Computers and Mathematics with Applications 65 (2013) 975–982

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Computers and Mathematics with Applications

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A numerical method for solving a fractional partialdifferential equation through converting it into anNLP problemMohammad Ali Mohebbi Ghandehari, Mojtaba Ranjbar ∗

Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

a r t i c l e i n f o

Article history:Received 12 May 2012Received in revised form 16 December 2012Accepted 1 January 2013

Keywords:Fractional partial differential equationsDiscretizationNonlinear programming

a b s t r a c t

In this paper, we propose a new approach for solving fractional partial differentialequations, which is very easy to use and can also be applied to equations of other types.The main advantage of the method lies in its flexibility for obtaining the approximatesolutions of time fractional and space fractional equations. Using this approach, we converta fractional partial differential equation into a nonlinear programming problem. Severalnumerical examples are used to demonstrate the effectiveness and accuracy of themethod.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The theory of fractional calculus was first proposed in the year 1695 by Marquis de l’Hopital and from then on manystudies were done and many important books were published on this field, among which we can especially mention thestudy of Oldham and Spanier [1].

Fractional differential equations have recently proved to be valuable tools for the modeling of many phenomena influid mechanics, physics, electrochemistry, mathematical biology, viscoelasticity and other sciences. Different fractionaldifferential equations have been used for solving the fractional telegraph equation [2], the space–time fractional diffusion-wave equation [3,4], the fractional KdV equation [5], and another fractional two-point boundary value problem [6]. In mostcases, these problems do not admit analytical solution, so these equations should be solved using special techniques. In thelast decade, several computational methods have been applied to solve fractional differential equations. Momani, Odibatand Yildirim [7,8] used the Adomian decomposition method and the homotopy perturbation method (HPM) to solve thespace–time fractional telegraph equation.

The variational iteration method is a relatively new approach for providing an approximation solution to linear andnonlinear problems [9]. Recently Odibat andMomani implemented the variational iterationmethod and the decompositionmethod to solve the space–time fractional diffusion-wave equation [10,11].

In this paper, rather than using these methods, we propose a new numerical approach for solving partial differentialequations of fractional order by using discretization and an interpolation method [12,13].

The organization of this paper is as follows: In Section 2, some theorems are presented that will be used in later sections.In Section 3, the method is discussed. Section 4 is devoted to numerical experiments and the results are compared with theexact solutions. Section 5 is the conclusion.

∗ Corresponding author. Tel.: +98 412 4327500; fax: +98 412 4327535.E-mail addresses:mohammadalimohebbi@yahoo.com (M.A. Mohebbi Ghandehari), m_ranjbar@azaruniv.edu, ranjbar633@gmail.com (M. Ranjbar).

0898-1221/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2013.01.003

976 M.A. Mohebbi Ghandehari, M. Ranjbar / Computers and Mathematics with Applications 65 (2013) 975–982

2. Preliminaries

In this section, we recall the basic definitions from fractional calculus and some theorems of integral calculus which weshall apply to formulate our new approach.

Definition 2.1. A real function f (t), t > 0, is said to be in the space Cµ, µ ∈ R, if there exists a real number p (> µ) suchthat f (t) = tpf1(t), where f1(t) ∈ C[0, ∞], and it is said to be in the space Cm

µ iff f (m)∈ Cµ,m ∈ N .

The Riemann–Liouville fractional integral and Caputo derivative are defined as follows.

Definition 2.2. The Riemann–Liouville fractional integral operator of order α ≥ 0, of a function f ∈ Cµ, µ ≥ −1, is de-fined as

Jα f (t) =1

Γ (α)

t

0(t − τ)α−1f (τ )dτ , α > 0, t > 0, J0f (t) = f (t).

Some of the most important properties of operator Jα for f ∈ Cµ, µ ≥ −1, α, β ≥ 0 and γ > −1 are as follows [14]:

1. Jα Jβ f (t) = J (α+β)f (t);2. Jα Jβ f (t) = Jβ Jα f (t);3. Jαtγ =

Γ (γ+1)Γ (α+γ+1) t

α+γ .

