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Journal of Computational Science 1 (2010) 146–149

Contents lists available at ScienceDirect

Journal of Computational Science

journa l homepage: www.e lsev ier .com/ locate / jocs

avelet method for a class of fractional convection-diffusion equation withariable coefficients

iming Chena,∗, Yongbing Wua, Yuhuan Cuib, Zhuangzhuang Wanga, Dongmei Jina

Science College of Yanshan University, No. 438, West Section of Hebei Street, Qinhuangdao 066004, PR ChinaCollege of Light Industry, Hebei Polytechnic University, Tangshan 063000, PR China

r t i c l e i n f o

rticle history:eceived 15 December 2009eceived in revised form 18 April 2010ccepted 10 July 2010

a b s t r a c t

A wavelet method to the solution for a class of space–time fractional convection-diffusion equation withvariable coefficients is proposed, by which combining Haar wavelet and operational matrix together anddispersing the coefficients efficaciously. The original problem is translated into Sylvester equation andcomputation became convenient. The numerical example shows that the method is effective.

eywords:ariable coefficientsonvection-diffusion equationractional derivativeaar waveletperational matrix

© 2010 Elsevier B.V. All rights reserved.

umerical solution

. Introduction

Wavelet analysis as a new branch of mathematics is widelypplied in signal analysis, image manipulation and numerical anal-sis, etc. It mainly studies the expression of functions, that is,unctions are decomposed into summation of “basic functions”,nd every “basic function” is obtained by compression and transla-ion of a mother wavelet function with good properties of localitynd smoothness, which makes people can analyze the proper-ies of locality and integer in the process of expressing functions1]. The Haar wavelets have the following features: (1) orthogonalnd normalization [2], (2) having close support, and (3) the sim-le expression. They are single rectangle wavelets in the supportegion, accordingly, the Haar wavelets are widely used in dealingith practical problems.

Fractional differential equations are generalized from classicalnteger-order ones, which are obtained by replacing integer-ordererivatives by fractional ones. Their advantages comparing with

nteger-order differential equations are the capability of simulat-

ng natural physical process and dynamic system more accurately3]. Therefore, fractional diffusion equations are largely used inescribing abnormal slowly-diffusion phenomenon, and fractionaliffusion equations are always used in describing abnormal con-

∗ Corresponding author. Tel.: +86 15076007210.E-mail address: wuyongbing.good@163.com (Y. Chen).

877-7503/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.jocs.2010.07.001

vection phenomenon of liquid in medium. Therefore, fractionalconvection-diffusion equations are increasingly studied, but it isdifficult to do theoretic analyzing and numerical solving for them[4]. Zhuang and Liu [5] show an explicit difference approximationsfor space–time fractional diffusion equation, there are a lot of work[6–9] in discussing boundary problems of fractional differentialequations, Zhou et al. [10–13] analyzed the existence and unique-ness of solutions for boundary problems of fractional differentialequations.

In this paper, making use of good properties of Haar waveletand the operational matrix, we consider the following space–timefractional convection-diffusion equation with variable coefficients:

∂˛u(x, t)∂t˛

= −b(x)∂u(x, t)

∂x+ a(x)

∂ˇu(x, t)∂xˇ

+ q(x, t) (1)

Here, ∂˛u(x, t)/∂ t˛ is fractional derivative of Caputo sense, ∂ˇu(x,t)/∂ xˇ is fractional derivative Riemann–Liouville sense [14].

2. Definitions of fractional derivatives and integrals

In the development of theories of fractional derivatives and inte-grals, it appears many definitions such as Riemann–Liouville andCaputo fractional differential–integral definition [14] as follows:

utatio

(

(

3

1ib∫

w�

otFf

Q

H

Q

wp

c

wr

s

y

Y. Chen et al. / Journal of Comp

1) Riemann–Liouville definition:

Ra D˛

t f (t) =

⎧⎨⎩

dmf (t)dtm

, ˛ = m ∈ N;

dm

dtm

1� (m − ˛)

∫ t

a

f (T)

(t − T)˛−m+1dT, 0 ≤ m − 1 < ˛ < m.

