IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Finite Difference Method for Two-Dimensional Fractional Partial Differential Equation with parameter I. I. Gorial Department of Mathematics, Ibn Al–Haitham College Education, University of Baghdad Received in: 27, June, 2010 Accepted in: 27, September, 2010 Abstract In this paper, we introduce and discuss an algorithm for the numerical solution of two- dimensional fractional partial differential equation with parameter. The algorithm for the numerical solution of this equation is based on implicit and an explicit difference method. Finally, numerical example is provided to illustrate that the numerical method for solving this equation is an effective solution method. Key words: Fractional derivative, two finite difference methods, fractional partial differential equation. Introduction In recent years there has been a great deal of interest in fractional partial differential equations [1, 2, 3, 4, 5]. These equations arise quite naturally in continuous time random walk with spatial and temporal memories. More and more works by researchers from various fields of science and engineering deal with dynamical systems described by fractional partial differential equations, which have been used to represent many natural processes in physics[6], finance[7,8], and hydrology[9,10]. In this paper, we find the numerical solution of two-dimensional fractional partial differential equation with parameter of the form:                    y tyxu yxb x tyxu yxa t tyxu ),,( ),( ),,( ),( ),,( ……. (1) subject to the initial condition u (x,y,0) = f(x,y), for x0  x  xR and y0  y yR ………..(2) and the boundary conditions u (x0,y,t) = 0, for y0  y yR and 0 t  Τ u (x,y0,t) = 0, for x0  x  xR and 0 t  Τ …......(3) u (xR,y,t) = g(y,t), for y0  y yR and 0 t  Τ u (x,yR,t) = k(x,t), for x0  x  xR and 0 t  Τ where a, b and f are known functions of x and y, g is a known function of y and t, k is a knwon function of x and t.  and  are given fractional number.  is a scalar parameters. We use a variation on the classical explicit and implicit Euler method. We prove that these methods by using a novel shifted version of the usual grunwaled finite difference an approximation for the non-local fractional derivative operator. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 1. Two Finite Difference Methods for Solving Two-Dimensional Fractional Partial Differential Equation with Parameter In this section, we propose two finite difference methods, i.e., an implicit finite difference method and explicit finite difference method for solving two-dimensional fractional partial differential equation with parameter (1)-(3). The finite difference method starts by dividing the x-interval [x0, xR] into n subintervals to get the grid points xi= x0 + ix, where   nxxx R 0 and i=0,1,…,n. And the y-interval [y0, yR] into m subintervals to get the grid points yj= y0 + jy, where   myyy R 0 and j=0,1,…,m. Also, the t-interval [0,T] is divided into M subintervals to get the grid points ts = st, s = 0, …,M, where MTt  . The problem here is to find the eigenpair (,u) which satisfy eq.(1)-(3). This equation can be written as an eigenvalue problem Au=Bu, where     y yxb x txaB t A          ),(),(, . Firstly, present the following implicit finite difference method for the initial-boundary value problem of the two-dimensional fractional partial differential equation with parameter. By the shifted Grunwald estimate to the  , - the fractional derivative, [11]: )(),,)1(( )( 1),,( 0 , xOtyxkxug xx tyxu M k k          ………. (4) )(),)1(,( )( 1),,( 0 , yOtykyxug yy tyxu M k k          to reduce eq.(1) as the following form s ji s kji j k kji i k s jkikji s ji uug y t bug x t au , 1 1, 1 0 ,, 1 0 1 ,1,, 1 ,                         1,...,1  ni , Msmj ,...,0,1,...,1  ………(5) Where ),,(, sji s ji tyxuu  , ),(, jiji yxaa  , ),(, jiji yxbb  , ! )1()1( )1(, k k g k k      ,k=0,1,2,… and ! )1()1( )1(, k k g k k      , k=0,1,2,… Secondly, present the following explicit finite difference method for solving the two- dimensional fractional partial differential equation with parameter eq.(1) with the boundary conditions eq.(3), and the initial condition (2), also by usimg the shifted Grunwald estimate to the  , -th fractional derivative given by eq.(4) to reduce as the following form:                     s kji j k k ji i k s jkik ji s ji s ji ug y b ug x a t uu 1, 1 0 , , 1 0 ,1, ,, 1 ,   , 1,...,1  ni , Msmj ,...,0,1,...,1  ……….(6) IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Where ),,(, sji s ji tyxuu  , ),(, jiji yxaa  , ),(, jiji yxbb  , ! )1()1( )1(, k k g k k      , k=0,1,2,… and ! )1()1( )1(, k k g k k      k=0,1,2,… After evaluating eq.(5) and eq.