THE DIRIHLET PROBLEM FOR THE FRACTIONAL POISSON S EQUATION WITH CAPUTO DERIVATIVES A Finite Difference Approximation and a Numerical Solution

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1 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation... THERMAL SIENE: Year, Vol. 6, No., pp THE DIRIHLET PROBLEM FOR THE FRATIONAL POISSON S EQUATION WITH APUTO DERIVATIVES A Finite Difference Approimation and a Numerical Solution Vetlugin D. BEIBALAEV and Ruslan P. MEILANOV Dagestan State Universit, Mahachkala, Dagestan, Russia Original scientific paper DOI:.98/TSI476B A finite difference approimation for the aputo fractional derivative of the 4, <order has een developed. A difference schemes for solving the Dirihlet s prolem of the Poisson s equation with fractional derivatives has een applied and solved. Both the stailit of difference prolem in its right-side part and the convergence have een proved. A numerical eample was developed appling oth the Lieman and the Monte-arlo methods. Ke words: fractional aputo derivative, approimation, differential equations, numerical methods, stailit Introduction Fractional partial differential equations are generalizations of classical partial differential equations replacing the integer-order derivatives fractional-order derivatives [, ]. Because fractional derivatives provide useful tools for a description of memor and hereditar properties, the utilit of fractional partial differential equations in mathematical modelling attracts more and more attention [3-5]. Different effective methods such as the fractional comple transform [6, 7], the homotop perturation method [8, 9], the variational iteration method [], the ep-function method [, ], the heat-alance integral method [3-6], and others [7-9]. In this paper we consider the Poisson equation with fractional derivatives in the domain D ={< < a, < < } concerning the solution u(, ) satisfing the equation: u Du f D (, ) (, ) (, ) (a) with a oundar condition u G Y(, ) () where G is the oundar of the area D, and the fractional order is <, and D u(, s ) u (, ) s ds G( ) ( ) (a) * orresponding author; kaspij_3@mail.ru

2 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation 386 THERMAL SIENE: Year, Vol. 6, No., pp D u(,) s u (, ) ( ) ( s) ds () G are fractional aputo derivatives [, ] with respect the space variales and, respectivel. The time and space-fractional partial differential equation descrie transport dnamics in comple sstems governed anomalous dispersion and non-eponential relaation [3]. Because of compleit in the theoretic analsis of numerical approimation of fractional sstems, the common approach is to appl the finite difference method to discretize fractional derivative operators, and then otain the numerical solutions of the fractional partial differential equations. There eists significant interest in developing numerical methods for their solutions. Meerschaert et al. [7, 8] considered one dimensional Riemann-Liouville (R-L) fractional advection dispersion equation with variale coefficients on a finite domain [6] and two-sides space-fractional partial differential equations [8] finite difference method. Muslih et al. [9] proposed a Fourier transform method to solve fractional Poisson s equation with Riesz fractional derivative. Tadjeran et al. [] developed a second-order accurate numerical approimation for the fractional diffusional equation. Goloviznin et al. [] developed a numerical method for solutions of some -D equations with fractional derivatives. Beialaev [, 3], developed a numerical method of the solution for heat transfer prolems in media with fractal structures. The outline of the paper is: () we consider a finite difference approimation of the aputo derivative, and () Develop a numerical method and investigating the right-side stailit and the convergence of the numerical scheme. (3) A numerical eample using the Lieman method [4] and the Monte arlo approach ased on the suggested finite difference approimation scheme is developed. The numerical method Finite difference approimation of the aputo derivatives From the definition of the aputo derivative [] in the interval [ n, n+ ] we otain: n () ( u s D u) s ( ) d (3) G ( s) n where G( a) a e d is the gamma function. Epressing of the second order derivative with respect to the space variale u( ) finite differences in the interval [ n, n + ] we get: d u u ( n ) u ( n) u ( n ) (4a) d h n Hence, the finite difference approimation of the fractional derivative of order is: ( ) ( ) ( ) ( un un un D u) n (4) G( 3 ) h Similarl, we have: ( u) u( m ) u( m) u( m ) G( 3 ) l D m (5)