(2)

Fractional integral of order ˛ is as follows:

Ra I˛

t f (t) = 1� (−˛)

∫ t

a

(t − T)−˛−1f (T) dT, ˛ < 0. (3)

2) Caputo definition:

Ca D˛

t f (t) =

⎧⎨⎩

dmf (t)dtm

, ˛ = m ∈ N;

1� (m − ˛)

∫ t

a

f (m)(T)

(t − T)˛−m+1dT, 0 ≤ m − 1 < ˛ < m.

(4)

. Proposed method

Chen and Hsiao [15] raised the ideology of operational matrix in975, and Kilicman and Al Zhour [16] investigated the generalized

ntegral operational matrix, that is, the integral of matrix �(t) cane approximated as follows:

t

0

�(t) dt ∼= Q��(t) (5)

here Q� is an operational matrix of one-time integral of matrix(t), similarly, we can get operational matrix Q n

�of n-time integral

f �(t). Wu and Hsiao [17] proposed a uniform method to obtainhe corresponding integral operational matrix of different basis.or example, the operational matrix of ˚(t) can be expressed byollowing:

˚ = ˚QB˚−1 (6)

ere QB is the operational matrix of the block pulse function:

Bm = 12m

⎡⎢⎢⎢⎢⎣

1 2 2 · · · 20 1 2 · · · 20 0 1 · · · 2...

......

......

0 0 0 0 1

⎤⎥⎥⎥⎥⎦ (7)

here m is the dimension of matrix ˚(t), and usually m = 2˛, ˛ isositive integer.

If ˚(t) is a unitary matrix, then Q˚ = ˚QB˚T, Q˚ is a matrix withharacteristic of briefness and profound utility.

For t ∈ [0, 1], Haar wavelet function is defined as follows:

h0(t) = 1√m

hi(t) = 1√m

⎧⎪⎪⎨⎪⎪⎩

2j/2,k − 1

2j≤ t <

k − 1/22j

−2j/2,k − 1/2

2j≤ t <

k

2j

0, otherwise

here, i = 0, 1, 2, . . ., m − 1, m = 2l, l is a positive integer. j and kepresent integer decomposition of the index i, i.e. i = 2j + k − 1.

For arbitrary function y(x, t) ∈ L2(R), it can be expanded into Haar

eries by

(x, t) ≈m−1∑i=0

m−1∑j=0

ci,jhi(x)hj(t) (8)

nal Science 1 (2010) 146–149 147

where ci,j =∫ 1

0y(x, t)hi(x) dx ·

∫ 10

y(x, t)hj(t) dt (i, j = 0, 1, 2, . . .,m − 1) are coefficients, discrete y(x, t) by choosing the same stepof x and t, we obtain

Y(x, t) = HT (x)CH(t) (9)

Here Y(x, t) is the discrete form of y(x, t), and

H =

⎡⎢⎣

h0,0 h0,1 · · · h0,m−1h1,0 h1, · · · h1,m−1· · · · · · · · · · · ·

hm−1,0 hm−1,1 · · · hm−1,m−1

⎤⎥⎦ ,

C =

⎡⎢⎣

c0,0 c0,1 · · · c0,m−1c1,0 c1,1 · · · c1,m−1· · · · · · · · · · · ·

cm−1,0 cm−1,1 · · · cm−1,m−1

⎤⎥⎦

C is the coefficient matrix of Y, and it can be obtained by formula:

C = (HT )−1 · Y · H−1.