(6) at i=1,…,n-1, j=1,…,m-1 and s=0,…,M one can get a system of algebraic equations which can be solved. Using any suitable method to get the eigenpair can solve   ),( ,. .. ,0 1,. .. ,1 1,. .. ,1 Ms mj niu    . Also, from the initial and boundary conditions one can get: jiji fu , 0 ,  , i=0,…, n 0,0 s ju , j=0,…, m and s=1,…,M 00, s iu , i=0,…, n and s=1,…,M s j s jR gu , , j=0,…, m and s=1,…,M s i s Ri ku , , i=0,…, n and s=1,…,M Where ),,(, sjiji tyxff  , ),( sj s j tygg  and ),( si s i txk  2. Numerical example In this section, present numerical example which confirm our theoretical results. Example: Consider the two-dimensional fractional partial differential equation with parameter:               8.1 8.18.1 5.1 5.15.1 ),,( 2 )2.1(),,( 6 )5.2(),,( y tyxuy x tyxux t tyxu  subject to the initial condition u (x,y,0) = x3y2, 0  x  1, 0 < y < 1 and the boundary conditions u (0,y,t) = 0, 0 < y < 1, 0  t 0.025 u (x,0,t) = 0, 0 < x < 1, 0  t 0.025 u (1,y,t) = e ty2, 0 < y < 1, 0  t 0.025 u (x,1,t) = etx3, 0 < x < 1, 0  t 0.025 This fractional partial differential equation together with the above initial and boundary condition is constructed such that the exact solution is u(x,y,t) = e tx3 y2. Table1, 2, 3 and 4 give the numerical solution using the two finite difference methods. From table 1, 2, 3 and 4, it can be seen that there isa good agreement between the numerical solution and exact solution. 3. Conclusions In this paper, a numerical method for solving the two-dimensional fractional partial differential equation with parameter has been described and demonstrated. Furthermore numerical example is presented to illustrate that good agreement between the numerical solution and exact solution has been noted. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 References 1. Gorenflo, R. and Mainardi , F. (1998), Fract. Cal. Appl. Anal., 1: 167-191. 2. Giona, M. and Roman, H. E. (1992), Physica A, 185 : 87-97. 3. Liu, F. ;Anh, V. and Turner, I. (2004) J. Comp. Appl. Math., 166 : 209-219. 4. Metzler, R. and Klafter, J. (2000) Phys. Rep., 339 : 1-77. 5. Meerschaert, M.,M.; Scheffler, H. P. and Tadjeran, C. (2006), J. Comput. Phys., 211: 249–261. 6. Metzler, R. and Klafter, J. (2004) J. Physics, A 37: R161-R208. 7. Sabatelli, L. ; Keating, S. Dudley, J. and Richmond, P. (2002) Eur. Phys. J., B 27: 273–275. 8. Scalas, E.; Gorenflo, R. and Mainardi, F.(2000), Phys., A 284: 376–384. 9. Benson, D.; Wheatcraft, S. and Meerschaert, M. (2000) Water Resour. Res., 36: 1403–1412. 10. Schumer, R.; Benson, D. A.; Meerschaert, M. M. and Baeumer, B. (2003), Water Resour. Res., 39: 1022–1032. 11. Meerschaert, M. M. and Tadjeran, C. (2004), J. Comput. Appl. Math., 172: 65–77. Table (1) The numerical solution of example using the implicit finite difference method for 2.0,2.0  yx and 0125.0t Numerical Solution Exact Solution Error 0.46300 0.50000 3.70000 E -2 3.61100E-4 3.24025 E -4 -3.70749 E -5 0.01100 1.03688 E -2 -6.31197 E -4 0.08200 7.87381 E -2 -3.26190 E -3 0.30700 0.33180 E -2 -0.30368 4.01100E-4 3.28101 E-4 -7.29992 E -5 0.01100 1.04992 E -2 -5.00773 E -4 0.07900 7.97285 E -2 7.28504 E -4 0.28800 0.33596 4.79753 E -2 Table (2) The numerical solution of example using the implicit finite difference method for 25.0,25.0  yx and 0125.0t Numerical Solution Exact Solution Error 0.39600 0.50000 0.04000 1.31700E-3 9.88846 E -4 -3.28154 E-4 0.03300 3.16431 E -2 -1.35692 E-3 0.22900 0.24029 1.12896 E-2 1.41900E-3 1.00128 E -3 -4.17716 E-4 0.03300 3.20411 E -2 -9.58902 E-4 0.22100 0.24331 2.23121 E-2 IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Table (3) The numerical solution of example using the explicit finite difference method for 2.0,2.0  yx and 0125.0t Numerical Solution Exact Solution Error 0.50000 0.50000 0.00000 3.20000E-4 3.24025 E -4 4.02510 E -6 0.01000 1.03688 E -2 3.68803 E -4 0.07800 7.87381 E -2 7.38100 E -4 0.32800 0.33180 3.80171 E -3 2.96300E-4 3.28101 E -4 3.18008 E -5 0.01000 1.04992 E -2 4.99227 E -4 0.07700 7.97285 E -2 2.72850 E -3 0.30400 0.33598 3.19753 E -2 Table(4) The numerical solution of example using the explicit finite difference method for 25.0,25.0  yx and 0125.0t Numerical Solution Exact Solution Error 0.50000 0.50000 0.00000 9.76600E-4 9.88846 E -4 1.22461 E-5 0.03100 3.16431 E -2 6.43077 E-4 0.23700 0.24029 3.28961 E-3 9.73200E-4 1.00128 E -3 2.80843 E-5 0.03100 3.20411 E -2 1.04110 E-3 0.22600 0.24331 1.73121 E-2 2011) 2( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد طریقة الفروق المنتهیةّ للمعادلة التفاضلیِة الجزئیةِ الكسریة الثنائیة األبعاِد مع متغیر لایرایمان ایشو كو جامعة بغداد، الهیثمابن -كلیة التربیة،قسم الریاضیات 2010 ،حزیران، 27: استلم البحث في 2010 ،ایلول، 27: قبل البحث في الخالصة وان . فـي هـذا البحـث قــدمنا وناقشـنا خوارزمیـة للحـل العــددي لمعادلـة التفاضـلیِة الجزئیـِة الكســریة الثنائیـة األبعـاِد مـع متغیــرِ . خوارزمیة الحل العددي لتلك المعادلة قائمة على اساس طریقة الفروق المنتهیة الضمنیة و الصریحة . و حل مؤثر فعالیقة العددیة لحل هذه المعادلة هي طریقة ذان الطر ا والذي وضحعددی اخیرا قدمنا مثاال .المعادلة التفاضلیِة الجزئیِة الكسریة، طریقتا الفروق المنتهیة،االشتقاق الكسري : الكلمات المفتاحیة