3 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation... THERMAL SIENE: Year, Vol. 6, No., pp Sustitution of the functions u( n +h) and u( n h) developed as Talor s series, in (4) ields: h4 hu ( u IV u ( n h) u ( n) u ( n h) n ) ( n ) ( D G( 3 ) h u) n ah4 (6) G( 3 ) h where a = M /G(3 ), M = mau IV ( n ). The numerical method For finding the solution of the prolem () in the area D ={ a, } let us introduce a grid W ={z n,m =( n, m ), n =,,...N, m =,,... M} where n = nh, m = ml, h = =a/n, l = /M. Then, using the equalities epressions (4) and (5) we get the difference scheme: u ( n, m) u ( n, m) u ( n, m) G( 3 ) h u ( n, m ) u ( n, m) u ( n, m ) f (, ) (7) n m G( 3 ) l ( n, m )w, u( n, m )= ( n, m ), ( n, m ) (8) where w ={z n, m =( n, m ),n =,,..., N,m =,,..., M } is a set of internal nodes of the grid W, g ={z,m, zn, m } M m {z n,, z n,m } N n is a set of oundar nodes. For seek convenience eq. (7) can e epressed in a form with respect to u n,m, i. e. in a form: un, m un, m unm, unm, G( 3 ) G( 3 ) u h l n,m f n,m (9) G( 3 ) h G( 3 ) l Let us denote as z, i. e. z n,m =( n, m ) the central point of a template, approimating eq. (a). Moreover, Z() denotes the entire template of five points z n,m, z n±,m, z n,m±, while denotes Z () all points of Z() ecept the central one z n,m =( n, m ), i. e. Z () is a template of four points z n±,m, z n, m±. Then, eq. (9) ma e epressed as: A(z) u(z) = Zz B(z,) u()+f() Az ( ) G( 3 ) h l B(, z zn, m), B( z, zn, m) G( 3 ) h G( 3 ) l A(z) >, B(z, >, F(z)= f( n, m ) (a) (, c) (c, d) (e) Denoting L (z)u(z) =A(z)u(z) Z( z ) B( z, ) u( ) we write down eq. (9) as: L (z)u(z) =F(z), zw, u(z) =(z), z ()

4 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation 388 THERMAL SIENE: Year, Vol. 6, No., pp Let us epress the solution () as a sum u(z)= u(z) ~ +u(z), where u(z) is the solution of a homogeneous equation with non-homogeneous oundar conditions: L u(z)=, ~ zw, u(z) ~ = (z), zg () and u (z) is the solution of non-inhomogeneous equation with a homogeneous oundar condition, i. e.: L u(z)=f(z), zw, u(z) =, z (3) With respect to solution (), all the conditions of the maimum principle are satisfied, and we have: u ~, u ma. u( z), ma. ( z) (4a,,c) u ( W) ( g) ( W) zw ( g) zg After simple transformations we read l u u h u u ( h l ) [ ( ),, (,, ) G( 3 ) hl f ] n,m (5) n,m n m n m nm nm Right-side stailit and convergence Stailit Let us construct a majorant function for solving the prolem (3) and appl the comparison theorem, namel: Y(z) =K(a + ) (6) where K is an aritrar constant, while a and are the length of the sides of the rectangle D.Itis ovious that Y(z) for all zw. Then, let us denote D(z)=A(z) Z( z ) B(, z ) and calculate the epression: LY(z)=D(z)Y(z)+ B(z,)[Y(z) Y()] (7) Z( z) using the function (6) for all zw. Then, the function LY (z) can e epressed as: LY(z)= DY DY (8) According to the definition of the fractional derivation in aputo sense for we have: K s ( ) K ( ) s t DY ds s G( ) s s ( s) G( ) ( ) d ds dt K ( ) t t K ( ) t t t t G( ) ( ) d ( ) d G( ) K ( ) K ( ) G( ) G( ) B(, ) K G( ) (9) G( ) G( ) G( ) where B(a, ) = ta t ( ) dt is the Euler s eta function.