H is a orthogonal matrix, then

C = H · Y · H−1 (10)

Consider the variable coefficient fractional convection-diffusionequation:

∂˛u(x, t)∂t˛

= −b(x)∂u(x, t)

∂x+ a(x)

∂ˇu(x, t)∂xˇ

+ q(x, t)

0 ≤ x ≤ 1, 0 < t ≤ 1

u(x, 0) = 0, 0 ≤ x ≤ 1

u(0, t) = u(1, t) = 0, t > 0

(11)

where a(x) > 0, b(x) > 0, and a(x), b(x) ∈ C[0, 1], 0 < ˛ ≤ 1, 1 < ˇ ≤ 2, q(x,t) ∈ C(D), D = [0.1] × [0, 1], and u(x, t) ∈ L2(R).

Since u(x, t) ∈ L2(R), we suppose

u(x, t) ≈m−1∑i=0

m−1∑j=0

cijhi(x)hj(t) (12)

Then we can obtain the discrete form of Eq. (12) by taking step� = 1/m of x, t, there is

U = HT (x)CH(t) (13)

Then combining Eq. (13) with Eq. (6), we get

∂˛u

∂t˛≈ ∂˛U

∂t˛= HT (x)C

∂˛

∂t˛H(t) = HT (x)CQ−˛

H H(t) (14)

∂ˇu

∂xˇ≈ ∂ˇU

∂xˇ=(

∂ˇ

∂xˇHT (x)

)CH(t) =

(∂ˇ

∂xˇH(x)

)T

CH(t)

= HT (x)(Q−ˇH )

TCH(t) (15)

Here, q(x, y) is a known function, discrete it, then we have

D = (q(xi, tj)), i, j = 0, 1, 2, . . . , m − 1. (16)

Coefficients a(x), b(x) can be dispersed into a(xi), b(xi) (i = 0, 1, 2, . . .,m − 1).

Let⎡ ⎤ ⎡ ⎤

A =⎢⎢⎣

a(x0) 0 · · · 0

0 a(x1). . . 0

.

.

.. . .

. . . 00 · · · 0 a(xm−1)

⎥⎥⎦ , B =⎢⎢⎣

b(x0) 0 · · · 0

0 b(x1). . . 0

.

.

.. . .

. . . 00 · · · 0 b(xm−1)

⎥⎥⎦ (17)

148 Y. Chen et al. / Journal of Computational Science 1 (2010) 146–149

Table 1Absolute error for ˛ = 0.5, t = 0.5 s, and different values of m for Example 1.

x m = 16 m = 32 m = 64

0.1 6.093e−003 6.093e−003 1.210e−0030.2 4.843e−003 4.843e−003 1.259e−0030.3 1.089e−002 2.750e−002 1.865e−0030.4 1.937e−002 1.937e−002 7.412e−0030.5 1.000e−006 1.000e−006 1.000e−006

S

H

T

H

E

4

e

Ed

Ht

u

Twi

srtcfmrt

Ed

w0

u

Fig. 1. Numerical solution of m = 16.

Fig. 2. Numerical solution of m = 32.

0.6 7.460e−e−003 4.359e−002 7.460e−0030.7 1.734e−002 1.734e−002 1.724e−0030.8 2.964e−002 7.750e−002 4.990e−0030.9 4.443e−002 4.443e−002 1.678e−002

ubstitute Eqs. (14)–(17) into Eq. (11), there is

T (x)CQ−˛H H = −B

(∂

∂xH(x)

)T

CH(t) + A

(∂ˇ

∂xˇH(x)

)T

CH(t) + D

(18)

hen we have(B

(∂

∂xH(x)

)T

− A

(∂ˇ

∂xˇH(x)

)T)

C + CQ ˛H = HDH−1 (19)

q. (20) is a Sylvester equation.

