Photovoltaic Cells and Systems: SQU Journal for Science, 17(2) (2012) 245-253 © 2012 Sultan Qaboos University 542 Error Analysis of an Explicit Finite Difference Approximation for the Space Fractional Wave Equations N.H. Sweilam* and T.A. Assiri** *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. KEYWORDS: Fractional order wave equation, Caputo's derivative, Stability condition, Stability matrix analysis. تحليل خطأ تقريب الفروق المحدودة الظاهرة عند حل المعادالت الموجية في فضاء الرتب الكسرية ناصر حسن سويلم و تغريد عبدالرحمن عسيري الرتب الكسرية، حيث تم تعريف المشتقة من لقد تم في هذا البحث دراسة عددية لمعادالت موجية في فضاء :ملخص ة للمعادلة الموجية. كما ييجاد حلول تقريبإلرة ظاهتب الكسرية باستخدام تعريف كابوتو. وتم استخدام الفروق المحدودة الالر بعض األمثلة ة، فقد تم عرض يالتقريب فعالية هذه الطريقة توضيحل الفروق.االستقرار وتحليل الخطأ لتلك تم مناقشة االختبارية. 1. 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 mailto:nsweilam@sci.cu.edu.eg mailto:n_sweilam@yahoo.com,**%20Faculty%20of%20Science,%20Department%20of%20Mathematics,%20University N.H. SWEILAM and T.A. ASSIRI 542 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:   2 2 , 0, 1 2 , ( , ) ( , ) , 0 ,x t t u x t u x t d x L t t             (1)     1 2( ,0) , ( ,0) , 0 ,tu x f x u x f x x L    (2)    (0, ) , ( , ) ,u t t u L t t   (3) where the variable coefficient ( , ) 0.d x t  The parameter  refers to the fractional order of spatial derivatives, and the Caputo's fractional derivative ( ),xD u x  is defined as follows (Podlubny, 1999).          1 0 , , , , ( , ) ,1 , 1 , m m x mx m m x t d u x t m N u x t dx D u x u t x d m m m                              (4) where  . is the gamma function. 2. 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 2m  that                                      2 1 2 0 211 1 2 0 11 1 2 0 1 2 2 0 , ,1 2 ,1 2 1 , 2 , 1 ,1 2 1 , 2 , 1 , 1 . 3 x x j hk j jh j hk j jh k j x t u t D u x d u x z t z dz z u x j h t u x jh t u x j h t z dz h h u x j h t u x jh t u x j h t j j                                                                       Let 0t    be the grid step in time, ,nt n 0 ,nt T  0, , 1,n N  N T  and 0x h   be the grid step in space, ,kx kh 0 ,kx L  1, , 1,k K  so that  , n ku u kh n and  , 0 .k kd d x Applying the forward finite difference formula to the initial conditions (2), we obtain ERROR ANALYSIS OF AN EXPLICIT FINITE DIFFERENCE APPROXIMATION 542      0 11 1 2, .k k k k ku f x u f x f x   Now the discrete form of (1) using the explicit finite difference scheme can be written as     1 1 1 2 2 1 12 0 2 [ 2 ][ 1 ] , 3 n n n k n n nk k k k k j k j k j j u u u d h u u u j j                          and 1 1 1 1 1 1 1 1 2 [ 2 ] [ 2 ] , k n n n n n n n n n k k k k k k j k j k j k j j u u u s u u u s g u u u                    (5) where   , 3 kd ss     2 ,s h      2 2 1 .jg j j       The general form of (5) with initial conditions, can take the following form  1 0 1 12 , , n n n kU U f x U AU U       (6) where  1 2 1, , , T n n n n kU u u u  and A is the coefficients matrix with elements ija obtained from (5). 3. 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        1 3 1 2 2 2 2 . 