Generalized Hat Functions Method for Solving Fractional ...Abstract—In this paper, a numerical scheme, based on fractional order generalized hat functions operational matrix of integration

$Page 1: Generalized Hat Functions Method for Solving Fractional ...Abstract—In this paper, a numerical scheme, based on fractional order generalized hat functions operational matrix of integration$
Abstract—In this paper, a numerical scheme, based on

fractional order generalized hat functions operational matrix of

integration for solving fractional integro-differential equations

of Bratu-type is presented. The operational matrix is used to

reduce the fractional integro-differential equation to a system of

algebraic equations. The error analysis of generalized hat

functions is also given. Numerical examples are provided to

demonstrate the validity and applicability of the method.

Moreover, comparing the methodology with the known

technique shows that our approach is more efficient and more

accurate.

Index Terms—Fractional Bratu-type equations; generalized

hat functions; operational matrix; error analysis; uniqueness;

numerical solution.

I. INTRODUCTION

RCTIONAL calculus is a 300 years old mathematical

discipline, and has been developed progressively up to

now. Compared with integer order differential equations,

fractional differential equations have the advantage that they

can better explain some natural physics processes and

dynamic system processes [1-5] because the fractional order

differential operators are non-local operators. Differential

equations involving fractional order derivatives are used to

describe a lot of systems, of which the important applications

lie in field of electrode-electrolyte polarization, heat

conduction, viscoelasticity, electromagnetic waves, diffusion

equations and so on [6-7]. In general, it is difficult to derive

the analytical solutions to most of the fractional differential

equations. Therefore, it is important to develop some reliable

and efficient techniques to solve fractional differential

equations. The numerical solutions of fractional differential

equations have attracted considerable attention from many

researchers. In recent years, an increasing number of

numerical methods are being developed. The most

commonly used methods are Variational Iteration Method

[8], Adomian Decomposition Method [9-10], Generalized

Manuscript received July 22, 2016; revised December 20, 2016.

Mulin Li is with the Inner Mongolia Vocational and Technical College of

Communications, Department of electronic and information engineering,

Inner Mongolia, Chifeng, China.

Lifeng Wang (Corresponding author) is with the School of Aeronautic

Science and Engineering, Beihang University, Beijing, China (e-mail:

[email protected])).

Yu Liu is with the Inner Mongolia Vocational and Technical College of

Communications, Department of electronic and information engineering,

Inner Mongolia, Chifeng, China.

Differential Transform Method [11-12], and Wavelet

Method [13-14].

Bratu’s problem is also discussed in all kinds of

applications, such as chemical reaction theory, the fuel

ignition model of the thermal combustion theory and

nanotechnology [15-17]. About Bratu’s problem, both

mathematicians and physicists have devoted a lot of effort. In

Ref. [18], Syam and Hamdan presented the Laplace Adomian

decomposition method for solving Bratu’s problem. Wazwaz

[19] proposed the Adomian decomposition method for

solving Bratu’s problem. Yigit Aksoy and Mehmt Pakdemirli

[20] had solved Bratu-type equation of new perturbation

iteration solutions. Yi et. al [21] used CAS wavelets method

to solve fractional integro-differential equations of

Bratu-type.

The main objective of the present paper is to introduce

generalized hat functions operational matrix to solve the

fractional integro-differential equations of Bratu-type. In this

study, we consider the following equation:

( )

0( ) ( , ) ( ) 0

1 , 0 , 1

xu tD u x k x t e dt g x

r r x t

(1)

with initial condition: ( ) (0)j

ju b , 0,1, , 1j n (2)

Dis the fractional derivative in the Caputo sense, r is a

positive integer. is a real constant. ( )u x is the solution to

be determined. 2 2[0,1), ([0,1) [0,1))g L k L are

given functions.

The structure of this paper is as follows: In Section 2, the

generalized hat functions are introduced. The generalized hat

functions operational matrix of fractional integration is also

introduced and the error analysis of generalized hat functions

is given in Section 3. In Section 4, we summarize the

application of generalized hat functions operational matrix

method to the solution of the fractional integro-differential

equations of Bratu-type. In Section 5, uniqueness theorem of

this equation is proposed. Two numerical examples are

provided to clarify the method in Section 6. The conclusion is

given in Section 7.

