Improved Rotated Finite Difference Method for Solving Fractional Elliptic Partial Differential Equations

Transcription

1 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) ISSN (Print) 33-44, ISSN (Online) Global Societ of Scientific Researc and Researcers ttp://asrjetsjournal.org/ Improved Rotated Finite Difference Metod for Solving Fractional Elliptic Partial Differential Equations Abdulafi Moammed Saeed* Department of Matematics, College of Science, Qassim Universit, Saudi Arabia Abstract Real life problems wit fractional partial differential equations (FPDE's) are of great importance, since fractional differential equations accumulate te wole information of te function in a weigted form. Tis as man applications in psics, cemistr, engineering, etc. For tat reason, we need a metod for solving suc equations, effectivel, eas use and applied for different problems. Te objective of tis paper is to solve fractional elliptic partial differential equations, b using new accelerated version of rotated five point s approimation metod. Eperiment results of te test problem are given in order to confirm te superiorit of our proposed metod. Kewords: Rotated Finite Difference Approimation Metod Fractional Elliptic Partial Differential Equations.. Introduction Fast computational metods for solving partial differential equations using finite difference scemes derived from sewed (rotated) difference operators ave been etensivel investigated over te ears. Tese Iterative metods based on te rotated finite difference approimations ave been sown to be muc faster tan te metods based on te standard five-point formula in solving te partial differential equations wic is due to te formers overall lower computational compleities ([,,3,4,5]). Fractional Partial Differential Equations (FPDE's) can be seen as a generalization of te classical partial differential equations (PDE's) in te sense tat it taes into account te memor and ereditar properties of te psical penomena ([6,7]). As it was in te classical PDE's tere is no general metod tat can be used in solving FPDE's. Numerical solution of FPDE's as received great progress in te recent ears ([8,9]) * Corresponding autor. 6

2 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp 6-7 Te time and space-fractional partial differential equation describe transport dnamics in comple sstems governed b anomalous dispersion and non-eponential relaation []. Because of compleit in te teoretic analsis of numerical approimation of fractional sstems, te common approac is to appl te finite difference metod to discretize fractional derivative operators, and ten obtain te numerical solutions of te fractional partial differential equations. Furtermore, Goloviznin and is colleagues [] developed a numerical metod for solving some -D equations wit fractional derivatives. Te paper is organized in five sections: Section describes te formulation of te Rotated Point Iterative Metod for solving te fractional Poisson s equation. In Section 3, te proposed accelerated version of rotated five point s approimation metod will be given. In Section 4, te numerical results are presented in order to sow te efficienc of te new proposed metod. Finall, te conclusion is given in Section 5.. Formulation of te Rotated Point Iterative Metod for solving te Fractional Poisson s equation Consider te Poisson s equation in te form: ( + ) u (, ) F (, ), (, ) D (.) Were: D {(, ) : (, ) [, L] [, L ]}. Subject to te Diriclet-boundar conditions: u(, ) u (,) ul (, ), and ul (, ) g ( ) Beibalaev and is colleagues [8] considered te fractional Poisson s equation in te form: ( + ) u (, ) F (, ), were, (.) subject to te same Diriclet-boundar conditions of equation (.). Now, we consider te corresponding fractional order Elliptic b te form: ( + ) u (, ) F (, ), were (, ) (.3) It can be seen tat equation (.3) is a generalization to equations (.) and (.). Te simplest standard five-point finite difference approimation of te Laplacian is 6

3 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp 6-7 u i+, j 4 4 u + ui, j ui, j + u + ui, j u u O( 4 4 ) f (.4) Here, u u, ). Anoter approimation to equation (.) can be derived from te rotated five-point finite ( i j difference approimation to give [] u + u + u + u 4u i+, j+ i, j i+, j i, j u u u 4 + O ( ) f (.5) In order to obtain te finite difference approimation of te fractional order equation (.), we use te treatment introduced in [8] for approimate Caputo s fractional derivative of order, ( ) in te form: c u, i+ j u + ui, j D + u ( i, j), i (3 ) (.6a) c u u + u D + u ( i, j) j (3 ) i, j+ i, j (.6b) We can observe tat for equation (.), te corresponding finite difference approimation of te Caputo s fractional order derivative of order is: 4 a (4) u ma ( ) i+, j u + u u i, j c a i D + u (, ) + a i j i (3 a) (3 a) (.7a) and 4 a (4) u ma ( ) i, j+ u+ u u i, j c a i D + u (, ) + a i j j (3 a) (3 a) (.7b) Terefore, te standard five-point finite difference approimation of equation (.) can be written as: ui+, j u + ui, j ui, j+ u + ui, j + f (3 ) (3 ) wic can be rearranged as in te form 63

