IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 Finite Difference Method for Two-Dimensional Fractional Partial Differential Equation with parameter I. I. Gorial Departme nt of Mathematics, Ibn Al–Haitham College Education, Unive rsity of Baghdad Received in: 27, June , 2010 Accepte d in: 27, Septe mber, 2010 Abstract In this p aper, we introduce and discuss an algorithm for the numerical solution of two- dimensional fractional p artial differential equation with p arameter. The algorithm for the numerical solution of this equation is based on imp licit and an exp licit difference method. Finally, numerical examp le is p rovided to illustrate that the numerical method for solving this equation is an effective solution method. Key words: Fractional derivative, two finite difference methods, fractional p artial differential equation. Introduction In recent y ears there has been a great deal of interest in fractional p artial differential equations [1, 2, 3, 4, 5]. These equations arise quite naturally in continuous time random walk with sp atial and temporal memories. M ore and more works by researchers from various fields of science and engineering deal with dy namical sy st ems described by fractional partial d ifferential equ ations, which have been used to represent many natural p rocesses in p hy sics[6], finance[7,8], and hydrology [9,10]. In this p aper, we find the numerical solution of two-dimensional fractional p artial differential equation with p arameter 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 y 0  y y R ………..(2) and the boundary conditions u (x0,y ,t) = 0, for y 0  y y R and 0 t  Τ u (x,y 0,t) = 0, for x0  x  xR and 0 t  Τ …......(3) u (xR,y ,t) = g(y ,t), for y 0  y y R and 0 t  Τ u (x,y R,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 p arameters. We use a variation on the classical exp licit and imp licit Euler method. We p rove that these methods by using a novel shifted version of the usual grunwaled finite difference an app roximation for the non-local fractional derivative op erator. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 1. Two Finite Di fference Methods for S olving Two-Di mensional Fractional Partial Di fferential Equati on with Parameter In this section, we p rop ose two finite difference methods, i.e., an imp licit finite difference method and exp licit finite difference method for solving two-dimensional fractional p artial differential equation with p arameter (1)-(3). The finite difference method st arts by dividing the x-interval [x0, xR] into n subintervals to get the grid p oints xi= x0 + ix, where   nxxx R 0 and i=0,1,…,n. And the y -interval [y 0, y R] into m subintervals to get t he grid points y j = y 0 + jy , where   myyy R 0 and j=0,1,…,m. Also, t he t-interval [0,T] is divided into M subintervals to get t he grid points ts = st, s = 0, …,M , where MTt  . The problem here is t o find the eigenpair (,u) which satisfy eq.(1)-(3). This equation can be written as an eigenvalue p roblem Au=Bu, where     y yxb x txaB t A          ),(),(, . Firstl y, p resent the following imp licit finite difference method for the initial-boundary value p roblem of the two-dimensional fractional p artial differential equation with p arameter. By the shifted Grunwald est imate 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,… S econdly, p resent the following exp licit finite difference method for solving the two- dimensional fractional p artial differential equation with p arameter 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. S CI. 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,… Aft er evaluating eq.(5) and eq.(6) at i=1,…,n-1, j=1,…,m-1 and s=0,…,M one can get a sy st em of algebraic equations which can be solved. Using any suitable method t o get t he 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 sj s jR gu , , j=0,…, m and s=1,…,M si 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, p resent numerical examp le which confirm our theoretical results. Example: Consider the two-dimensional fractional p artial differential equation with p arameter:               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) = x 3 y 2 , 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 t y 2 , 0 < y < 1, 0  t 0.025 u (x,1,t) = e t x 3 , 0 < x < 1, 0  t 0.025 This fractional p artial differential equation together with the above initial and boundary condition is const ructed such that the exact solution is u(x,y ,t) = e t x 3 y 2 . Table1, 2, 3 and 4 give the nu merical solution usin g the two finite d ifference 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 p aper, a numerical method for solving the two-dimensional fractional p artial differential equation with p arameter has been described and demonst rated. Furt hermore numerical examp le is p resented to illustrate that good agreement between the numerical solution and exact solution has been noted. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 Re ferences 1. Gorenflo, R. and M ainardi , F. (1998), Fract. Cal. App l. 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. App l. M ath., 166 : 209-219. 4. M etzler, R. and Klafter, J. (2000) Phy s. Rep ., 339 : 1-77. 5. M eerschaert, M .,M .; Scheffler, H. P. and Tadjeran, C. (2006), J. Comput. Phy s., 211: 249–261. 6. M etzler, R. and Klafter, J. (2004) J. Physics, A 37: R161-R208. 7. Sabatelli, L. ; Keating, S. Dudl ey, J. and Richmond, P. (2002) Eur. Phy s. J., B 27: 273–275. 8. Scalas, E.; Gorenf lo, R. and M ainardi, F.(2000), Phy s., A 284: 376–384. 9. Benson, D.; Wheatcraft, S. and M eerschaert, M . (2000) Water Resour. Res., 36: 1403–1412. 10. Schumer, R. ; Benson, D. A.; M eerschaert, M . M . and Baeumer, B. (2003), Water R esour. Res., 39: 1022–1032. 11. M eerschaert, M . M . and Tadjeran, C. (2004), J. Comp ut. App l. M ath., 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 S olution Exact S olution 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 S oluti on Exact S oluti on 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. S CI. 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 S oluti on Exact S oluti on 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 S oluti on Exact S oluti on 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