II. GENERALIZED HAT FUNCTIONS AND THEIR PROPERTIES

The interval [0, ]T is divided into n subintervals

[ ,( 1) ]ih i h , 0,1,2, , 1i n , of equal lengths

h where /h T n . The generalized hat functions’ family of

Generalized Hat Functions Method for Solving

Fractional Integro-Differential Equations of

Bratu-type

Mulin Li1, Lifeng Wang2, Yu Liu1

F

IAENG International Journal of Computer Science, 44:1, IJCS_44_1_12

(Advance online publication: 22 February 2017)

______________________________________________________________________________________

first 1n hat functions is defined as follows [22]:

0

, 0 ,( )

0, ,

h xx h

x h

otherwise

(3)

( 1), ( 1) ,

( 1)( ) , ( 1) ,

0, ,

1,2, , 1,

i

x i hi h x ih

h

i h xx ih x i h

h

otherwise

i n

(4)

( ), ,

( )

0, .n

x T hT h x T

x h

otherwise

(5)

Using the definition of generalized hat functions, we have

following observations

1, ,( )

0,i

i kkh

i k

(6)

and

( ) ( ) 0, 2i jx x i j (7)

Any arbitrary function 2[0, ]u L T is approximated in

vector form as

1 1

0

( ) ( ) ( )n

T

i i n n

i

u x u x U x

(8)

where 1 0 1[ , , , ]T

n nU u u u and

1 0 1( ) [ ( ), ( ), , ( )]T

n nx x x x .

The important aspect of using the generalized hat functions

in the approximation of function ( )u x , lies in the fact that

the coefficients iu in Eq.(8) are given by

( ), 0,1,2, ,iu u ih i n (9)

III. OPERATIONAL MATRIX OF THE INTEGRATION FOR

GENERALIZED HAT FUNCTIONS

A. Fractional calculus

Before we derive the generalized hat functions operational

matrix of the fractional integration, we first review some

basic definitions of fractional calculus, which have been

given in [2].

Definition 1. The Riemann-Liouville fractional integral

operator J of order is given by

1

0

1( ) ( ) ( ) , 0

( )

x

J u x x u d

(10)

0 ( ) ( )J u x u x (11)

Definition 2. The Caputo definition of fractional differential

operator is given by

( )

10

( ), ;

1 ( )( ) ,

( ) ( )

0 1 .

r

r

rx

r

d u xr N

dx

uD u x d

r x

r r

(12)

The Caputo fractional derivatives of order is also defined

as ( ) ( )r rD u x J D u x , where rD is the usual integer

differential operator of order r . The relation between the

Riemann- Liouville operator and Caputo operator is given by

the following expressions:

( ) ( )D J u x u x (13)

1( )

0

( ) ( ) (0 ) , 0!

krk

k

xJ D u x u x u x

k

(14)

B. Fractional order generalized hat functions operational

matrix of integration.

In this part, we may simply introduce the operational

matrix of fractional integration of generalized hat functions,

more detailed introduction can be found in the Ref. [23].

If J is fractional integration operator of generalized hat

functions, we can get:

1 1 1( ) ( )n n nJ x P x

(15)

where

1 2 3

1 2 1

1 2

1

3

( 1) ( 1)

0

0 1

0 0 1

0 0 0 1( 2)

0 0 0 0 1

n

n

n

n

n

n n

hP

(16)

where 1( 1) ( 1) , 1,2, ,k k k k k n (17)

and 1 1 1( 1) 2 ( 1)

1,2, , 1

k k k k

k n

(18)

1nP

is called the generalized hat functions operational matrix

of fractional integration.

Apart from the generalized hat functions, we consider

another basis set of block pulse functions. The set of these

functions, over the interval [0, )T , is defined as:

1, ( 1)( )

0, ,i

ih x i hb x

otherwise

0,1,2, , 1i n (19)

with a positive integer value for n and T

hn

.

The following properties of block pulse functions will be

used in this paper

0,( ) ( )

( ),i j

i

i jb x b x

b x i j

(20)



______________________________________________________________________________________

0

0,

( ) ( ),

T

i j

i j

b x b x dx Ti j

n

(21)

Let 0 1 1( ) [ ( ), ( ), , ( )]T

n nB t b t b t b t . We suppose

( ) ( )n n nJ B x F B x , then nF is called the block

pulse operational matrix of fractional integration , here

1 2 1

1 2

3

1

0 11

0 0 1( 2)

0 0 0 1

n

n

n nF h

and

1 1 1( 1) 2 ( 1)k k k k ,

1,2, , 1k n .