4 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp u u u u u (3 ) f (.8) i+, j i, j i, j+ i, j B te same manner, te rotated five-point finite difference approimation can be written as: 4 u u u u u (3 ) f (.9) i+, j+ i, j i, j+ i+, j Tere are two was to approimating equation (.3), te first one b using Caputo s formula wic is replaced b a finite sum of integrals at te discretization points, and approimate te second order derivative b using te standard five-point finite difference formula (.8) or rotated five-point finite difference formula (.9). If te standard five-point finite difference is used, ten te finite difference formula of Caputo s fractional derivative will tae te form: i c + Γ( ) s us (, ) D u(, ) ( s) ds c Dui, j b ui +, j ui, j+ u i, j (3 ) Γ i ( ) (.) were b + let, [( ) ] b * s b b bs b Γ(3 b ), b Γ(3 ) Ten, te finite difference sceme for equation (.3) will be given in te form: i * ( +,, +, ) + j b ui j ui j ui j bs( ui, j s+ ui, j s+ ui, j s ) f s were f f(, ). If, we can see tat: i j * s b b b Γ(3 ). Te second wa to approimating equation (.3) b using Grunwald-Letniov (G-L) approimation [] as te following: (, ) lim Γ, ( w ) Γ ( + ) Γ ( ) R L u i j, ( ) N wu N j N u (, ) c R L Du (, ) Du (, ), Γ ( + ) 64

5 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp i G L i j i +, j Du (, ) g u + o ( ). B using te standard five-point finite difference approimation (.8), equation (.3) can be written in te form: i+ + +, + j g ui j gs uj s+, i f s (.) were g rz g s rz r r z z γ z γ γ γ γγ ( )...( γ + ) z >! Furtermore, b using te rotated five-point finite difference approimation (.9), equation (.3) can be written in te form: i + + +, + + j g ui j gs u j s +, i + f s (.) were g rz g s rz r r z z γ z γ γ γ γγ ( )...( γ + ) z >! 3. Te proposed accelerated version of rotated five point s approimation metod It s well nown tat in te finite difference treatment te PDE's or te FPDE's are replaced b an algebraic sstem of equations wic can be written as te form Au f, (3.) were, A is (N ) (N ) coefficients matri, u and f are two (N ) matrices, were T T u [ uj,, uj,,..., ujn, ] and j, j, jn, f [ f, f,..., f ], j,,..., N. It is well nown tat te computational molecule of standard finite difference approimation for te classical (integer) case of PDE's can be represented as in figure wereas te computational molecule of standard finite difference approimation for FPDE's can be represented as in figure. 65

6 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp 6-7 i,j+ i,j+,j i,j i+,j i-,j i,j i+,j i, i,j- Figure : Computational molecule of Eq. (.4) Figure : Computational molecule of Eq. (.8) Also, we can observe tat for te rotated five-point finite difference approimation te following transformations tae place i,j± i ±,j± i ±,j i ±,j. Terefore, te computational molecule of te rotated five-point finite difference approimation for PDE and FPDE can be sown in figure 3 and figure 4 respectivel. i-, j+ i+,j+, j+ i+,j+ i,j i,j i-, i-,j- i+,j- i+,j- Figure 3: Computational molecule of Eq. (.5) Figure 4: Computational molecule of Eq. (.9) It is clear tat te coefficients matri for te fractional order case f equations (.8) and (.9) ave te same structure as in equations (.4) and (.5) ecept te free column f in te rigt side of sstem (3.). Teoreticall, it can be seen tat te coefficients matrices resulting from sstems (.4) and (.5) are nonsingulars, so te sstem (3.) as unique solution ([5]). In addition to tat te coefficients matri A is strictl diagonal dominant, ten A is non-singular and sstems (.8) and (.9) ave a unique solution for ( <, < ). Since it is well nown tat preconditioners pla a vital role in accelerating te convergence rates of iterative metods, several preconditioned strategies ave been used for improving te convergence rate of te iterative metods derived from te standard and sewed (rotated) finite difference operators ([4], [5]). A well-designed 66

