An efficient method for solving fractional Sturm–Liouville problems

Chaos, Solitons and Fractals 40 (2009) 183–189 www.elsevier.com/locate/chaos

An efficient method for solving fractional Sturm–Liouville problems Qasem M. Al-Mdallal UAE University, Department of Mathematical Sciences, College of Science, P.O. Box 17551, Al-Ain, United Arab Emirates Accepted 16 July 2007

Abstract The numerical approximation of the eigenvalues and the eigenfunctions of the fractional Sturm–Liouville problems, in which the second order derivative is replaced by a fractional derivative, is considered. The present results can be implemented on the numerical solution of the fractional diffusion-wave equation. The results show the simplicity and efficiency of the numerical method. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction The fractional derivatives have been proved to be a powerful technique for solving integral and differential equations resulted from several physical modeling such as the fractional diffusion-wave equation, for more details see Liang and Su [11], Ahmad and El-Khazali [1], Syam et al. [16], Diethelm and Ford [3], Diethelm and Freed [5], Gaul et al. [8], Glockle and Nonnenmacher [9], Mainardi [12], and Miller and Ross [14]. However, some other researchers worked on the existence and uniqueness of solutions to some fractional differential equations (see Diethelm and Ford [4], Podlubny [15], and Samko et al. [17]). In this paper, we consider the following class of eigenvalue problems of the form: Da ½pðxÞy 0 ðxÞ þ kqðxÞyðxÞ ¼ 0;

x 2 ð0; 1Þ; 0 < a 6 1;

ð1Þ

subject to ayð0Þ þ by 0 ð0Þ ¼ 0;

cyð1Þ þ dy 0 ð1Þ ¼ 0;

ð2Þ a

where a; b; c; d 2 R, and q(x), p(x) > 0 with q(x) and p(x) are smooth functions. Here, D denotes the fractional differential operator of order a and given by Z x 1 Da yðxÞ ¼ ðx  tÞka1 y ðkÞ ðtÞ dt; ð3Þ Cðk  aÞ 0 where k 2 N and satisfies the relation k  1 < a < k. Within the following text there will be referred to Eqs. (1) and (2) as ‘‘fractional Sturm–Liouville problems’’. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.041

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A very interesting application on the fractional Sturm–Liouville problems is the fractional diffusion-wave equation given by b oa u 2o u ðx; tÞ ¼ c ðx; tÞ; ota oxb

ð4Þ

0 < x < L; t > 0;

subject to uð0; tÞ ¼ 0;

uðL; tÞ ¼ 0; t > 0 ou uðx; 0Þ ¼ f1 ðxÞ; ðx; 0Þ ¼ f2 ðxÞ; 0 < x < L; ot

where 0 6 a 6 2, 1 6 b 6 2 and c > 0. Clearly, Eq. (4) is a homogenous linear integro partial equation obtained from the classical wave equation by replacing the second-order time derivative term by a fractional derivative of order a 2 [1, 2]. Essentially, letting u(x, t) = /(x)T(t) then taking the appropriate derivatives and separating Eq. (4) yields the spatial eigenvalue problem db /ðxÞ þ k/ðxÞ ¼ 0; dxb

ð5Þ

subject to /ð0Þ ¼ 0; /ðLÞ ¼ 0: It is worth mentioning that many of the universal electromagnetic acoustic, and mechanical responses can be modeled accurately using Eq. (4), see Fujita [7], Metzler and Klafter [13] and Kulish and Large [10]. The method of solution used to solve Eq. (1) based on an efficient numerical technique named Adomian decomposition method which has been used by many authors in order to solve several types of differential equations, see for example Adomian [2], Wazwaz [18] and El-Wakil and Abdou [6]. The rest of the paper is organized as follows. The implementation of Adomian decomposition method on problem (1) with a brief review of the fractional calculus are discussed in Section 2. The numerical results will be given in the last section.