Definition 2.3. The fractional derivative of f (t) in the Caputo sense is defined as

Dα f (t) = Jm−αDmf (t) =1

Γ (m − α)

t

0(t − τ)m−α−1f (m)(τ )dτ ,

for m − 1 < α ≤ m,m ∈ N, t > 0, f ∈ Cm−1.

Definition 2.4. For m to be the smallest integer that exceeds α, the Caputo time-fractional derivative operator of orderα > 0 is defined as

Dαt u(x, t) =

∂αu(x, t)∂tα

=

1

Γ (m − α)

t

0(t − τ)m−α−1 ∂mu(x, τ )

∂τmdτ form − 1 < α < m

∂mu(x, t)∂tm

for α = m.

Now, we state some theorems of calculus and optimization.

Theorem 2.1. Let f (x, t) be a given function and a, b, c, d are constants, and let t1, . . . , tm and x1, . . . , xn be sets ofsupporting points in [a, b] and [c, d] respectively, where a = t1 < · · · < tm = b and c = x1 < · · · < xn = d; then b

a

d

cf (x, t)dxdt = lim

m,n→∞

n−1i=1

m−1j=1

f (τi, ζj)xitj, (1)

where xi = xi+1 − xi and tj = tj+1 − tj, and τi, ζj are arbitrary points in the intervals [xi, xi+1] and [tj, tj+1] respectively.

Proof. As we know from calculus, b

a

d

cf (x, t)dxdt = lim

m,n→∞

n−1i=1

m−1j=1

xi+1

xi

tj+1

tjf (x, t)dxdt. (2)

Notice that the subintervals are of unequal length. The right-hand components of (2) contain integrals with small intervals,so we can substitute each integral by xi+1

xi

tj+1

tjf (x, t)dxdt = f (τi, ζj)xitj, (3)

and now we use (3) in the right-hand side of (2) to obtain

limm,n→∞

n−1i=1

m−1j=1

xi+1

xi

tj+1

tjf (x, t)dxdt = lim

m,n→∞

n−1i=1

m−1j=1

f (τi, ζj)xitj. (4)

On considering (2) and (4) together, the proof is completed.

M.A. Mohebbi Ghandehari, M. Ranjbar / Computers and Mathematics with Applications 65 (2013) 975–982 977

Remark 2.1. If we choose the same distance between support points, we obtain the following formula: b

a

d

cf (x, t)dxdt = lim

m,n→∞hk

n−1i=1

m−1j=1

f (τi, ζj), (5)

where h =b−an and k =

d−cm , and τi, ζj are arbitrary points in the intervals [xi, xi+1] and [tj, tj+1] respectively.

Theorem 2.2. Let y = f (x) be a convex function on a convex set; then any local minimum of f is a global one.

Proof. See [15].

Theorem 2.3. Consider n convex functions f1, . . . , fn; then g(x, y) =n

i=1 αifi(x, y) is also a convex function for αi ≥ 0 (i =

1, . . . , n).

Proof. We have that fi(x, y) are convex functions for i = 1, . . . , n. This means that

fi((1 − λ)(x, y) + λ(x′, y′)) ≤ (1 − λ)fi(x, y) + λfi(x′, y′),

and now from

g(x, y) =

ni=1

αifi(x, y),

we take

g((1 − λ)(x, y) + λ(x′, y′)) =

ni=1

αifi(1 − λ)(x, y) + λ(x′, y′)

≤

ni=1

αi(1 − λ)fi(x, y) + λfi(x′, y′)

= (1 − λ)

ni=1

αifi(x, y) + λ

ni=1

αifi(x′, y′) = (1 − λ)g(x, y) + λg(x′, y′).

3. An approach for solving fractional partial differential equations

In this section, we propose our method for finding the numerical solution of a partial differential equation of fractionalorder of the form

∂αu(x, t)∂tα

+ β∂u(x, t)

∂x+ γ

∂2u(x, t)∂x2

= f (x, t), t ∈ Ω, x ∈ Ω ′, 0 < α ≤ 1, (6)

where β, γ are real parameters. We assume that Ω = Ω ′= [0, 1], and also that we have a bounded initial condition

u(x, 0) = u0(x) and boundary conditions u(0, t) = g1(t) and u(1, t) = g2(t) for all t ∈ Ω , and that f is a continuousfunction.

