Vladimir G. Pimenov     (Ural Federal University; Krasovskii Institute of Mathematics and Mechanics, Ekaterinburg, Russian Federation)
Ahmed S. Hendy     (Ural Federal University, Ekaterinburg, Russian Federation)


For two sided space fractional diffusion equation with time functional after-effect, an implicit numerical method is constructed and the order of its convergence is obtained. The method is a fractional analogue of the Crank–Nicholson method, and also uses interpolation and extrapolation of the prehistory of model with respect to time.


Fractional partial differential equation, Grunwald-Letnikov approximations, Grid schemes, Functional delay, Interpolation, Extrapolation, Convergence order

Full Text:



Wu J. Theory and applications of partial functional differential equations. New York: Springer–Verlag, 1996. 438p.

Zhang B., Zhou Y. Qualitative Analysis of Delay Partial Difference Equations. New York: Hindawi Publishing Corporation, 2007. 375 p.

Tavernini L. Finite difference approximations for a class of semilinear Volterra evolution problems // SIAM J. Numer. Anal., 1977. Vol. 14, no. 5. P. 931–949.

Van Der Houwen P.J., Sommeijer B.P., Baker C.T.H. On the stability of predictor-corrector methods for parabolic equations with delay // IMA J. Numer. Anal., 1986. Vol. 6. P. 1–23.

Zubik-Kowal B. The method of lines for parabolic differential-functional equations // IMA J. Numer. Anal., 1997. Vol 17. P. 103–123.

Kropielnicka K. Convergence of Implicit Difference Methods for Parabolic Functional Differential Equations // Int. Journal of Mat. Analysis, 2007. Vol. 1, no. 6. P. 257–277.

Garcia P., Castro M.A., Martin J.A., Sirvent A. Convergence of two implicit numerical schemes for diffusion mathematical models with delay // Mathematical and Computer Modelling, 2010. Vol. 52. P. 1279–1287.

Pimenov V.G., Lozhnikov A.B. Difference schemes for the numerical solution of the heat conduction equation with aftereffect // Proc. Steklov Inst. Math., 2011. Vol. 275. Suppl. 1. P. 137–148.

Samarskii A.A. Theory of difference schemes. Moscow: Nauka, 1989. 656 p. [In Russian]

Pimenov V.G. General linear methods for the numerical solution of functional-differential equations // Differential Equations. 2001, Vol. 37, no. 1. P. 116–127.

Kim A.V., Pimenov V.G. i-smooth calculus and numerical methods for functional differential equations. Moscow–Izhevsk: Regular and Chaotic Dynamics, 2004. 256 p. [In Russian]

Pimenov V.G., Tashirova E.E. Numerical methods for solving a hereditary equation of hyperbolic type // Proc. Steklov Inst. Math., 2013. Vol. 281. Suppl. 1. P. 126–136.

Lekomtsev A.V., Pimenov V.G. Convergence of the Alternating Direction Methods for the Numerical solution of a Heat Conduction Equation with Delay // Proc. Steklov Inst. Math., 2011. Vol. 272. Suppl. 1. P. 101–118.

Lekomtsev A., Pimenov V. Convergence of the scheme with weights for the numerical solution of a heat conduction equation with delay for the case of variable coefficient of heat conductivity // Appl. Math. Comput., 2015. Vol. 256. P. 83–93.

Pimenov V.G. Difference methods for the solution of the equations in partial derivatives with heredity. Ekaterinburg: Ural Federal University, 2014. 232 p. [In Russian]

Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications. Boca Raton: CRC Press, 1993. 1006 p.

Miller K., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993. 384 p.

Podlubny I. Fractional differential equations. San Diego: Acad. Press, 1999. 368 p.

Khader M.M., Danaf T.E., Hendy A.S. A computational matrix method for solving systems of high order fractional differential equations// Appl. Math. Modell., 2013. Vol. 37, no. 6. P. 4035–4050.

Pimenov V., Hendy V. Numerical studies for fractional functional differential equations with delay based on BDF-type shifted Chebyshev approximations // Abstract and Applied Analysis. 2015. Article ID 510875. P. 1–12.

Alikhanov A.A. Numerical methods of solutions boundary value problems for multi-term veriable-distributed order diffusion equations // Appl. Math. Comput., 2015. Vol. 268. P. 12–22.

Meerschaert M.M.,Tadjeran C. Finite difference approximations for two sided space fractional partial differential equations // Applied numerical mathematics, 2006. Vol. 65. P. 80–90.

Tadjeran C., Meerschaert M.M., Scheffler H.P. A second-order accurate numerical approximation for the fractional diffusion equation // Journal of Computational Physics, 2006. Vol. 213. P. 205–214.

Pimenov V.G., Hendy A.S. Numerical methods for the equation with fractional derivative on state and with functional delay on time // Bulletin of the Tambov university. Series: Natural and technical science, 2015. Vol. 20, no. 5. P. 1358–1361.

Wang H., Wang K., Sircar T. A direct O(Nlog2N) finite difference method for fractional diffusion equations // Journal of Computational Physics, 2010. Vol. 229. P. 8095–8104.

Isaacson E., Keller H.B. Analysis of Numerical Methods. New York: Wiley, 1966. 541 p.

DOI: http://dx.doi.org/10.15826/umj.2016.1.005

Article Metrics

Metrics Loading ...


  • There are currently no refbacks.