2. Analysis of the Adomian decomposition method The following is a brief derivation of the algorithm serves to solve the fractional Sturm–Liouville two point boundary value problems. Consider again the eigenvalue problem given by (1), that is Da ½pðxÞy 0 ðxÞ þ kqðxÞyðxÞ ¼ gðxÞ;

x 2 ð0; 1Þ:

ð6Þ

For convenient, we may rewrite Eq. (6) in the form LyðxÞ ¼ N;

ð7Þ a

1

0

where L ¼ Lðx; D ; D Þ is chosen to be an invertible operator and N ¼ Nðk; x; y; y Þ is a linear operator contains all d other terms. Here D1 ¼ dx . Clearly, we may choose the operator L as Lðx; Da ; D1 Þ ¼ Da pD1 :

ð8Þ

Before discussing the inverse of the operator L, we will present some essential information about fractional calculus theory that will be used strongly herein. Firstly, we introduce the Riemann–Liouville definition of fractional derivative operator J aa . Definition 1. The Riemann–Liouville fractional integral operator of order a is defined by Z x 1 J aa yðxÞ ¼ ðx  tÞa1 yðtÞ dt CðaÞ a where y 2 L1[a, b], and a 2 Rþ . The following lemma is also important in our discussion. Lemma 1. For k 2 N, a 2 Rþ , if k  1 < a < k, and y 2 L1[a, b] then

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185

Daa J aa yðxÞ ¼ yðxÞ and J aa Daa yðxÞ ¼ yðxÞ 

k1 X

y ðkÞ ð0þ Þ

m¼0

xm m!

where b > a P 0 and x > 0. As a result of the previous lemma, we may introduce the inverse of the operator L to have the form Z x 1 a J 0 ðÞ dt: L1 ¼ pðtÞ 0 Applying L1 on left-hand side of Eq. (7) one obtain Z x Z x Z x 1 a a 1 1 1 ðpy 0 ðtÞ  pð0Þy 0 ð0ÞÞ dt ¼ yðxÞ  yð0Þ  pð0Þy 0 ð0Þ J 0 D pD yðtÞ dt ¼ dt; ðL1 LÞðyðxÞÞ ¼ pðtÞ pðtÞ pðtÞ 0 0 0 where the initial conditions y(0) and y 0 (0) should be known. Hence, applying L1 on the whole Eq. (7) should give Z x 1 yðxÞ ¼ yð0Þ þ pð0Þy 0 ð0Þ dt þ L1 Nðk; x; y; y 0 Þ 0 pðtÞ Z x Z x 1 1 a ¼ yð0Þ þ pð0Þy 0 ð0Þ ð9Þ dt þ J 0 Nðk; t; y; y 0 Þ dt: pðtÞ pðtÞ 0 0 The Adomian’s decomposition method assumes the solution y(x) can be represented by an infinite series in the form 1 X y n ðxÞ; ð10Þ yðxÞ ¼ n¼0

and the term N by an infinite series of polynomials 1 X An ; N¼

ð11Þ

n¼0

where An are the Adomian polynomials given by " !# 1 X 1 dn i N l yi : An ¼ n! dln i¼0

ð12Þ

l¼0

Combining Eqs. (9) and (11) gives Z 1 X y n ðxÞ ¼ yð0Þ þ pð0Þy 0 ð0Þ n¼0

0

x

1 X 1 L1 An ðxÞ: dt þ pðtÞ n¼0

ð13Þ

It should be noted that the linearity property of the operator L1 has been used in the above equation. Consequently, we may introduce the following recursive relations from Eq. (13): Z x 1 y 0 ðxÞ ¼ yð0Þ þ pð0Þy 0 ð0Þ dt; pðtÞ ð14Þ 0 1 y nþ1 ðxÞ ¼ L An ðxÞ; n P 0; where An ðxÞ ¼ kqðxÞy n ðxÞ: The series solution of y(x) follows directly and the accuracy of the solution definitely depends on the number of the calculated terms. In the following calculations, the number of terms in the Adomian series (10) did not exceed 25.

3. Numerical results In this section, two regular and singular fractional eigenvalue problems are solved using the method discussed above.

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Example 1. Consider the regular fractional eigenvalue problem D1=2 y 0 ðxÞ þ kyðxÞ ¼ 0;

x 2 ð0; 1Þ;

ð15Þ

subject to y 0 ð0Þ ¼ 0;

yð1Þ ¼ 0:

ð16Þ

Eq. (15) can be written in an operator form as Ly ¼ kyðxÞ:

ð17Þ

Applying L1 on Eq. (17), and using the initial condition at x = 0, yields Z x k J 1=2 yðtÞ dt; yðxÞ ¼ A  Cð1=2Þ 0 0 Z xZ s k ¼A ðs  tÞ1=2 yðtÞ dt ds; Cð1=2Þ 0 0 where A = y(0). Applying the decomposition series (10) and then using the analysis of the previous section gives y 0 ðxÞ ¼ A; 4 y 1 ðxÞ ¼  pffiffiffi Akx3=2 ; 3 p 1 2 3 y 2 ðxÞ ¼ Ak x ; 6 32 pffiffiffi Ak3 x9=2 ; y 3 ðxÞ ¼  945 p .. . y 25 ðxÞ ¼ 4:90247  1047 Ak26 x39 : This gives the approximation of y(x) in a series form by yðxÞ ffi yðxÞ ¼

