Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space. (English) Zbl 1223.35112

Jiang, Wei; Lin, Yingzhen

Commun. Nonlinear Sci. Numer. Simul. 16, No. 9, 3639-3645 (2011).

Summary: The reproducing kernel theorem is used to solve the time-fractional telegraph equation with Robin boundary conditions. The time-fractional derivative is considered in the Caputo sense. We discuss and derive the exact solution in the form of series with easily computable terms in the reproducing kernel space.

MSC:

35C20	Asymptotic expansions of solutions to PDEs
35L20	Initial-boundary value problems for second-order hyperbolic equations
35R11	Fractional partial differential equations

Keywords:

Captuo fractional derivative; reproducing kernel space; Robin boundary conditions

References:

[1]	Chen, Chang-Ming; Liu, F.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J comput phys, 227, 886-897, (2007) · Zbl 1165.65053
[2]	Chen, Chang-Ming; Liu, F.; Burrage, K., Finite difference methods and a Fourier analysis for the fractional reaction – subdiffusion equation, Appl math comput, 198, 754-769, (2008) · Zbl 1144.65057
[3]	Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of difference methods for the space – time fractional advection – diffusion equation, Appl math comput, 191, 12-20, (2007) · Zbl 1193.76093
[4]	Zhang, P.; liu, F., Implicit difference approximation for the time fractional diffusion equation, J appl math comput, 22, 3, 87-99, (2006) · Zbl 1140.65094
[5]	Liu, Q.; Liu, F.; Turner, I.; Anh, V., Approximation of levy – feller advection – dispersion process by random walk and finite difference method, J comput phys, 222, 57-70, (2007) · Zbl 1112.65006
[6]	Yuste, S.B.; Acedo, L., An explicit finite difference method and a new von neumman-type stability analysis for fractional diffusion equations, SIAM J numer anal, 42, 5, 1862-1874, (2005) · Zbl 1119.65379
[7]	Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., Application of a fractional advection – dispersion equation, Water resour res, 36, 6, 1403-1412, (2000)
[8]	Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., The fractional-order governing equation of levy motion, Water resour res, 36, 6, 1413-1423, (2000)
[9]	So, F.; Liu, K.L., A study of the subdiffusive fractional fokker – plank equation of bistable systems, Physica A, 331, 378-390, (2004)
[10]	Me1zler, R.; Klafter, J., The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys rep, 339, 1-77, (2000)
[11]	Tan, Wenchang; Fu, Chaoqi; Fu, Ceji; Xie, Wenjun; Cheng, Heping, Subdiffusion model for calcium spark in cardiac myocytes, Appl phys lett, 91, 183901, (2007)
[12]	Eckstein, E.C.; Goldstein, J.A.; Leggas, M., The mathematics of suspensions: Kac walks and asymptotic analyticity, Electron J differ eqs, 3, 39-50, (1999) · Zbl 0963.76090
[13]	Eckstein, E.C.; Leggas, M.; Ma, B.; Goldstein, J.A., Linking theory and measurements of tracer particle position in suspension flows, Proc ASME FEDSM, 251, 1-8, (2000)
[14]	Orsingher, E.; Beghin, L., Time-fractional telegraph equation and telegraph processes with Brownian time, Probab theory related fields, 128, 141-160, (2004) · Zbl 1049.60062
[15]	Beghin, L.; Orsingher, E., The telegraph process stopped at stable-distributed times connection with the fractional telegraph equation, Fract calc appl anal, 6, 187-204, (2003) · Zbl 1083.60039
[16]	Chen, J.; Liu, F.; Anh, V., Analytical solution for the time-fractional telegraph equation by the method of separating variables, J math anal appl, 338, 1364-1377, (2008) · Zbl 1138.35373
[17]	Momani, S., Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl math comput, 170, 1126-1134, (2005) · Zbl 1103.65335
[18]	Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[19]	Cui, Minggen; Lin, Yingzhen, Nonlinear numerical analysis in the reproducing kernel space, (2009), Nova Science Publisher New York · Zbl 1165.65300
[20]	Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space – time fractional advection – diffusion equation, Appl math comput, 191, 12-20, (2007) · Zbl 1193.76093