Error Analysis of an Explicit Finite Difference Approximation for the Space Fractional Wave Equations

*Faculty of Science, Department of Mathematics, University of Cairo, Giza, Egypt, Email: nsweilam@sci.cu.edu.eg, n_sweilam@yahoo.com. **Faculty of Science, Department of Mathematics, University of Um-Alqura, Saudi Arabia, Email: r_ieda@hotmail.com. ABSTRACT: In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.


Introduction
ractional derivatives in mathematics are natural extension of integer-order derivatives, where the order is non integer.Fractional order differential equations have been the focus of many studies due to their frequent appearance in various applications especially in the fields of fluid mechanics, viscoelasticity, biology, physics and engineering (Bagley and Torvik, 1984;Mainardi, 1995;Mainardi and Paradisi, 1997;Podlubny, 1999;2002).Consequently, considerable attention has been given to the solutions of fractional ordinary/partial differential equations (Sweilam et al., 2011).Numerical approximations are the main tool to simulate and study the behaviour of the solutions of such model problems (Fix and Roop, 2004;Meerschaert and Tadjeran, 2004;Sweilam et al., 2007;Sweilam and Khader, 2010;Tenreiro Machado, 2003;Yuste, 2011;Yuste and Acedo, F 2005).Difference methods and, in particular, explicit finite difference methods, are an important class of numerical methods for solving fractional differential equations (Morton and Mayers, 1994;West and Seshadri, 1982;Xu et al., 2001).The usefulness of the explicit method and the reason why they are widely employed is based on their particularly attractive features (Yuste, 2011;Yuste and Acedo, 2005).
In this paper, an EFDA scheme is designed for solving a fractional order wave equation where the fractional derivative is in the Caputo sense.Moreover, since the explicit methods may be unstable, then, it is crucial to determine under which conditions, if any, these methods are stable.We will use here a kind of fractional von Neumann stability analysis to derive the stability conditions.We consider in this paper the following SFWE model: where the variable coefficient ( , ) 0.
The parameter  refers to the fractional order of spatial derivatives, and the Caputo's fractional derivative ( ), x D u x  is defined as follows (Podlubny, 1999).


is the gamma function.

Explicit finite difference approximation for SFWE
Let us consider , h L K  where K is a positive integer, by using a second order difference approximation and (4), we get for 2 m  that Applying the forward finite difference formula to the initial conditions (2), we obtain Now the discrete form of (1) using the explicit finite difference scheme can be written as where The general form of ( 5) with initial conditions, can take the following form where and A is the coefficients matrix with elements ij a obtained from (5).

Stability analysis of EFDA
It is well known that the explicit difference schemes are not always stable for integer order differential equations.Then, for any ,  there are always choices of t  and x  for which the numerical schemes may become unstable.Therefore, it is important to determine under which conditions, if any, the explicit method presented here is stable.To analyze the stability of the numerical scheme (6), we will use here a kind of fractional von Neumann stability analysis.
Theorem 1 The explicit finite-difference scheme (6) for SFWE is conditionally stable if where q is a real spatial wave-number.Inserting this expression we get where () x  means the Riemann zeta function.The stability will be determined by the behaviour of . n and assume that () q   is independent of time, then we can obtain Inserting the extrema value 1   into this equation, we obtain the following stability bound on s: or, in terms of the Riemann zeta function ( , ) Neglecting the truncation error term   ,, kn T x t we get the explicit difference scheme (5).From ( 1) and ( 7), we ) From this result and from ( 8), we claim that

Numerical results
Example 1.Consider the space fractional wave equation   (instead of 1.8 in ( 9)), the exact solution is ( , ) sin 2 (cos 2 sin 2 ) The numerical studies are given as follows: the exact solutions for 2   (as given by ( 10)) and the EFDA solution for 1.8   at 0.05 t  when 0.005 h  and 0.0025 are given in Table 1.In order to test the numerical scheme, we also plot in Figure 1 the exact and approximate solutions for integer case 2.

 
Moreover, the approximate solution for 1.8   respectively.Figure 3 shows the unstable solutions behaviour when 0.157 h  and , 0.001

 
where the value of s is larger than the stability bound .
x s For more details see Theorem 1.

 
The numerical studies for Example 2 can be presented as follows: the exact solutions for 2   (as given by ( 12)) and the EFDA solution for 1.6  2. In order to test the numerical scheme, we also plot in Figure 4 the exact and approximate solutions for integer case 2.
  Moreover, the approximate solution for   respectively.Figure 6 shows the unstable solutions' behaviour when 0.008 h  and 0.001,

 
where the value of s is larger than the stability bound .
x s For more details see Theorem 1.

Conclusions
An explicit finite difference approximation for SFWE has been explored, where the fractional derivative was in the Caputo sense.Error analysis and stability of the explicit numerical method for SFWE were discussed by means of a fractional version of the von Neumann stability analysis.Finally, some numerical results of EFDA were presented.These numerical results demonstrate that the EFDA is a computationally simple and efficient method for SFWE.
Figure 1.EFDA solutions when 0.005 h  and 0.0025   : (left) comparison with the exact solution for 2   at 0.05 t  , (right) for 1.8  To study the behaviour of these solutions, Figure2is plotted to show the 3D-EFDA solutions for 2

Figure
Figure 2. 3D-EFDA solutions for: (left) 2,   (right) in Figure 4. To study the behaviour of these solutions, Figure 5 is plotted to show the 3D-EFDA solutions for 2   and 1.6

Table 2 .
The exact and EFDA solutions at