There is a relation between the block pulse functions and

generalized hat functions, namely

1( ) ( )n nt B t (22)

where 1 0 1( ) [ ( ), ( ), , ( )]T

n nx x x x , and

( 1)

1/ 2 0 0 0

1/ 2 1/ 2 0 0

0 1/ 2 1/ 2 0

0 0 1/ 2 1/ 2

0 0 1/ 2n n

.

C. Error analysis

In this section, from Eq.(8), we suppose

0

( ) ( ) ( )n

n i

i

u x u ih x

(23)

and

1

0

1( ) ( ) ( )

( )

x

n nJ u x x u d

(24)

where ( )nJ u xdenotes the n th approximate of order

Riemann-Liouville fractional integral of ( )u x .

Let ( ) ( ) ( )n nx J u x J u x , then we have the

following theorem.

Theorem 3.1. If ( ), [0, ]u x x T is approximated by the

Eq.(8), then

(i) ( ) ( ) 0nu jh u jh , for 0,1,2, ,j n ;

(ii)2 3

1 1( ) ( ) ( )

2nu x u x u jh O

n n

,

for ( 1) , 0,1,2, , 1jh t j h j n ;

(iii) For ( 1)jh t j h ,

2

2 3

1( )

2 ( 1)n

MTx O

n n

,

where ( )u jh M , 0M .

Proof. (i) From Eq.(6), the value of ( )nu x at j th point

, 0,1,2, ,x jh j n is given by

0

( ) ( ) ( ) ( )n

n i

i

u jh u ih jh u jh

, then

( ) ( ) 0nu jh u jh , for 0,1,2, ,j n .

(ii) If ( 1) , 0,1,2, , 1jh t j h j n , then from

Eq.(3)-(5) and Eq.(23), we have

0

1

( ) ( ) ( )

( ) ( ) (( 1) ) ( )

( 1)( ) ( )

( ) ( ) ( ) ( )( )

( ) ( )( ) ( )

n

n i

i

j j

u x u ih x

u jh x u j h x

j h x x jhu jh u jh h

h h

u jh h u jh u jh h u jhu jh jh x

h h

u jh h u jhu jh x jh

h

(25)

when 0h , we obtain

( ) ( ) ( ) ( )nu x u jh x jh u jh (26)

Using the Taylor’s series of ( )u x , in the powers of

( )x jh , we have

( )

0

( )( ) ( )

!

kk

k

x jhu x u jh

k

(27)

where ( )ku denotes the k th order derivative of ( )u x . From

Eq.(26) and Eq.(27), we get

( )

2

23

( )( ) ( ) ( )

!

( )( ) ( )

2

kk

n

k

x jhu x u x u jh

k

x jhu jh O x jh

(28)

Because ( ) ,x jh h nh T , so we have

2

2 3

1( ) ( ) ( )

2n

Tu x u x u jh O

n n

(29)

(iii) According to the definition of the absolute error ( )n x ,

we obtain

1

0

( ) ( ) ( )

1( ) ( ) ( )

( )

n n

x

n

x J u x J u x

x u u d

(30)

For ( 1)jh t j h , we get

1

0

1( 1)

1

0

1

1( ) ( ) ( ) ( )

( )

( ) ( ) ( )1

( )( ) ( ) ( )

x

n n

jr h

nrh

r

x

njh

x x u u d

x u u d

x u u d

(31)

Substituting Eq.(29) into Eq.(31), we have



______________________________________________________________________________________

21( 1)

1

2 30

21

2 3

1 1( ) ( ) ( )

( ) 2

1( ) ( )

2

jr h

nrh

r

x

jh

Tx x u rh O d

n n

Tx u rh O d

n n

(32)

If ( )Max u kh M , 0,1,2, ,k j , then we obtain

1( 1)

12

02 3

1

2

2 3

2

2 3

( )1 1

( )( ) 2

( )

1

( 1) 2

1

2 ( 1)

jr h

rhr

nx

jh

x dMT

x On n

x d

x MTO

n n

MTO

n n

(33)

This completes the proof.

