demonstrating super-diffusion. Having chosen a variable order that increases linearly from 0.5 at t = 0

to 0.75, the diffusion rate gradually changes to the rate of the constant order γ = 0.75, at t = 10. With

different order functions it is possible to model decelerating diffusion processes or an even more com-

plex situation containing both behaviours. In the case of a decreasing linear order γ2(t), the behaviour

is analogous. The solution starts by behaving similarly to initial order, gradually slowing down with time.

It appears that, for the case of a linear order variation with time, the solution is bounded by the constant

order solutions equal to the maximum and minimum of the variable order solution. Attending now to

γ3(t), which corresponds to a variable order that increases until t = 5 decreasing afterwards up to t = 10

it is possible to conclude that the rate at which the derivative changes need careful consideration in the

modelling process. In Figure 3.7 it is possible to see that a sudden decrease of the derivative order may

produce non-physical results, with the solution increasing with time at its maximum value in space near

t = 10. This example, however simple, demonstrates that variable order fractional calculus is a viable

58

tool to model situations where the diffusive behaviour changes with time.

3.2.3 Space dependent fractional order

To evaluate the impact of a space varying order, equation (3.2.1) will become

0Dγ(x)t u(x, t) = K

∂2u(x, t)

∂x2, x ∈ [0, 10]t ∈ [0, T ] (3.2.10)

To better compare the effects of space varying order, the following three space dependent orders will

be modelled, as represented in Figure 3.8

γ1(x) = 0.25 +∣∣∣0.75

( xL− 0.5

)∣∣∣ (3.2.11)

γ2(t) =

0.25 + 0.75

x

L, 0 < x < L/2

0.25 + 0.75(

1− x

L

), L/2 ≤ x < L

(3.2.12)

γ3(x) = 0.25 + 0.25 cos

(4πx

10

)+∣∣∣0.75

( xL− 0.5

)∣∣∣ (3.2.13)

0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

γ(x)

γ1(x)

γ2(x)

γ3(x)

Figure 3.8: The three space dependent fractional orders considered.

Figure 3.9 shows the evolution of the solution at x = 5, the point where the highest value of the

solution occurs. It can be seen that super-diffusion also occurs on the initial time steps for this case.

Although the evolution with time tends to follow a constant order closer to the local value of the space

59

0 2 4 6 8 100.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

t

u(5,

t)

γ=0.25γ=0.65γ1(x)

γ2(x)

γ3(x)

Figure 3.9: Time evolution at x = 5 modelled with different space dependent fractional orders.

60

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

u(x,

10)

γ=0.25γ=0.65γ1(x)

γ2(x)

γ3(x)

Figure 3.10: Solution at t = 10 modelled with different space dependent fractional orders.

61

varying order, the effects of the order on the remaining domain are clearly noticeable. With the local

fractional order assuming a value γ1(5) = 0.25, the increase in order towards the boundaries of the

interval causes a higher effective diffusion rate at x = 5. Conversely, the order γ2(x) has a maximum

at x = 5, decreasing linearly towards the boundaries. Consequently, at x = 5, the effective diffusion is

rate lower than what would be observed if the order was constant, with the value corresponding to the

local value at x = 5. Also in Figure 3.9, one can see that at x = 5, the behaviour of an oscillating space

variable order can be seen with the solution for γ3(x). In this case, because of the oscillating effect, the

effective order at x = 5 is very close to the local fractional order γ3(5) = 0.5.

Figure 3.10 depicts the solution at t = 10. With this figure, the way the space varying order affects

the solution becomes even clearer. One can notice for instance that for x . 2 the solution with order

γ1(x) is higher than with a constant fractional order γ = 0.25, even if the solution tends to follow more

closely this order, corresponding to the local order γ1(x) at x = 5. The solution with γ3(x) may also be

found in Figure 3.10, allowing further conclusions into the behaviour with an oscillating variable order in

space. Near the boundaries the variable order γ3(x) assumes higher values than the order γ1(x), giving

higher solutions at the boundaries in respect to γ1(x). In the center of the interval γ3(x) is higher than

γ1(x) leading to a lower solution with a more rounded shape, in accordance with the behaviour of the

derivative of order γ3(x) in the center of the space interval.