(Note that we are choosing 0 ≤ x ≤ 1 and 0 ≤ t ≤ 1, since every interval such as [a, b] can be transformed into thisinterval by a linear transformation).

Let

Eu(x, t) =∂αu(x, t)

∂tα+ β

∂u(x, t)∂x

+ γ∂2u(x, t)

∂x2− f (x, t) = 0, (7)

where Eu(x, t) is a functional and depends on the unknown function u(x, t), so Eu : PC[0, 1] × [0, 1] → R, where PC meansthat they are piecewise continuous on the interval [0, 1] × [0, 1].

We can convert fractional partial differential equation (6) into an equivalent optimization problem, as follows:

Minu∈PC[0,1]×[0,1]∥Eu(x, t)∥1 = Min 1

0

1

0|Eu(x, t)|dxdt. (8)

Theorem 3.1. Let u(x, t) be a continuous function on [0, 1] × [0, 1] and a solution for (6); then u(x, t) is the optimal solutionof (8) with zero objective function and vice versa.

Proof. Let u1(x, t) be a solution for (6), which is continuous on [0, 1] × [0, 1]; then we have

∂αu1(x, t)∂tα

+ β∂u1(x, t)

∂x+ γ

∂2u1(x, t)∂x2

− f (x, t) = 0,

978 M.A. Mohebbi Ghandehari, M. Ranjbar / Computers and Mathematics with Applications 65 (2013) 975–982

so ∂αu1(x, t)∂tα

+ β∂u1(x, t)

∂x+ γ

∂2u1(x, t)∂x2

− f (x, t) = 0.

Since u1, f are continuous on their domains, by integrating both sides of the last equation on [0, 1] × [0, 1] we obtain 1

0

1

0

∂αu1(x, t)∂tα

+ β∂u1(x, t)

∂x+ γ

∂2u1(x, t)∂x2

− f (x, t) dxdt = 0,

and, thus, ∥Eu1∥1 = 0, and this means that u1(x, t) is the optimal solution of (8) with zero objective function.For the converse part, we let u1(x, t) be the optimal solution of (8) with zero objective function; then 1

0

1

0

∂αu1(x, t)∂tα

+ β∂u1(x, t)

∂x+ γ

∂2u1(x, t)∂x2

− f (x, t) dxdt = 0.

Since ∂αu1(x,t)

∂tα + β∂u1(x,t)

∂x + γ∂2u1(x,t)

∂x2− f (x, t)

is an absolute function, by using Lebesgue integral theoremswe see thatthe following equality must hold:∂αu1(x, t)

∂tα+ β

∂u1(x, t)∂x

+ γ∂2u1(x, t)

∂x2− f (x, t)

= 0,

with

∂αu1(x, t)∂tα

+ β∂u1(x, t)

∂x+ γ

∂2u1(x, t)∂x2

= f (x, t),

and thus u1(x, t) is the solution of (6).

By applying Theorem 2.1 to the double integrals of (8) we obtain 1

0

1

0

∂αu(x, t)∂tα

+ β∂u(x, t)

∂x+ γ

∂2u(x, t)∂x2

− f (x, t) dxdt

= limm,n→∞

ni=1

mj=1

xi

xi−1

tj

tj−1

∂αu(x, t)∂tα

+ β∂u(x, t)

∂x+ γ

∂2u(x, t)∂x2

− f (x, t) dxdt

= limm,n→∞

ni=1

mj=1

δxδt∂αu(τi, ζj)

∂tα+ β

∂u(τi, ζj)∂x

+ γ∂2u(τi, ζj)

∂x2− f (τi, ζj)

, (9)

where τi, ζj are arbitrary points in the intervals [xi−1, xi] and [tj−1, tj] respectively, tj = jδt , xi = iδx, and δx =1n , δt =

1m .

