Numerical Simulations for the Space-Time Variable Order Nonlinear Fractional Wave Equation

The explicit finite-difference method for solving variable order fractional space-time wave equation with a nonlinear source term is considered.The concept of variable order fractional derivative is considered in the sense of Caputo.The stability analysis and the truncation error of the method are discussed. To demonstrate the effectiveness of the method, some numerical test examples are presented.


Introduction
It is well known that the fractional calculus definitions are extensions of the usual calculus definitions [1][2][3][4][5][6][7][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, Grünwald, Caputo, Riesz [3,[12][13][14][15][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][18][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.
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: subject to initial conditions and the following boundary conditions where 0 ≤  < , 0 ≤  < , (, ) > 0 is a constant, Ψ 1 (), Ψ where the constant  > 0 is called a Lipschitz constant for .

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: Then, Now, pick two positive integers ,  and define the step size of space and time by ℎ, , respectively, where ℎ = / and  = /.Also we introduce the following notations: By the same way, we have For simplicity, let us define Then, we can rewrite (2) in the following form: that is, The previous equation can be expressed in the following matrix form: and for  ≥ 2 where , and   = (   ) is a matrix with the following coefficients: for  = 1, 2, . . .,  − 1, and  = 1, 2, . . .,  − 1.Also, we note that then ‖‖ ∞ = 1.

The Stability Analysis and the Truncation Error
Let us consider  +1 and  +1 to be two different numerical solutions of (15) with initial values given by  0 and  0 , respectively.
Proof.Let us define  +1 −  +1 =  +1 .From (15) we have where and Δ  = diag ( x u(x, t) Proof.In terms of standard centered difference formula, we have By the integral mean value theorem, we have where Combining the pervious two formulae, we have Now, by using Lemma 4, we can derive the truncation error of explicit finite difference scheme (14).It has a local truncation error of () (from the left side) and (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 () + (ℎ).Therefore, according to the Lax Equivalence Theorem [2], it converges at this rate.
In Figure 1, a comparison between the numerical and the exact solutions when  = 2 at  = 0.416 is presented.
In Figures 6(b) and 6(c), we report the approximate solution at  = 0.052 and  = 0.78, respectively.

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).

Figure 3 with
Figure 2 Figure 4