The fractional order integration of the function t was

selected to verify the correctness of matrix 1nP

. The

fractional order integration of the function ( )u t t is easily

obtained as follows

1(2)( )

( 2)J u x x

(34)

When 0.5, 32m , the comparison results for the

fractional integration is shown in Fig. 1.

Fig.1 0.5-order integration of the function ( )u x x .

IV. NUMERICAL SOLUTION OF EQ.(1)-(2)

In this section, the generalized hat functions method is

used for solving the fractional order integro-differential

equation of Bratu-type in the form Eq.(1). For this aim, we

approximate functions * ( )D u x, ( )g x and ( , )k x t in the

matrix forms.

Let 1 1( ) ( )T

n nD u x U x

, 1 1( ) ( )T

n ng x G x ,

1 1( , ) ( ) ( )T

n nk x t x K t . In this paper, we suppose

( )

0

( )

!

il

u t

i

u te

i

, then

1( )

1 1 1

0

1 1 1 1

( ) ( ) (0)!

( )

knT k

n n n

k

T T

n n n n

xu x U P x u

k

U P Q x

(35)

where coefficient 1nQ

is known and can be obtained by

using the initial conditions.

Substituting Eq.(22) into Eq.(35), we obtain

1 1 1 1

1 1 1

( ) ( )

( )

T T

n n n n

T T

n n n n

u x U P Q x

U P Q B x

(36)

Define 1 1 1 1 2[ , , , ]T T

n n n nU P Q a a a A

, then

( ) ( )nu x AB x .

Applying the properties of block pulse functions, we have

1 2[ ( )] [ , , , ] ( ) ( )i i i i

n n i nu x a a a B x A B x (37)

By substituting the above expanded forms into Eq.(1), we get

( )

0

1 1 1 10

0

1 1

1 1 10

0

1 1

1 1 1

0

( ) ( , ) ( )

[ ( )]( ) ( ) ( )

!

( )

( )( ) ( ) ( )

!

( )

1( ) ( ) ( ) (

!

xu t

jlxT T

n n n n

j

T

n n

T Tlx n jT T

n n n n

j

T

n n

lT T

n n n j n

j

D u x k x t e dt g x

u tU x x K t dt

j

G x

B t AU x x K B t dt

j

G x

U x x K diag A Bj

0

1 1

1

1 1 1

0

1 1

1

1

0

1

1 1

)

( )

1( ) ( ) ( ) ( )

!

( )

1( ) ( ) ( ) ( )

!

( )

( ) ( ) ( ) 0

x

T

n n

lT T

n n n j n n

j

T

n n

lT T T

n n n j n n

j

T

n n

T T T

n n n n n n

t dt

G x

U x x K diag A F B xj

G x

U B x B x K diag A F B xj

G B x

U B x W B x G B x

(38)

where nW is a n -vector with elements equal to the diagonal

entries of the following matrix

1

0

1( )

!

lT

j n

j

W K diag A Fj

(39)

Putting the collocation points 1

0

n

i ix

into Eq. (38), Eq.(38)

will be

1 1 0T T T

n n nU Q G (40)

Solving the system of equations given by Eq.(40), the

numerical solution ( )u x is obtained. The Eq.(40) can be

solved by iterative numerical technique such as Newton’s

method. Also the Matlab function “fsolve” is available to

deal with such a nonlinear system of algebraic equations.



______________________________________________________________________________________

V. EXISTENCE OF UNIQUENESS

Theorem1. (Uniqueness theorem) Eq. (1) has a unique

solution whenever 0 1 , where =( ) ( 1)

ML

，

0 , 1

sup ( , )x t

M k x t

。

Proof. Suppose 0,1t , then ( )u x is bounded. Therefore

the nonlinear term ( )u te in Eq. (1) is Lipschitz continuous

with *

*

uue e L u u , 0L .

