THE DIRIHLET PROBLEM FOR THE FRACTIONAL POISSON'S EQUATION WITH CAPUTO DERIVATIVES: a finite difference approximation and a numerical solution

Abstract

A finite difference approximation for the Caputo fractional derivative of the 4 - β, 1 < β ≤ 2 order has been developed. A difference schemes for solving the Dirihlet's problem of the Poisson's equation with fractional derivatives has been applied and solved. Both the stability of difference problem in its right-side part and the convergence have been proved. A numerical example was developed by applying both the Liebman and the Monte-Carlo methods.

Dates

  • Submission Date2011-04-21
  • Revision Date2011-07-14
  • Acceptance Date2011-07-18

DOI Reference

10.2298/TSCI110421076B

References

  1. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Translation from the Russian), Gordon and Breach, Amsterdam, 1993.
  2. Nakhushev A.M. Elements of fractional calculation and their application. Nalchik, 2003. 299p. (In Russian).
  3. R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1,pp. 1-77.
  4. Chen, C., Liu, F., Burrage, K. Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation, Applied Mathematics and Computation, 198(2008), 2, pp. 754-769.
  5. S.B. Yuste, L. Acedo, K. Lindenberg, Reaction front in an A + B ?C reaction-subdiffusion process, Phys. Rev. E , 69 (2004) 036126.
  6. Li, Z.B., He, J.H., Fractional Complex Transform for Fractional Differential Equations, Mathematical and Computational Applications, 15(2010), 5, pp. 970-973
  7. Li, Z.B., An Extended Fractional Complex Transform, Journal of Nonlinear Science and Numerical Simulation, 11(2010), s, pp. 0335-0337
  8. Jafari, H., Kadkhoda, N., Tajadodi, H., et al. Homotopy Perturbation Pade Technique for Solving Fractional Riccati Differential Equations, Int. J. Nonlinear Sci. Num., 11(2010) ,2, pp. 271-275
  9. Golbabai, A., Sayevand, K., The Homotopy Perturbation Method for Multi-order Time Fractional Differential Equations, Nonlinear Science Letters A, 1(2010), 2, pp.147-154
  10. He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167(1998), 1-2, pp. 57-68
  11. He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167(1998), 1-2, pp. 57-68
  12. Zhang, S., Zhang, H.Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, 375(2011), 7, pp. 1069-1073
  13. Hristov, J.,Approximate solutions to fractional subdiffusion equations, European Physical Journal, 193(2011), 1, pp. 229-243
  14. Hristov, J., Starting radial subdiffusion from a central point through a diverging medium (A sphere): Heat-Balance Integral Method, Thermal Science, 15(2011),s, pp. S5-S20
  15. Hristov, J., Heat-balance integral to fractional (half-time) heat diffusion sub-model, Thermal Science, 14(2010), 2, pp. 291-316
  16. Hristov, J., A Short-Distance Integral-Balance Solution to a Strong Subdiffusion Equation: A Weak Power-Law Profile, Int. Rev.Chem. Eng., (2010), 5, pp. 555-563.
  17. M.M. Meerschaert, C. Tadjeran. Finite difference approximations for fractional advection- dispersion flow equations, J. Comput. Appl. Math. , 172 (2004), 1,pp. 65-77. doi: 10.1016/j.cam.2004.01.033.
  18. M.M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sides space- fractional partial .differential equations, Appl. Num. Math., 56( 2006), 1,pp. 80-90. doi: 10.1016/j.jcp.2005.08.008.
  19. S.I. Muslih, O.P. Agrawal, Riesz fractional derivatives and fractional dimensional space, Int J Theor Phys , 49 (2010),2 ,pp. 270-275. , DOI: 10.1007/s10773-009-0200-1.
  20. C.Tadjeran, M.M. Meerschaert, H-P Scheffler. A second-order accurate numerical approximation for the fractional diffusional equation, J Comput Phy., 213(2006), 1,pp. 205-213. doi: 10.1016/j.jcp.2005.08.008.
  21. V.M.Goloviznin, I.A.Korotkin, Methods of the numerical solutions some one-dimensional equations with fractional derivatives, Differential Equations, 42 (2006), 7, pp.21-130. (In Russian).
  22. V.D. Beibalaev ,A numerical method of the mathematical model solution for heat transfer in media with fractal structure, Fundamental research (Moscow),5 (2007), 12, pp..249-251. (In Russian).
  23. V.D. Beibalaev, A mathematical model of transfer in mediums with fractal structure, Math. Model. (Moscow), 21(2009), 5, pp.55-62. (In Russian).
  24. Samarsky A.A., Gulin A.V. Numerical methods, Nauka, 1989, Moscow. (In Russian).
Volume 16, Issue 2, Pages385 -394