. Numerical examples

We applied the method presented in this paper and solved somexample.

xample 1. Consider the initial-boundary problem of fractionalifferential equation of order ˛(0 < ˛ ≤ 1) [18]

∂˛u

∂t˛+ x

∂u

∂x+ ∂2u

∂x2= 2t˛ + 2x2 + 2, 0 < t < 1, 0 < x < 1

u(0 · t) = 2� (˛ + 1)

� (2˛ + 1)t2˛, u(x, 0) = x2

ere, � ( · ) is gamma function. When ˛ = 0.5, the exact solution ofhe above problem is

(x, t) = x2 + 2� (˛ + 1)

� (2˛ + 1)t2˛

aking m = 16, m = 32, m = 64 and making use of MATLAB (Table 1),e obtain the results of computations of u(x, t) for ˛ = 0.5 are shown

n Figs. 1–4.From Figs. 1–4 we can see that with m increasing, the numerical

olution more and more close to the exact solution. The calculatingesults show that combining with wavelet matrix, the method inhis paper can be effectively used in numerical calculus for variableoefficient fractional differential equations, and that the method iseasibility. At the same time with the sparse nature of Haar wavelet

atrix, compared with the [18], using the above method can greatlyeduce the computation and from the above results, we can see thathe numerical solutions are in good agreement with exact solution.

xample 2. Consider the space–time fractional convection-iffusion equation [19]:

∂˛u(x, t)∂t˛

= −b(x)∂u(x, t)

∂x+ a(x)

∂ˇu(x, t)∂xˇ

+ q(x, t),

0 ≤ x ≤ 1, 0 < t ≤ 1

here a(x) = � (2.8)x/2, b(x) = x0.8, q(x, t) = 2x2(1 − x)t1.2/� (2.2) +.2x1.8(1 + t2)

When ˛ = 0.8, ˇ = 1.5, the exact solution of the above problem is(x, t) = x2(1 − x)(1 + t2). Fig. 3. Numerical solution of m = 64.

Y. Chen et al. / Journal of Computatio

Fig. 4. Exact solution of ˛ = 0.5.

Table 2Exact solution at t = 0.2 s, and absolute error for different values of m for Example 2.

x m = 16 m = 32 m = 64 Exact solution

0.1 4.60e−003 1.00e−004 1.00e−004 0.00940.2 2.60e−003 2.60e−003 2.60e−003 0.03330.3 1.61e−002 6.00e−003 9.00e−004 0.06550.4 8.90e−003 8.80e−003 3.80e−003 0.09980.5 1.60e−003 1.50e−003 1.00e−003 0.1300

Smca

wsm

F

[

[

0.6 7.60e−003 2.80e−003 2.50e−003 0.14980.7 3.60e−e−003 1.50e−003 9.00e−004 0.15290.8 1.19e−002 4.80e−003 2.00e−004 0.13310.9 1.63e−002 1.61e−002 7.10e−003 0.0842

We solved the problem, by applying the technique described inection 3. The absolute error for ˛ = 0.8, ˇ = 1.5, t = 0.2 s, and m = 16,= 32, m = 64, are shown in Table 2. From Table 2, we see that we

an achieve a good approximation with the exact solution by using

bove method.

From Fig. 5 we can find the numerical solutions which is goodith exact solution that it is not only effective to get numerical

olutions with good agreement, and by Table 2 we can see that withincreasing, the absolute error became more and more small.

ig. 5. Comparison of numerical solution and exact solution t = 0.2 s and m = 64.

[

[

[[

[

[

[

[

nal Science 1 (2010) 146–149 149

5. Conclusion

This paper presents a numerical method by combining waveletfunction with operational matrix and dispersing variable coef-ficients effectively. Taking advantage of good properties oforthogonal and sparseness of wavelet matrix, we transform vari-able coefficient fractional differential equation into Sylvesterequation which is easily to be solved. Numerical examples showthat the method above is an efficient algorithm.

Acknowledgments

This work is supported by the Natural Foundation of HebeiProvince (E2009000365).

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Yiming Chen obtained his Doctor’s degree from YanshanUniversity. He has been a Post Doctoral from 2005 to 2007

in Glamorgan University, United Kingdom. He is now aProfessor in College of Science, Yanshan University, China.His research interests include Boundary Element Methodin Contact Problem, Boundary Element Solution for theVariation Inequality and Numerical solution of differentialEquations.