3.2.4 Solution dependent fractional order

To evaluate the impact of a space varying order, equation 3.2.1 will become

0Dγ[u(x,t)]t u(x, t) = K

∂2u(x, t)

∂x2, x ∈ [0, 10]t ∈ [0, T ] (3.2.14)

The order γ[u(x, t)] was assumed to follow the following function, represented in Figure 3.11.

γ[u(x, t)] = 0.5 + 0.3u

Umax(3.2.15)

The value Umax is taken to be Umax = 1, the highest value given by the initial condition.

An iterative procedure was carried out to calculate the solution with a solution dependent time deriva-

tive. As an initial guess, the variable order was calculated with the results from the initial time step. The

final solution can then be used to calculate the derivative order, solving equation (3.2.14) a second time.

Following the same procedure more iterations can be made until the solution converges. After three

iterations, the maximum difference to the previous iteration had order 10−6 the iterative procedure was

terminated.

A solution dependent fractional order poses an even more complex problem, mixing the effects of

a space and time varying fractional orders. Figure 3.12, shows the solution at x = 5 which appears to

begin by following the constant fractional order γ = 0.8 but then, as the solution decreases, the diffusion

process slows down with the solution reaching t = 10 at approximately the same diffusion rate as the

constant order γ = 0.6. The combined effect of a space varying order can also seen on Figure 3.13. At

62

0

5

10

0

5

100.5

0.6

0.7

0.8

0.9

tx

γ[u(

x,t)

]

Figure 3.11: Solution dependent fractional order.

0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1

t

u(5,

t)

γ=0.6γ=0.8γ[u(x,t)]

Figure 3.12: Time evolution in x = 5 with a solution dependent variable order model.

63

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

u(x,

10)

γ=0.6γ=0.8γ[u(x,t)]

Figure 3.13: Solution t = 10 with a solution dependent variable order model.

x = 0.5 the solution assumes a value lower than the the local fractional order because of decay with time

of the solution, but higher that it would be if the solution did not vary with space, because the fractional

order decreases towards the boundaries, the same effect observed in space varying fractional order.

64

Chapter 4

Conclusions

This work presents intends to contribute in the building of a road between fractional calculus and its

engineering applications. A total of nine finite difference schemes are implemented and compared within

three different types of fractional diffusion equations, providing the tools and knowledge to properly apply

finite differences in the solution of fractional partial differential equations. An intuitive grasp on how the

order of the time fractional derivative impacts the solution of a sub-diffusive system was also achieved,

both for the constant and variable order cases, greatly increasing the odds of a successful application.

The result of this effort resulted in two papers [110, 111], currently under submission. The mastery of

a numerical solution method and the understanding of the effects of constant and variable order have

gathered the necessary and sufficient conditions to proceed with a real world application.

In chapter 2 the solutions time, space and time-space fractional diffusion equations were calculated

each with three different finite difference schemes. Numerical examples have validated the implemen-

tations and allowed the comparison of the different schemes in terms of accuracy and computing time.

Looking at time fractional diffusion equations, all schemes have shown to follow the theoretical pre-

dictions of their order of convergence. It was seen that weighted and shifted methods increase greatly

increase computing times by considering two time levels in the time fractional derivative and that high or-

der schemes for the solution of the time fractional diffusion equation are the best choice and the ability of

reaching the same order of magnitude with less time steps is an enormous advantage, with significant

reduction of computing times. The three schemes developed for space fractional diffusion equations

have also shown to follow the theoretical predictions regarding their accuracy. Since the matrices used

to compute the left and right space fractional derivatives are full, the first, second and fourth schemes

have shown little difference in computing times. However, higher order approximations for time fractional

derivatives come with increased smoothness requirements and, as it was seen in numerical examples,

these requirements change considerably the accuracy of these schemes. Despite these requirements,

the fourth order scheme has shown the smallest errors and since the computing time is almost the same

as the lower order schemes and therefore is considered the best for the solution of these equations.