5 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation... THERMAL SIENE: Year, Vol. 6, No., pp Similarl, with D Y = KG( + ) we have LY(z)=KG( + ) and it ma e considered that Y(z) is a solution of the oundar prolem: LY ( z) F ( z), z w (a) Y( z) ( z), zg () where F(z) =LY(z) =KG( +)and(z) are the values of function (6) for zg. If K = [/G( + )] F ( w) then all conditions of the comparison theorem [4] concerning the prolems () and (3) are satisfied. From the comparison theorem it follows that: u ma. Y ( z ) a K ( a ) and u F (a,) ( W) zw ( W) ( w) G( ) Notice that eq. () follows from the choice of K defined aove. Moreover, according to the triangle inequalit and (4a,c), concerning the solutions of (7) and (8) we get the following estimations aout the stailit of the difference scheme of the right-side part f and the oundar conditions Y, namel a z f () ( W ) ( w) ( g) G( ) The constants in () are independent of the grid steps h and l, and the difference scheme (7) is stale. onvergence and error estimation Let us denote h n,m = z n,m u( n, m ). Here Z n,m is a solution of the difference prolem (7), and u(, ) is a solution of the (a). Then, sustituting z n,m = h n,m + u( n, m ) in eq. (7) we see that the error satisfies the equation: hn, m hnm, hn, m hnm, hn,m hnm- h G(3 )l n,m, G(3 ) (3a) nm, u u u G( 3 ) h u u u G( 3 ) l n, m nm, n, m nm, n,m nm, G( 3 ) hl f n,m (3) (, ) w, h, (, ) g (3c, d) n m n, m n m Let us impose some constraints to the function u(, ). Firstl, let the fourth order derivative of the solution u(, ) are constrained. Then the approimation error has an order of magnitude 4.Thus, there eists a constant M with is independent of h and l, so that ( W) M (h 4 + l 4 ). Notice that the prolem (3) differs from the difference scheme (7) onl with respect to the right-side parts. Therefore, the following estimation is correct, namel: h a G( ) ( W) ( w) ( W) h M ( h4 l4) (4a, ) where M = [/G( + )]M (a + ) is a constant independent of h and l. Therefore, the difference scheme (8) is convergent and has an order of approimation h 4 + l 4.

6 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation 39 THERMAL SIENE: Year, Vol. 6, No., pp The Monte-arlo method to the set of difference equations developed Solution scheme Let us consider a case when h=lthat transforms the set (7) as: un,m [( un, m un, m ) ( un, m un, m) G( 3 ) h fn,m 4 ] (5) where u n,m is the approimated value of u(nh, mh) for n and m L. If the point (nh, mh) is at the oundar, then the value of u n,m is known and equals (nh, mh), f n,m = f(nh, mh). The relationship (5) is, in fact, a complete mathematical epectation if to assume f n,m =. Hence, the average values n,m meet the following: () A point matching the node (nh, mh), suggests the initial value of the counter to e [G(3 )h f n,m ]/4; () With almost equal proailities we can move the point to one of neighoring nodes, adding to the counter the proper value of [G(3 )h f n,m ]/4. Then eecute () again, and move to the net node till reaching the oundar. After reaching the oundar the appropriate value of is added to counter, and trajector stops. The effective value of the counter provides a selective casual value n,m. It is ovious N that the values n,m satisf the set (5), i. e. M n,m = u n,m. Thus, we get, un,m i i /N, where is the numer of trajectories ( i is the result of i th trajector), simulated from the initial point (nh, mh). As an eample we consider a prolem solved the Lieman and the Monte arlo methods. Eample prolem: a numerical solution Find the solution of: D u (, ) Du (, ) [ 4 ( )]sin p p (6) defined in D = { < <.5,< < } and oeing the oundar condition u G =. Two methods, those of Lieman [4] and the Monte-arlo method were used. Numerical data are summarized in tas. (-4), while graphical eample are illustrated figs. (-4) for two fractional orders for = and =.5. In the figures there are the diagrams of the prolem (6) solution for = and =.5. In all cases illustrated the increase in the fractional order reduces the magnitudes of the peaks. Tale. Solution of the prolem = the Lieman's method (numerical data)

7 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation... THERMAL SIENE: Year, Vol. 6, No., pp Tale. ontinuation Tale. Solution of the prolem for = the Monte arlo method (numerical data) Tale 3. Solution of the prolem for =.5 the Lieman's method (numerical data)

8 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation 39 THERMAL SIENE: Year, Vol. 6, No., pp Tale 4. Solution of the prolem for =.5 the Monte arlo method (numerical data) Figure. Solution of the prolem (6) the Lieman's method for = Figure. Solution of the prolem (6) the Monte arlo method for = Figure 3. Solution of the prolem (6) the Lieman's method for =.5 Figure 4. Solution of the prolem (6) the Monte arlo method for =.5