25 X

~y n ðxÞ:

ð18Þ

n¼0

We explore the first three eigenvalues (k1,i,k2,i and k3,i) numerically in Table 1 where i represents the number of terms used in the Adomian series, i.e. yðxÞ ffi yðxÞ ¼

i X

~y n ðxÞ:

n¼0

The numerical evidence in Table 1 suggests that the first three eigenvalues are k1 ¼ 2:11027708;

k2 ¼ 13:76538223

and

k3 ¼ 24:24328676:

The eigenfunctions corresponding to the above eigenvalues are shown in Fig. 1.

Table 1 The approximations to the first three eigenvalues i

k1,i

k2,i

k3,i

20 21 22 23 24 25

2.11027708 2.11027708 2.11027708 2.11027708 2.11027708 2.11027708

13.76538223 13.76538223 13.76538223 13.76538223 13.76538223 13.76538223

24.24321596 24.24329538 24.24328578 24.24328687 24.24328675 24.24328676

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187

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Fig. 1. The approximate eigenfunctions corresponding to the first, second and third eigenvalues: ——, Y1; - - - - - -, Y2; and , Y3.

Example 2. Consider the singular fractional eigenvalue problem   1 D1=2 y 0 ðxÞ þ þ k yðxÞ ¼ 0; x 2 ð0; 1Þ; x

ð19Þ

subject to yð0Þ ¼ 0; y 0 ð1Þ ¼ 0:

ð20Þ

Eq. (19) may be rewritten in the following operator form as:   1 Ly ¼  þ k yðxÞ: x

ð21Þ

Applying L1 on Eq. (21), and using the initial condition y(0) = 0 gives    Z x 1 1 J 1=2 þ k yðtÞ dt; 0 Cð1=2Þ 0 t   Z xZ s 1 1 ðs  tÞ1=2 ¼ Ax  þ k yðtÞ dt ds; Cð1=2Þ 0 0 t

yðxÞ ¼ Ax 

where B = y 0 (0) is to be determined. Substituting the decomposition series (10) for y(x) and then using the analysis of Section 2 gives the following recursive relation: y 0 ðxÞ ¼ Ax; 3

4Ax2 ð5 þ 2xkÞ pffiffiffi ; 15 p   Ax2 40 þ 28xk þ 5x2 k2 y 2 ðxÞ ¼ ; 120   5 8Ax2 1155 þ 1122xk þ 363x2 k2 þ 40x3 k3 pffiffiffi ; y 3 ðxÞ ¼  51975 p y 1 ðxÞ ¼ 

and so on until y25(x). This gives the approximation of y(x) in a series form by yðxÞ ffi yðxÞ ¼

25 X

~y n ðxÞ:

n¼0

The first three eigenvalues (k1,i, k2,i and k3,i) are presented in Table 2.

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Table 2 The approximations to the first three eigenvalues i

k1,i

k2,i

k3,i

20 21 22 23 24 25

1.66091840 1.66091840 1.66091840 1.66091840 1.66091840 1.66091840

13.55041801 13.55041792 13.55041793 13.55041793 13.55041793 13.55041793

20.51422606 20.51448518 20.51445151 20.51445562 20.51445514 20.51445520

0.35

1.2

0.3

1

0.25

0.8

0.2

0.6

0.15

0.4

0.1

0.2

0.05

0 -0.2

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Fig. 2. The approximate eigenfunctions (left) corresponding to the first, second and third eigenvalues and their derivatives (right): ——, Y1 and Y 01 ; - - - - - -, Y2 and Y 02 ; , Y3 and Y 02 .

Consequently, the first three eigenvalues should be k1 ¼ 1:66091840;

k2 ¼ 13:55041793

and

k3 ¼ 20:51445520:

The eigenfunctions corresponding to the above eigenvalues and their derivatives are shown in Fig. 2.

4. Conclusion In this paper, we have proposed the numerical solution of the fractional Sturm–Liouville problems. The Adomian decomposition method proved to be very efficient for computing the eigen-elements of the present problem. We show that the present technique could be used to solve the fractional diffusion-wave equation.

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