Regarding time-space fractional diffusion equations, three different different schemes in increas-

ing spatial order of accuracy were implemented, using the same approximation for the time fractional

65

derivative. The numerical examples, analysing time and space refinement with different fractional orders

in time and space, have proved that the theoretical predictions for the order of accuracy were correct.

However the error for the first order scheme shows to be insensitive to the refinement of the space in-

terval and order γ, with a fixed time step. This situation was overcome by using h = τ2−γ . Due to the

similarities in the construction and solution of the schemes the first and second order in time schemes

show similar computing times for the same grid. On the other hand, the fourth order scheme shows

computing times more than ten times larger than the previous. This huge increase in the computing time

for the high order scheme is caused by the application of the averaging operator of the compact scheme

to the time fractional derivative, greatly increasing the computational cost. The results indicate that the

most efficient choice for time-space fractional diffusion equation is the second order scheme. To unlock

all the potential of a high order scheme, a high order approximation should also be used for the time

fractional derivative, enabling the use a coarser time grid for the same error magnitude.

In chapter 3, a scheme able to solve time and space dependent variable order fractional diffusion

equations was implemented and tested. The matrix forms of the scheme were also constructed, allowing

for immediate implementation. With the help of this scheme, the effect of the order dependence on time,

space and the solution itself on the modelling of time fractional diffusion equations was studied.

The examples of variable order time fractional diffusion equations helped in achieving better under-

standing of how the order of the derivative affects the behaviour of a physical system that are relevant

for modelling real world problems. A comparison of the constant fractional order case with standard

diffusion provided an intuitive grasp on the effect a fractional order effect. A clear evolution towards

the standard diffusion equation was seen with the increase of the time fractional derivative for orders

0 ≤ γ ≤ 1. A super-diffusive behaviour was noticed at initial times and dimensional analysis led to a

time fractional diffusion coefficient based for the modelling based on the standard diffusion coefficient,

eliminating this behaviour. A linear time evolution of the fractional order led to solutions bounded by the

constant order solutions with the initial and final values of time variable order. It also seen that care has

rapid changed in the time variable order may lead to non-physical solutions. For a variable order that is

function of space it was seen that the solution at each point in space follows an effective order that is

lower than its local value that depends of the order in the remaining of space. For variable orders that

are function of the solution, the combined effects of space and time dependent orders were observed.

All the developed algorithms were coded and the programs developed from scratch by the author. In

addition the numerical methods are presented in matrix form that may help others to programme these

schemes more quickly.

There is plenty future work to be done in the realm of fractional calculus. Schemes regarding other

types of equation can be solved and the problem can be extended to the two-dimensional case. A higher

order scheme in time and space should also be developed for the time-space fractional diffusion equa-

tion. Regarding variable order diffusion equations, a higher order scheme could also be implemented

and the analysis of the order effects could be extended to the space derivative or to the fractional

diffusion-wave equation. As the purpose of this work is also paving the way to future engineering ap-

plications one could now start by a simple study of diffusion, for instance in porous media, using the

66

results obtained with a commercial software to do a parametric study of the order of the derivative in the

diffusion equation.

