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Numerical simulations for the space-time variable order nonlinear fractional wave equation.

1. Introduction

It is well known that the fractional calculus definitions are extensions of the usual calculus definitions [1-8], where the orders need not to be positive integers. On the other hand, the variable order calculus is a natural extension of the constant order (integer or fractional) calculus. In this sense, the order may function in any variable such as time and space variables or a system of other variables [9,10]. In general, one can say that this extension is introduced by Samko and Ross in [11], where Marchaud fractional derivative and Riemann-Liouville derivative are extended to the variable order cases the order in this case is a function in the space variable only. Many authors have introduced different definitions of variable order differential operators, each of these with a specific meaning to suit desired goals. These definitions such as Riemann-Liouville, Grunwald, Caputo, Riesz [3,12-16], and some notes as Coimbra definition [17,18].

Coimbra in [17] used Laplace transform of Caputo's definition of the fractional derivative as the starting point to suggest a novel definition for the variable order differential operator. Because of its meaningful physical interpretation, Coimbra's definition is better suited for modeling physical problems. The variable order differentials are an important tool to study some systems such as the control of nonlinear viscoelasticity oscillator (for more details see [17-19] and the references cited therein), where the order changes with respect to a parameter or more parameters.

In the following, we present the basic definition for the variable order fractional derivatives which we will use in this paper.

Definition 1 (see [14]). The Caputo space variable order derivative is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

where 0 < [varies] (x,t) < 1.

The main aim of this work is to use the explicit finite difference method (EFDM) to study numerically the following nonlinear space-time variable order wave equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

subject to initial conditions

u (x, 0) = [[phi].sub.1] (x), [u.sub.t](x, 0) = [[phi].sub.2] (x), (3)

and the following boundary conditions

u (0, t) = [[psi].sub.1] (x), u (a, t) = [[psi].sub.2](x), (4)

where 0 [less than or equal to] x < a, 0 [less than or equal to] t < T, B(x, t) > 0 is a constant, [[psi].sub.1] (x), [[psi].sub.2](x) are smooth functions, and f(u,x,t) is a non-linear scour term that satisfies the Lipschitz condition, that is,

[absolute value of (f ([u.sub.1],x,t) - f([u.sub.2],x,t))] [less than or equal to] L[absolute value of ([u.sub.1] - [u.sub.2])], (5)

where the constant L > 0 is called a Lipschitz constant for f.

2. Discretization for EFDM

In this section, EFDM is used to study the model problem (2), then the space-time solutions domain will be discretized. The discrete form for the pervious Caputo derivative can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Now, pick two positive integers N, M and define the step size of space and time by h, [tau], respectively, where h = a/M and [tau] = T/N. Also we introduce the following notations:

[x.sub.i] = ih, for i = 1,2, ..., N, (8)

[t.sub.j] = j[tau], for j = 1, ..., M,

[u.sup.j.sub.i] [approximately equal to] u([x.sub.i],[t.sub.j]), [B.sup.j.sub.i] = B([x.sub.i],[t.sub.j]), and [f.sup.j.sub.i] = f([u.sup.j.sub.i], [x.sub.i], [t.sub.j]). Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

By the same way, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

For simplicity, let us define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Then, we can rewrite (2) in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The previous equation can be expressed in the following matrix form:

[U.sup.0.sub.i] = [[phi].sub.1], [U.sup.1.sub.i] = [U.sup.0.sub.i] + [tau][[phi].sub.2], (14)

and for j [greater than or equal to] 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

and [A.sup.j] = ([a.sup.j.sub.lm]) is a matrix with the following coefficients:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

for n= 1,2, ..., K - 1, and m = 1, 2, ..., K - 1. Also, we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

then [[parallel]A[parallel].sub.[infinity]] = 1.

Lemma 2. The coefficients [G.sup.j.sub.k] and [H.sup.k.sub.i] satisfy the following conditions:

(1) [G.sup.j.sub.0] = 1, and [H.sup.0.sub.i] = 1,

(2) [G.sup.j.sub.k] > [G.sup.j.sub.k+1], and [H.sup.k.sub.i] > [H.sup.k+l.sub.i], for k = 0,1, ...

3. The Stability Analysis and the Truncation Error

Let us consider [W.sup.j+1] and [U.sup.j+1] to be two different numerical solutions of (15) with initial values given by [W.sup.0] and [U.sup.0], respectively.

Theorem 3. The explicit method approximation defined by (15) to the variable order space-time wave equation (2) is unconditionally stable, that is,

[absolute value of ([W.sup.j+1] - [U.sup.j+1])] [less than or equal to] C [absolute value of (W0 - U0)], for any j. (19)

Proof. Let us define [W.sup.j+1] - [U.sup.j+1] = [[epsilon].sup.j+1].