3 xs s              Proof. Let us analyze the stability of (5) by substituting in a separated solution n iqj x j nU e   where q is a real spatial wave-number. Inserting this expression we get 1 1 1 1 1 1 2 [ 2 ] [ 2 ] k iq x iq x n n n n j n j n j n j j s e e s g                           , where ( )x means the Riemann zeta function. The stability will be determined by the behaviour of .n If we write 1n n   and assume that ( )q  is independent of time, then we can obtain 1 1 1 1 2 ( 2 ) ( 2 ) k j j j iq x iq x j j s g s e e                      . Inserting the extrema value 1   into this equation, we obtain the following stability bound on s: N.H. SWEILAM and T.A. ASSIRI 542 2 2 2 1 sin ( ) 1 ( 1) [( 1) ] 2 nn j x j q x s s j j            , with     2 2 1 1 1 , j x j s j j              or, in terms of the Riemann zeta function    32 1 2 2 .xs        Then, the method is stable when        1 3 1 2 2 2 2 . 3 xs s              □ Table 1. The exact and EFDA solutions at 0.05t  when 0.005, 0.0025.h   ix 2  1.8  0.0000 0.00000000 0.00000000 0.0500 0.43696211 0.49019405 0.1000 0.83115133 0.92270888 0.1500 1.14398166 1.25716135 0.2000 1.34483109 1.46300554 0.2500 1.41403909 1.52125246 0.3000 1.34483109 1.42692326 0.3500 1.14398166 1.18975019 0.4000 0.83115133 0.83331667 0.4500 0.46696211 0.39279598 0.5000 0.00000000 -0.08846503 0.5500 -0.43696211 -0.56317267 0.6000 -0.83115133 -0.98470522 0.6500 -1.14398166 -1.31166940 0.7000 -1.34483109 -1.51194717 0.7500 -1.41403909 -1.56583600 0.8000 -1.34483109 -1.46797473 0.8500 -1.14398166 -1.22786629 0.9000 -0.83115133 -0.86894593 0.9500 -0.43696211 -0.43396401 1.000 -0.00000000 0.00000000 Theorem 2 The truncation error of SFWE is       2 , .T x t O t O x    Proof. Evaluating (1) at the point  ,k nx t gives 2 ( , )2 [ ] 0x tk n u u d t x         , by the difference equation 2 1 ( , ) n n t k x k k nu d u T x t    . (7) ERROR ANALYSIS OF AN EXPLICIT FINITE DIFFERENCE APPROXIMATION 542 Neglecting the truncation error term  , ,k nT x t we get the explicit difference scheme (5). From (1) and (7), we get 2 2 12 ( , ) ( , ) [ ] [ ] ( , ) n n t k x k k n x t x tk n k n u u u d u T x t t x            , 2 2 2 2 ( , ) ( , ) ( )k n t k nu x t u x t O t t      , (8) 2 1 ( , ) ( ) n x k x tk n u u O x x         , 2 ( , ) ( , ) ( , )1 ( )x x t x t x tk n k n k n u u d u O x dxx x x                   . So that 2 1 ( ) ( ) n n x k x ku u O x O x      . From this result and from (8), we claim that       2 , .T x t O t O x    □ 4. Numerical results Example 1. Consider the space fractional wave equation 2 1.8 2 1.8 0 1 , ( , ) ( , ) , 0 1, t u x t u x t x t x          (9)        ,0 sin 2 , ,0 2 sin 2 ,tu x x u x x    (0, ) (1, ) 0 .u t u t  When 2  (instead of 1.8 in (9)), the exact solution is ( , ) sin 2 (cos 2 sin 2 )u x t x t t    . (10) Figure 1. EFDA solutions when 0.005h  and 0.0025  : (left) comparison with the exact solution for 2  at 0.05t  , (right) for 1.8  at 0.125.t  N.H. SWEILAM and T.A. ASSIRI 522 The numerical studies are given as follows: the exact solutions for 2  (as given by (10)) and the EFDA solution for 1.8  at 0.05t  when 0.005h  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  at 0.125t  when 0.005h  and 0.0025  is also shown in Figure 1. To study the behaviour of these solutions, Figure 2 is plotted to show the 3D-EFDA solutions for 2  and 1.8  respectively. Figure 3 shows the unstable solutions behaviour when 0.157h  and , 0.001  where the value of s is larger than the stability bound .xs For more details see Theorem 1. Figure 2. 