67

68

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80

Appendix A

Schemes for fractional diffusion

equations in matrix form

A.1 Time Fractional Diffusion Equations

A.1.1 First Order Weighted Average Scheme

Equation (2.4.9) can be given in matrix form by

(E − µ(1− θ)S)un+1 = un + µ

n∑k=0

[θω(1−γ)k + (1− θ)ω(1−γ)

k ]Sun−k

+ µθ

n∑k=0

ω(1−γ)k Xun−kb + µ(1− θ)

n+1∑k=0

ω(1−γ)k Xun+1−k

b + τθfn + τ(1− θ)fn+1 (A.1.1)

where µ =Kγτ

γ

h2, ω(1−γ)

k is given by equation (2.2.3), un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u

nM ]T ,

fn = [fni , fn2 , · · · , fnM−1]T , E = IM−1 and

S =

−2 1 0 · · · 0 0

1 −2 1 · · · 0 0

0 1 −2 · · · 0 0...

......

. . ....

...

0 0 0 · · · −2 1

0 0 0 · · · 1 −2

(M−1)×(M−1)

X =

1 0

0 0...

...

0 0

0 1

(M−1)×2

A.1.2 Second Order Finite Difference Scheme

Equation (2.4.25) can be written in matrix form as

A.1

[E − rυ(γ)0 S]un+1 = un + rυ(γ)0 Xun+1

b

+ r

n−1∑k=0

(ω(γ)k − ω(γ)

k−1 + υ(γ)k+1 − υ

(γ)k )[Sun−k +Xun−kb ] + r(ω(γ)

n − ω(γ)n−1[Su0 −Xu0b ] +

τ

2(fn + fn+1]

(A.1.2)

with r , ω(γ)k and υ

(γ)k defined as in equations (2.4.20), (2.4.21) and (2.4.22), respectively. Additionally

un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u

nM ]T , fn = [fni , f

n2 , · · · , fnM−1]T , E = IM−1 and

S =

−2 1 0 · · · 0 0

1 −2 1 · · · 0 0

0 1 −2 · · · 0 0...

......

. . ....

...

0 0 0 · · · −2 1

0 0 0 · · · 1 −2

(M−1)×(M−1)

X =

1 0

0 0...

...

0 0

0 1

(M−1)×2

A.1.3 Third Order Finite Difference Scheme

Equation (2.4.41) can be written in matrix form as

(E +h2

12S)u1 = −h

2

12Xu1b =

2Kγτγ

Γ(γ + 3)(Su1 +Xu1b) + (E +

h2

12S)(F 1 +XF 1

b) (A.1.3)

and equation (2.4.40) as

gγ0 (E +h2

12S)un = −

n∑k=1

gγk (E +h2

12S)un−k − (

h2

12)

n∑k=0

gγkXun−kb +

Kγτγ

h2(Sun + unb ) + τγ(E +

h2

12S)fn +

h2τγ

12Xfnb (A.1.4)

where gγk is defined as in equation (2.2.17), un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u

nM ]T , fn = [fni , f

n2 , · · · , fnM−1]T ,

fnb

= [fn0 , fnM ]T ,Fn = [Fni , F

n2 , · · · , FnM−1]T ,Fnb = [Fn0 , F

nM ]T , E = IM−1 and

S =

−2 1 0 · · · 0 0

1 −2 1 · · · 0 0

0 1 −2 · · · 0 0...

......

. . ....

...

0 0 0 · · · −2 1

0 0 0 · · · 1 −2

(M−1)×(M−1)

X =

1 0

0 0...

...

0 0

0 1

(M−1)×2

A.2

A.2 Space Fractional Diffusion Equations

A.2.1 First Order Finite Difference Scheme

Equation (2.5.6) can be written in matrix form as

(E − µ1A− µ2AT )un+1 = un + τfn+1 + µ1

ω(α)2 0...

...

ω(α)M ω

(α)0

un+1b + µ2

ω(α)0 ω

(α)M

......

0 ω(α)2

un+1b (A.2.1)

where µ1 =K1τ

hα, µ2 =

K2τ

hα, ω(1−γ)

k is given by equation (2.2.3), un = [uni , un2 , · · · , unM−1]T , unb =

[un0 , unM ]T , fn = [fni , f

n2 , · · · , fnM−1]T , E = IM−1 and

A =

ω(α)1 ω

(α)0 0 · · · 0 0

ω(α)2 ω

(α)1 ω

(α)0 · · · 0 0

ω(α)3 ω

(α)2 ω

(α)1 · · · 0 0

......