We can choose τi = xi, ζj = tj (the upper bounds in each interval): thus, the above problem changes to the following:

limm,n→∞

ni=1

mj=1

1mn

∂αu(xi, tj)∂tα

+ β∂u(xi, tj)

∂x+ γ

∂2u(xi, tj)∂x2

− f (xi, tj) . (10)

Using the Caputo fractional partial derivative of order α for the time fractional derivative in the Eq. (6), we have

∂αu(x, t)∂tα

=1

Γ (1 − α)

t

0

∂u(x, s)∂s

(t − s)−αds, 0 < t < 1, 0 < α < 1. (11)

Considering n points x1, . . . , xn in the bounded domain [0, 1] and the grid points t1, . . . , tm in the time interval [0, 1],where xi = iδx =

in and tj = jδt =

jm , we can approximate the time fractional derivative as

∂αu(xi, tj)∂tα

=1

Γ (1 − α)

tj

0

∂u(xi, s)∂s

(tj − s)−αds =1

Γ (1 − α)

j−1k=0

(k+1)δt

kδt

∂u(xi, s)∂s

(tj − s)−αds

≃1

Γ (1 − α)

j−1k=0

(k+1)δt

kδt

u(xi, tk+1) − u(xi, tk)δt

(tj − s)−αds

=1

Γ (1 − α)

j−1k=0

ui,k+1 − ui,k

δt

(k+1)δt

kδt(tj − s)−αds

M.A. Mohebbi Ghandehari, M. Ranjbar / Computers and Mathematics with Applications 65 (2013) 975–982 979

=1

Γ (1 − α)

j−1k=0

ui,k+1 − ui,k

δt

(j − k)1−α

− (j − k − 1)1−α

1 − α

(δt)1−α,

=(δt)−α

Γ (2 − α)

j−1k=0

ui,j−k − ui,j−k−1

(k + 1)1−α

− (k)1−α. (12)

Substituting (12) in (10), we get

limm,n→∞

1mn

ni=1

mj=1

(δt)−α

Γ (2 − α)

j−1k=0

ui,j−k − ui,j−k−1

(k + 1)1−α

− (k)1−α

+ β∂u(xi, tj)

∂x+ γ

∂2u(xi, tj)∂x2

− f (xi, tj)

. (13)

Considering the assumptions, we can rewrite (13) as follows:

limm,n→∞

1mn

ni=1

mj=1

(m)α

Γ (2 − α)

j−1k=0

ui,j−k − ui,j−k−1

(k + 1)1−α

− (k)1−α

+ β∂u(xi, tj)

∂x+ γ

∂2u(xi, tj)∂2x

− f

in,

jm

. (14)

On the other hand, the space derivatives in (14) will be replaced by the following finite difference approximation:

∂u(xi, tj)∂x

≃(ui+1,j − ui−1,j)

2δx=

n2(ui+1,j − ui−1,j),

∂2u(xi, tj)∂2x

≃(ui+1,j − 2ui,j + ui−1,j)

(δx)2= n2(ui+1,j − 2ui,j + ui−1,j),

(15)

for i = 1, . . . , n − 1 and j = 1, . . . ,m, and

∂u(xn, tj)∂x

≃(un,j − un−2,j)

2δx=

n2(un,j − un−2,j),

∂2u(xn, tj)∂2x

≃(un,j − 2un−1,j + un−2,j)

(δx)2= n2(un,j − 2un−1,j + un−2,j),

(16)

for j = 1, . . . ,m. To solve (8), we find ui,j = u(xi, tj) for i = 1, . . . , n and j = 1, . . . ,m, so (14) is minimized.We are now dealing with an NLP problem and can use different software systems to find a solution for this problem,

such as LINGO. By Theorem 2.3, relation (14) is a convex function; hence, if zero is a local minimum of (8), it is also a globalminimum of (14).

With discretization of the initial condition and boundary conditions, as constraints on the objective function we obtain

ui,0 = u0

in

,

u0,j = g1

jm

,

un,j = g2

jm

.

Remark 3.1. In this method, we can control the accuracy of the results. For example, if we want the total error to be lessthan a given number ϵ, we solve the following:

Minu∈PC[0,1]×[0,1]∥Eu(x, t)∥pp

s.t.∥Eu(x, t)∥p

p < ϵ,

(17)

where 0 ≤ x, t ≤ 1, and for any p ≥ 1.

980 M.A. Mohebbi Ghandehari, M. Ranjbar / Computers and Mathematics with Applications 65 (2013) 975–982

Table 1Error norms corresponding to Example 4.1, for α = 0.5 in the interval [0, 1].