9 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation... THERMAL SIENE: Year, Vol. 6, No., pp onclusions Because of compleit in the theoretic analsis of numerical approimation of fractional sstems, the common approach is to appl the finite difference method to discretize fractional derivative operators, then otain the numerical solutions of the fractional partial differential equations. In this paper we discussed a finite difference approimation of time-space-fractional PDE () with aputo fractional derivatives. The stailit and the convergence of the right-side of the approimated prolem were proved. omputations the suggested finite difference scheme were performed allowing appling oth the Lieman and the Monte arlo methods for numerical solutions. References [] Samko, S. G., Kilas, A. A., Marichev, O. I., Fractional Integrals and Derivatives: Theor and Applications (translation from Russian), Gordon and Breach, Amsterdam, 993 [] Nakhushev, A. M., Elements of Fractional alculation and Their Application (in Russian), Nalchik, Russia, 3 [3] Metzler, R., Klafter, J., The Random Walk s Guide to Anomalous Diffusion: A Fractional Dnamics Approach, Phs. Rep., 339 (),, pp. -77 [4] hen,., Liu, F., Burrage, K., Finite Difference Methods and a Fourier Analsis for the Fractional Reaction-Sudiffusion Equation, Applied Mathematics and omputation, 98 (8),, pp [5] Yuste, S. B., Acedo, L., Lindenerg, K., Reaction Front in an A + B? Reaction-Sudiffusion Process, Phs. Rev. E,69(4), 3, 366 [6] Li, Z. B., He, J.-H., Fractional omple Transform for Fractional Differential Equations, Mathematical and omputational Applications, 5 (), 5, pp [7] Li, Z. B., An Etended Fractional omple Transform, Journal of Nonlinear Science and Numerical Simulation, (), Suppl., pp [8] Jafari, H., et al., Homotop Perturation Pade Technique for Solving Fractional Riccati Differential Equations, Int. J. Nonlinear Sci. Num., (),, pp [9] Golaai, A., Saevand, K., The Homotop Perturation Method for Multi-Order Time Fractional Differential Equations, Nonlinear Science Letters A, (),, pp [] He, J.-H., Approimate Analtical Solution for Seepage Flow with Fractional Derivatives in Porous Media, omputer Methods in Applied Mechanics and Engineering, 67 (998), -, pp [] Tadjeran,., Meerschaert, M. M., Scheeffler, H.-P., A Second-Order Accurate Numerical Approimation for the Fractional Diffusion Equation, Journal of omputational Phsics, 3 (6),, pp. 5-3 [] Zhang, S., Zhang, H. Q., Fractional Su-Equation Method and Its Applications to Nonlinear Fractional PDEs, Phsics Letters A, 375 (), 7, pp [3] Hristov, J., Approimate Solutions to Fractional Sudiffusion Equations, European Phsical Journal, 93 (),, pp [4] Hristov, J., Starting Radial Sudiffusion from a entral Point through a Diverging Medium (A Sphere): Heat-Balance Integral Method, Thermal Science, 5 (), Suppl., pp. S5-S [5] Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Su-Model, Thermal Science, 4 (),, pp [6] Hristov, J., A Short-Distance Integral-Balance Solution to a Strong Sudiffusion Equation: A Weak Power-Law Profile, Int. Rev.hem. Eng., (), 5, pp [7] Meerschaert, M. M., Tadjeran,., Finite Difference Approimations for Fractional Advection-Dispersion Flow Equations, J. omput. Appl. Math., 7 (4),, pp [8] Meerschaert, M. M., Tadjeran,., Finite Difference Approimations for Two-Sides Space-Fractional Partial Differential Equations, Appl. Num. Math., 56 ( 6),, pp. 8-9 [9] Muslih,S. I., Agrawal, O. P., Riesz Fractional Derivatives and Fractional Dimensional Space, Int J Theor Phs, 49 (),, pp [] Tadjeran,., Meerschaert, M. M., Scheffler, H.-P., A Second-Order Accurate Numerical Approimation for the Fractional Diffusional Equation, J omput Ph., 3 (6),, pp. 5-3

10 Beialaev, V. B., et al: The Dirihlet Prolem for the Fractional Poisson's Equation 394 THERMAL SIENE: Year, Vol. 6, No., pp [] Goloviznin, V. M., Korotkin, I. A., Methods of the Numerical Solutions Some One-Dimensional Equations with Fractional Derivatives (in Russian), Differential Equations, 4 (6), 7, pp. -3 [] Beialaev,V. D., A Numerical Method of the Mathematical Model Solution for Heat Transfer in Media with Fractal Structure (in Russian), Fundamental Research (Moscow), 5 (7),, pp [3] Beialaev, V. D., A Mathematical Model of Transfer in Mediums with Fractal Structure (in Russian), Math. Model. (Moscow) (9), 5, pp [4] Samarsk, A. A., Gulin, A. V., Numerical Methods (in Russian), Nauka, Moscow, 989 Paper sumitted: April, Paper revised: Jul 4, Paper accepted: Jul 8,