.... . .

......

ω(α)M−2 ω

(α)M−3 ω

(α)M−4 · · · ω

(α)1 ω

(α)0

ω(α)M−1 ω

(α)M−2 ω

(α)M−3 · · · ω

(α)2 ω

(α)1

(M−1)×(M−1)

A.2.2 Second Order Finite Difference Scheme

Equation (2.5.9) can be written in matrix form as

[E − τ

2hα(K1A+K2A

T )]un+1 = [E +τ

2hα(K1A+K2A

T )]un +τ

2(fn+1 + fn) +Hn (A.2.2)

where un = [uni , un2 , · · · , unM−1]T , fn = [fni , f

n2 , · · · , fnM−1]T , E = IM−1. Furthermore

A =

g(α)1 g

(α)0 0 · · · 0 0

g(α)2 g

(α)1 g

(α)0 · · · 0 0

g(α)3 g

(α)2 g

(α)1 · · · 0 0

......

.... . .

......

g(α)M−2 g

(α)M−3 g

(α)M−4 · · · g

(α)1 g

(α)0

g(α)M−1 g

(α)M−2 g

(α)M−3 · · · g

(α)2 g

(α)1

(M−1)×(M−1)

A.3

with g(α)k defined in equation (2.2.36) and

H =τ

2hα

K1g(α)2 +K2g

(α)0

K1g(α)3

...

K1g(α)M−1

K1g(α)M

(un0 + un+1

0 ) +τ

2hα

K2g(α)M

K2g(α)M−1...

K2g(α)3

K2g(α)2 +K1g

(α)0

(unM + un+1

M )

A.2.3 Fourth Order Finite Difference Scheme

Equation (2.5.15) can be written in matrix form as

(E+cαS−µ1A−µ2AT )un+1 = (E+cαS+µ1A+µ2A

T )un+(τ

2)(E+cαS)(fn+1+fn)+(

τ

2)(cαX)(fn+1

b+fn

b)

(A.2.3)

where µ1 =K1τ

2hα,µ2 =

K2τ

2hα, cα is defined in equation (2.2.41), un = [uni , u

n2 , · · · , unM−1]T , fn =

[fni , fn2 , · · · , fnM−1]T , fn

b= [fn0 , f

nM ]T , E = IM−1 and

S =

−2 1 0 · · · 0 0

1 −2 1 · · · 0 0

0 1 −2 · · · 0 0...

......

. . ....

...

0 0 0 · · · −2 1

0 0 0 · · · 1 −2

(M−1)×(M−1)

X =

1 0

0 0...

...

0 0

0 1

(M−1)×2

A =

g(α)1 g

(α)0 0 · · · 0 0

g(α)2 g

(α)1 g

(α)0 · · · 0 0

g(α)3 g

(α)2 g

(α)1 · · · 0 0

......

.... . .

......

g(α)M−2 g

(α)M−3 g

(α)M−4 · · · g

(α)1 g

(α)0

g(α)M−1 g

(α)M−2 g

(α)M−3 · · · g

(α)2 g

(α)1

(M−1)×(M−1)

where the coefficients g(α)k were defined in equation (2.2.46).

A.4

A.3 Time-space Fractional Diffusion Equations

A.3.1 First Order Finite Difference Scheme

Equation (2.6.8) can by given in matrix form as

(b0E − µ1A− µ2AT )un = b0u

n−1 −n−1∑k=1

(bn−k−1 − bn−k)uk + bn−1u0

+ µ1

ω(α)2 0...

...

ω(α)M ω

(α)0

unb + µ2

ω(α)0 ω

(α)M

......