Time 0.1 0.5 1

L∞ 1.2000e−003 1.0000e−003 6.7800e−004L2 9.1791e−004 6.7802e−004 4.4082e−004

Table 2Absolute errors corresponding to Example 4.1, for α = 0.5 in the interval [0, 1].

xi t = 0.1 t = 0.5 t = 1

0.1 3.8678e−004 3.7450e−004 3.1600e−0040.2 6.9640e−004 6.5970e−004 5.2900e−0040.3 9.3370e−004 8.5790e−004 6.4100e−0040.4 1.1000e−003 9.8740e−004 6.7800e−0040.5 1.2000e−003 1.0000e−003 6.2700e−0040.6 1.2000e−003 8.9350e−004 4.6800e−0040.7 1.1000e−003 6.4050e−004 2.0100e−0040.8 9.9630e−004 2.3080e−004 1.9700e−0040.9 7.4310e−004 1.8500e−004 4.0000e−005

4. Numerical examples

In this section, we solve some examples by our method and compare the numerical results with the exact solutions andsome earlier work. To illustrate the accuracy of the method, we compute the error norms L2 and L∞:

L2 = ∥u − U∥2 =

1N

Nj=1

(u(xj, tj) − U(xj, tj))21/2

,

L∞ = ∥u − U∥∞ = maxj|u(xj, tj) − U(xj, tj)|,where N is the number of collocation points.

Example 4.1. Consider Eq. (6) when β = 1, γ = −1 and f (x, t) =2t2−α

Γ (3−α)+ 2x − 2, which is a one-dimensional linear

inhomogeneous fractional equation:

∂αu(x, t)∂tα

+∂u(x, t)

∂x−

∂2u(x, t)∂x2

=2t2−α

Γ (3 − α)+ 2x − 2,

u(x, 0) = x2,u(0, t) = t2,u(1, t) = 1 + t2.

The exact solution of the above problem is u(x, t) = x2 + t2.

Now we can use our method to solve this equation. Letm = n = 10, α = 0.5 and 0 ≤ x, t ≤ 1; by (8), the fractional partialdifferential equation is converted into the following optimization problem:

Min1

100

10i=1

10j=1

(10)α

Γ (2 − α)

j−1k=0

ui,j−k − ui,j−k−1

(k + 1)1−α

− (k)1−α

+∂u(xi, tj)

∂x−

∂2u(xi, tj)∂2x

−2(j)2−α

102−αΓ (3 − α)−

i5

+ 2

s.t.

ui,0 =

i10

2

,

u0,j =

j10

2

,

u10,j = 1 +

j10

2

,

where ∂u(xi,tj)∂x and ∂2u(xi,tj)

∂2xare obtained from relations (15) and (16).

The results are displayed in Tables 1, 2 and Fig. 1, which show that the numerical solutions agree with the exact solutionand our results are more accurate than the results from the RBFs method [16].

M.A. Mohebbi Ghandehari, M. Ranjbar / Computers and Mathematics with Applications 65 (2013) 975–982 981

Fig. 1. Numerical solution of Example 4.1.

Fig. 2. Numerical solution of Example 4.2, when α = 0.7.

Fig. 3. Numerical solution of Example 4.2, when α = 0.5.

Example 4.2. We consider the following special case of problem (6):

∂αu(x, t)∂tα

=∂2u(x, t)

∂x2,

with the initial condition u(x, 0) = 4x(1 − x) and boundary conditions u(0, t) = u(1, t) = 0. The exact solution of thisproblem is not known. For the purpose of comparison we compute the results for α = 0.5, α = 0.7 with m = 100 andn = 10. The results are shown in Figs. 2 and 3. It should be noted that our results are very closely identical with resultsobtained in [16] using the radial basis function approximation method.

982 M.A. Mohebbi Ghandehari, M. Ranjbar / Computers and Mathematics with Applications 65 (2013) 975–982

5. Conclusion

In this paper, we propose a new method for the solution of time fractional order partial differential equations. Theresults show that this scheme is accurate and efficient. In this work, we just need to use some approximate formulas forthe derivatives of the unknown function u. By using our method, we reach a discrete problem. Then, we solve a nonlinearprogramming problem instead of solving the main fractional partial differential equation.

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