From (15) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

and [DELTA][F.sup.j] = diag [([O.sup.j.sub.m-l] [L.sup.j.sub.m-1], ..., [Q.sup.j.sub.1] [L.sup.j.sub.1]).sup.T].

Noting that [absolute value of ([L.sup.j.sub.i])] [less than or equal to] L, for any i, j.

Let [bar.Q] = max{[Q.sup.j.sub.m-1], ..., [Q.sup.j.sub.1]}. From (20), we have [[paragraph][A.sup.j] + [DELTA][F.sup.j][paragraph].sub.m] [less than or equal to] (2 + [bar.Q]L), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Now, we analyze the stability via mathematical induction [10]. From (14) we have [[parallel][[epsilon].sup.1.sub.i][parallel].sub.[infinity]] [less than or equal to] C[[parallel][[epsilon].sup.0.sub.i][parallel].sub.[infinity]], where C is a constant.

Now, assume that [[parallel][[epsilon].sup.j.sub.i][parallel].sub.[infinity]] [less than or equal to] C[[parallel][[epsilon].sup.0.sub.i][parallel].sub.[infinity]], then from (22), we

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Then, the theorem holds.

Lemma 4. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

be a smooth function; then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Proof. In terms of standard centered difference formula, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

By the integral mean value theorem, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [[xi].sub.j] [member of] [jh, (j+ 1)h]. Combining the pervious two formulae, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Now, by using Lemma 4, we can derive the truncation error of explicit finite difference scheme (14). It has a local truncation error of O([tau]) (from the left side) and O(h) (from the right side).

Remark 5. The pervious explicit method was shown to be stable. This method is consistent with a local truncation error which is O([tau]) + O(h). Therefore, according to the Lax Equivalence Theorem [2], it convergesat thisrate.

4. Numerical Examples

Example 1. Consider the following variable-order linear fractional wave equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

subjected to the following initial conditions:

u (x, 0) = [phi] (x) = [chi square] (8-x),

[u.sub.t] (x,0) = [PSI](x) = 0, (31)

where [X.sub.a] = 0, [X.sub.b] = 8, and T=1.

The exact solution of this problem when [beta] = 2 is u(x, t) = [chi square] (8 - x)([t.sup.2] + 1).

In Figure 1, a comparison between the numerical and the exact solutions when [beta] = 2 at t = 0.416 is presented.

In Figures 2(a) and 2(b), we report the approximate solutions at t = 0.052 and t = 0.78, respectively.

In Figures 3, 4, and 5, respectively, we report the approximate solutions in three dimensions, where the axis's are (t, x, u), (alfa, x, u), and (beta, x, u), respectively.

Example 2. Consider the following variable-order nonlinear fractional wave equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and u(x,0) = [phi](x) = 1 + sin x, [u.sub.t](x,0) = [PSI](x) = 0, where 0 [less than or equal to] x [less than or equal to] 10, T=1, and f(u, x, t) = [u.sup.2] - [sin.sup.2] (x) - [cos.sup.2](t).

This problem has the following exact solution, when [varies] = 2

u (x, t) = sin x + cos t. (33)

In Figure 6(a), we report the numerical solution when o (x, t), [beta](x, t) are variables at t = 0.52 and the exact solutions when [varies] = [beta] = 2.

In Figures 6(b) and 6(c), we report the approximate solution at t = 0.052 and t = 0.78, respectively.

5. Conclusions

In this paper, numerical studies using a simple explicit FDM for solving the variable order space-time wave equation are presented. The stability analysis and the truncation error of the proposed method are proved. Some test examples are given, and the results obtained by the method are compared with the exact solutions in integer order cases. Several figures are presented to simulate the solutions behaviors when the variable orders change with respect to space and time. The comparison certifies that FDM gives good results. Summarizing these results, we can say that the finite difference method in its general form gives reasonable calculations, easy to use, and can be applied for the variable order differential equations in general form. All results were obtained by using MATLAB version 7.6.0 (R2008a).

http://dx.doi.org/10.1155/2013/586870

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[19] C. F. M. Coimbra, C. M. Soon, and M. H. Kobayashi, "The variable viscoelasticity oscillator," Annalen der Physik, vol. 14, no. 6, pp. 378-389, 2005.

Nasser Hassan Sweilam (1) and Taghreed Abdul Rahman Assiri (2)

(1) Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

(2) Department of Mathematics, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia

Correspondence should be addressed to Nasser Hassan Sweilam; n_sweilam@yahoo.com

Received 3 March 2013; Revised 29 April 2013; Accepted 3 May 2013

Academic Editor: Chein-Shan Liu
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Title Annotation:Research Article
Author:Sweilam, Nasser Hassan; Assiri, Taghreed Abdul Rahman
Publication:Journal of Applied Mathematics
Article Type:Report
Date:Jan 1, 2013
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