0 ω(α)2

unb + τγfn (A.3.1)

where µ1 =K1τ

γ

hα, µ2 =

K2τγ

hα, ω(1−γ)

k is given by equation (2.2.3), bk is given by (2.2.6), un =

[uni , un2 , · · · , unM−1]T , unb = [un0 , u

nM ]T , fn = [fni , f

n2 , · · · , fnM−1]T , E = IM−1 and

A =

ω(α)1 ω

(α)0 0 · · · 0 0

ω(α)2 ω

(α)1 ω

(α)0 · · · 0 0

ω(α)3 ω

(α)2 ω

(α)1 · · · 0 0

......

.... . .

......

ω(α)M−2 ω

(α)M−3 ω

(α)M−4 · · · ω

(α)1 ω

(α)0

ω(α)M−1 ω

(α)M−2 ω

(α)M−3 · · · ω

(α)2 ω

(α)1

(M−1)×(M−1)

A.3.2 Second Order Finite Difference Scheme

Equation (2.6.15) can by given in matrix form as

(b0E − µ1A− µ2AT )un = b0u

n−1 −n−1∑k=1

(bn−k−1 − bn−k)uk + bn−1u0

+ µ1

g(α)2 0...

...

g(α)M g

(α)0

unb + µ2

g(α)0 g

(α)M

......

0 g(α)2

unb + τγfn (A.3.2)

where µ1 =K1τ

γ

hα, µ2 =

K2τγ

hα, g(α)k is defined in equation (2.2.36), bk is given by (2.2.6), un =

[uni , un2 , · · · , unM−1]T , unb = [un0 , u

nM ]T , fn = [fni , f

n2 , · · · , fnM−1]T , E = IM−1 and

A =

g(α)1 g

(α)0 0 · · · 0 0

g(α)2 g

(α)1 g

(α)0 · · · 0 0

g(α)3 g

(α)2 g

(α)1 · · · 0 0

......

.... . .

......

g(α)M−2 g

(α)M−3 g

(α)M−4 · · · g

(α)1 g

(α)0

g(α)M−1 g

(α)M−2 g

(α)M−3 · · · g

(α)2 g

(α)1

(M−1)×(M−1)

A.5

A.3.3 Fourth Order Finite Difference Scheme

Equation (2.6.24) can be written in matrix form as

(E + cαS −K1µA−K2µAT )un = (E + cαS)

n−1∑k=0

(bn−k−1 − bn−k)uk

+ (E + cαS)bn−1u0 − cαXunb + cαX

n−1∑k=0

(bn−k−1 − bn−k)ukb

+ cαbn−1Xu0b + µ(E + cαS)fn + µcαXf

n

b

+K1µ

g(α)2 0...

...

g(α)M g

(α)0

unb +K2µ

g(α)0 g

(α)M

......

0 g(α)2

unb (A.3.3)

where µ = τγΓ(2 − γ), cα is defined in equation (2.2.41), bk is given by (2.2.6), the coefficients g(α)k

were defined in equation (2.2.46), un = [uni , un2 , · · · , unM−1]T , unb = [un0 , u

nM ]T , fn = [fni , f

n2 , · · · , fnM−1]T ,

fnb

= [fn0 , fnM ]T , E = IM−1 and

S =

−2 1 0 · · · 0 0

1 −2 1 · · · 0 0

0 1 −2 · · · 0 0...

......

. . ....

...

0 0 0 · · · −2 1

0 0 0 · · · 1 −2

(M−1)×(M−1)

X =

1 0

0 0...

...

0 0

0 1

(M−1)×2

A =

g(α)1 g

(α)0 0 · · · 0 0

g(α)2 g

(α)1 g

(α)0 · · · 0 0

g(α)3 g

(α)2 g

(α)1 · · · 0 0

......

.... . .

......

g(α)M−2 g

(α)M−3 g

(α)M−4 · · · g

(α)1 g

(α)0

g(α)M−1 g

(α)M−2 g

(α)M−3 · · · g

(α)2 g

(α)1

(M−1)×(M−1)

A.6