7 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp 6-7 preconditioning of te PDE and FPDE problems reduces te number of iterations to reac convergence. Dramatic improvements are possible, but te difficult is to construct te suitable preconditioner. In general, a good preconditioner sould satisf te following prosperities: te first one is tat, te preconditioned sstem sould be eas to solve and te second one is tat te preconditioner sould be ceap to construct and appl. Usuall te sstem (3.) is large and te matri A is sparse. Furtermore, matri A can be write as A D L U (3.) were D is diagonal matri A, L is strictl lower triangular parts of A and U is strictl upper triangular parts of A. A preconditioner ( I + ML ) were < original sstem (3.) to te following sstem: M is used to modif te ( I + ML) Au ( I + ML) f (3.3) A preconditioner P ( I + ML) is a matri tat transforms te original sstem (3.) into new sstem (3.3) tat is equivalent in te sense tat it as te same solution, but tat as more favourable spectral properties. 4. Numerical Results and Discussion In te first part of tis section, we ave compared between te spectral radiuses of te iteration matri corresponding to te resulting sstem of rotated five-point finite difference (original sstem) and te preconditioned sstem for different values of, suc tat (, ). Table sows te comparison of te spectral radius between te original and te preconditioned sstems. Clearl it can be seen tat te spectral radius of te preconditioned sstem is smaller compared to te original sstem, tus justifing te superiorit of te preconditioned sstem against te original sstem wic is quite agreeing wit te results obtained in te previous wor ([3,4,8]). Table : Comparison of te spectral radiuses of te iteration matri corresponding to te original and te preconditioned sstems N order Original sstem Preconditioned sstem ρ ( A) ρ ( PA)

8 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp 6-7 Te second part of tis section as discussed te numerical solution of te following modal problem using te proposed iterative metod: c c + + D u (, ) + D u (, ) [4 + π ( )]sinπ defined in D { < <.5, < < } wit te boundar condition u te area D, and te fractional order is <. Γ suc tat: Γ is te boundar of Table : Solution of te modal problem for.5 b te original and te preconditioned sstems Original sstem Preconditioned sstem Numerical data of te original and te preconditioned sstems are summarized in tables (-3) for two fractional orders for.5 and. reduces te magnitudes of te peas. It can be observed tat in all cases illustrated te increase in te fractional order Trougout te two sections of our eperiments, te results reveal tat te proposed preconditioned is superior to te original sstem in solving fractional elliptic partial differential equations. 68

9 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp 6-7 Table (3): Solution of te modal problem for b te original and te preconditioned sstems Original sstem Preconditioned sstem Conclusion and Future Wors In tis paper, we ave formulated new preconditioned iterative metod based on rotated finite difference metod for solving fractional elliptic partial differential equations. From observation of all eperimental results, it can be concluded tat te proposed sceme ma be a good alternative to solve fractional elliptic partial differential equation and man oter numerical problems. Te idea of tis proposed metod can be etended to group iterative solver wic will be reported separatel in te future. Acnowledgements Financial support provided b Qassim Universit for te completion of tis researc is gratefull acnowledged. References [] A. R. Abdulla, Te Four Point Eplicit Decoupled Group (EDG) Metod: A Fast Poisson Solver International Journal of Comp. Mat. vol. 38, pp. 6-7, 99. [] A. Gunawardena, S. Jain, L. Snder, Modified iterative metods for consistent linear sstems. Linear Algebra Appl.vol. 54/56, pp 3 43,

10 American Scientific Researc Journal for Engineering, Tecnolog, and Sciences (ASRJETS) (6) Volume 6, No, pp 6-7 [3] A. Ibraim and A. R. Abdulla, Solving te Two Dimensional Diffusion Equation b te Four Point Eplicit Decoupled Group (EDG) Iterative Metod.,International Journal of Comp. Mat. vol. 58, pp 53-63, 994. [4] A. M. Saeed and N. H. M. Ali, Preconditioned Modified Eplicit Decoupled Group Metod In Te Solution Of Elliptic PDEs. Applied Matematical Sciences.vol. 4 (4), pp. 65-8,. [5] A. M. Saeed, N. H. M. Ali, On te Convergence of te Preconditioned Group Rotated Iterative Metods In Te Solution of Elliptic PDEs, Applied Matematics &Information Sciences, vol. 5(), pp ,. [6] S. I. Musli, O. P. Agrawal, Riesz Fractional Derivatives and Dimensional Space, Int J Teor Ps, vol. 49(), pp. 7-75,. [7] J. Hristov, Approimate Solutions to Fractional Subdiffusion Equations, European Psical Journal, vol. 93() pp. 9-43,. [8] V. D. Beibalaev, P. Ruslan, Meilanov, Te Dirilet Problem for Te Fractional Poisson s Equation Wit Caputo Derivatives: Afinite Difference Approimation and A Numerical Solution, Termal Science, vol. 6(), pp ,. [9] Z. B. Li, J. H. He, Fractional Comple Transform for Fractional Differential Equations, Matematical and Computational Applications, vol. 5(5), pp ,. [] R. Metzler, J. Klafter, Te Random Wal s Guide to Anomalous Diffusion: A Fractional Dnamics Approac. Ps. Rep., vol.339(), pp.-77,. [] V. M. Goloviznin, I. A. Korotin, Metods of te Numerical Solutions Some One-Dimensional Equations wit Fractional Derivatives, Differential Equations, vol.4(7), pp. -3, 6. [] K. B. oldam, J. Spanier, Te Fractional Calculus, Academic Press, New Yor, USA,