Letu and*u be two different solutions of Eq. (1) , then we

can get

( )

0( ) ( , ) ( )

xu tD u x k x t e dt g x (41)

* ( )

*0

( ) ( , ) ( )x

u tD u x k x t e dt g x (42)

Using Rieman-Liouville fractional integration, we have

1

0

1 ( )

0 0

1( ) ( ) ( )

( )

( ) ( , )( )

x

xu t

J D u x x g d

x k t e dtd

(43)

*

1

*0

( )1

0 0

1( ) ( ) ( )

( )

( ) ( , )( )

x

xu t

J D u x x g d

x k t e dtd

(44)

Because

1( )

0

( ) ( ) (0 )!

knk

k

tJ D x t x t x

k

, so Eq. (43)

and Eq. (44) can be transformed into 1

( ) 1

00

1 ( )

0 0

1( ) (0) ( ) ( )

! ( )

( ) ( , )( )

jn xj

j

xu t

xu x u x g d

j

x k t e dtd

(45)

*

1( ) 1

* *0

0

( )1

0 0

1( ) (0) ( ) ( )

! ( )

( ) ( , )( )

jn xj

j

xu t

xu x u x g d

j

x k t e dtd

(46)

Then we have

*

*

*

1 ( )

0 0

*

( )1

0 0

( )1 ( )

0 0

( )1 ( )

0 0

1

*0

( ) ( , )( )

( ) ( , )( )

( ) ( , )( )

( ) ( , )( )

( )( )

( )

xu t

xu t

xu tu t

xu tu t

x

x k t e dtd

u u

x k t e dtd

x k t e e dtd

x k t e e dtd

MLu u x d

MLu

1 1

*

*

1

( ) ( 1)

x xu

MLu u

Therefore * 1 0( ) ( 1)

MLu u

.

This implies that* (1 ) 0u u , where

=( ) ( 1)

ML

.

As 0 1 ,* 0u u , imply

*u u , so we can prove

Eq.(1) has the uniqueness solution.

VI. NUMERICAL EXAMPLES

To test the efficiency and accuracy of the proposed

generalized hat functions operational matrix of fractional

integration method, here we consider two examples and

suppose l n .

Example 6.1. Consider the following equation [21]:

0.5 ( )

0

1( ) ( ) ( ) 0

2

xu tD u x x t e dt g x (47)

where

21.5 0.5(3) (2)( ) ( 1)

(2.5) (1.5) 2

x xxg x x x e

, and

subject to the initial condition (0) 0u . The exact solution

of this problem is 2( )u x x x . Table I shows the

absolute errors obtained by generalized hat functions and

CAS wavelets (see [21]) method, respectively. The

comparison between the numerical solutions and the exact

solution by generalized hat functions and CAS wavelets

method are shown in Fig.2-4.



______________________________________________________________________________________

TABLE I.

ABSOLUTE ERRORS AT DIFFERENT POINTS FOR DIFFERENT n .

t 6n 12n 24n

Ours CAS Ours CAS Ours CAS

0 0 0.05345 0 0.02590 0 0.00591

1/6 0.02885 0.06527 0.00899 0.03850 0.00303 0.00861

2/6 0.01991 0.05142 0.00658 0.01273 0.00222 0.00925

3/6 0.01690 0.02376 0.00556 0.00782 0.00186 0.00174

4/6 0.01518 0.03786 0.00496 0.00811 0.00165 0.00314

5/6 0.01404 0.08725 0.00456 0.04440 0.00150 0.03718

Fig.2. The comparison between numerical and exact solution for 6n .

Taking a closer to Table I and Fig. 2-4, with k increasing,

we see that the approximate solutions converge to the exact

solution. From the comparison between two methods, we

find that generalized hat functions method can reach higher

degree of accuracy when solving the same equation.



Example 6.2. Consider this equation:

( )

0( ) 3 ( ) ( ) 0,

xu tD u x x t e dt g x (48)

such that the initial conditions (0) 0u ,

and2( ) 1/ 1 3 ( 1)/2g x x x x . The value of

1 is the only case for which we know the exact solution

( ) ln( 1)u x x . The numerical results are calculated

when the parameter 24n . The numerical solutions for

various are presented in Fig.5.

Fig.5. Numerical solution and exact solution of 1 .

The numerical results in Fig.5 corresponding to 1 are

in good agreement with the exact results. Therefore, we hold

that the solutions for 0.75 and 0.85 are also

credible.

VII. CONCLUSION

In this work, we introduce the generalized hat functions

and operational matrix of the fractional integration. Using

this matrix and block pulse functions to solve the fractional

integro- differential equations of Bratu-type numerically.

The error analysis of generalized hat functions and the

uniqueness theorem of this equation are proposed. The

present method is better than CAS wavelets method.

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Generalized Hat Functions Method for Solving Fractional ...Abstract—In this paper, a numerical scheme, based on fractional order generalized hat functions operational matrix of integration