3D-EFDA solutions for: (left) 2,  (right) 1.8.  Figure 3. Unstable EFDA solutions when 0.157h  and 0.001  : (left) comparison with the exact solution, (right) 3D-EFDA solutions. Example 2. Consider the space fractional wave equation 2 1.6 2 1.6 ( , ) ( , ) , 0 5, 0 1 , u x t u x t x t t x          (11) ( , 0) sin , ( , 0) 0,tu x x u x     (0, ) 0, (5, ) sin 5 cos .u t u t t  ERROR ANALYSIS OF AN EXPLICIT FINITE DIFFERENCE APPROXIMATION 522 When 2  (instead of 1.6 in (11)) the exact solution is ( , ) sin cosu x t x t . (12) Figure 4. EFDA solutions when 0.002h  and 0.001  : (left) comparison with the exact solution for 2  at 0.01,t  (right) for 1.6  at 0.01.t  Figure 5. 3D-EFDA solutions for: (left) 2  , (right) 1.6.  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  at 0.01t  where 0.002h  and 0.001  are given in Table 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 1.6  at 0.01t  when 0.002, 0.001h   is also shown 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  respectively. Figure 6 shows the unstable solutions’ behaviour when 0.008h  and 0.001,  where the value of s is larger than the stability bound .xs For more details see Theorem 1. N.H. SWEILAM and T.A. ASSIRI 525 Figure 6. Unstable EFDA solutions when 0.008h  and 0.001  : (left) comparison with the exact solution, (right) 3D-EFDA solutions. Table 2. The exact and EFDA solutions at 0.01t  when 0.002, 0.001.h   ix 2  1.6  0.0000 0.00000000 0.00000000 0.2000 0.19866128 0.19866128 0.4000 0.38940257 0.38940432 0.6000 0.56461961 0.56462215 0.8000 0.71732704 0.71733027 1.0000 0.84143691 0.84144069 1.2000 0.93200134 0.93200553 1.4000 0.98540982 0.98541426 1.6000 0.99953312 0.99953762 1.8000 0.97380819 0.97381257 2.0000 0.90926060 0.90926469 2.2000 0.80846366 0.80846730 2.4000 0.67543582 0.67543886 2.6000 0.51548049 0.51548281 2.8000 0.33497458 0.33497609 3.0000 0.14111429 0.14111493 3.2000 -0.05837178 -0.05837204 3.4000 -0.25553075 -0.25553190 3.6000 -0.44250252 -0.44250451 3.8000 -0.61183311 -0.61183586 4.0000 -0.75677158 -0.75677525 ERROR ANALYSIS OF AN EXPLICIT FINITE DIFFERENCE APPROXIMATION 522 5. 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. 6. Acknowledgments The authors wish to express their gratitude to the referees for their valuable suggestions and comments that have improved the paper. MatLab has been used for computations in this paper. 7. References BAGLEY, R.L. and TORVIK, P.J. 1984. On the Appearance of the Fractional Derivative in the Behavior of Real Materials. J. Appl. Mech., 51: 294-298. FIX, G.J. and ROOP, J.P. 2004. Least Squares Finite Element Solution of a Fractional Order Two-point Boundary Value Problem. Comput. Math. Appl., 48: 1017-1044. 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On an Explicit Finite Difference Method for Fractional Diffusion Equations. SIAM J. Numer. Anal., 42: 1862-1874. Received 14 January 2011 Accepted 20 November 2011 http://landau.unex.es/public_html/santos/PUBLICATIONS/a11QuiYus.JCND.pdf