zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Analytical solution for the time-fractional telegraph equation by the method of separating variables. (English) Zbl 1138.35373
Summary: A method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE). We discuss and derive the analytical solution of the TFTE with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin boundary conditions.

35L70Nonlinear second-order hyperbolic equations
35S05General theory of pseudodifferential operators
35L20Second order hyperbolic equations, boundary value problems
Full Text: DOI
[1] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) · Zbl 0292.26011
[2] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[3] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[4] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[5] Schneider, W. R.; Wyss, W.: Fractional diffusion and wave equations, J. math. Phys. 30, 134-144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578
[6] Mainardi, F.: The fundamental solutions for the fractional diffusion -- wave equation, Appl. math. Lett. 9, 23-28 (1996) · Zbl 0879.35036 · doi:10.1016/0893-9659(96)00089-4
[7] Anh, V. V.; Leonenko, N. N.: Spectral analysis of fractional kinetic equations with random data, J. stat. Phys. 104, 1349-1387 (2001) · Zbl 1034.82044 · doi:10.1023/A:1010474332598
[8] Anh, V. V.; Leonenko, N. N.: Renormalization and homogenization of fractional diffusion equations with random data, Probab. theory related fields 124, 381-408 (2002) · Zbl 1031.60043 · doi:10.1007/s004400200217
[9] Liu, F.; Anh, V.; Turner, I.: Numerical solution of the space fractional Fokker -- Planck equation, J. comput. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[10] Lin, R.; Liu, F.: Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear anal. 66, 856-869 (2007) · Zbl 1118.65079 · doi:10.1016/j.na.2005.12.027
[11] Liu, F.; Shen, S.; Anh, V.; Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, Anziam j. 46(E), 488-504 (2005)
[12] Liu, Q.; Liu, F.; Turner, I.; Anh, V.: Approximation of the Lévy -- Feller advection-dispersion process by random walk and finite difference method, J. phys. Comput. 222, 57-70 (2007) · Zbl 1112.65006 · doi:10.1016/j.jcp.2006.06.005
[13] Meerschaert, M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations, J. comput. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[14] Zhuang, P.; Liu, F.: Implicit difference approximation for the time fractional diffusion equation, J. appl. Math. comput. 22, 87-99 (2006) · Zbl 1140.65094 · doi:10.1007/BF02832039
[15] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space -- time advection -- diffusion equation, Appl. Math. Comput. (2007), in press · Zbl 1193.76093
[16] Eckstein, E. C.; Goldstein, J. A.; Leggas, M.: The mathematics of suspensions: Kac walks and asymptotic analyticity, Electron. J. Differential equations 3, 39-50 (1999) · Zbl 0963.76090 · emis:journals/EJDE/conf-proc/03/e1/abstr.html
[17] 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)
[18] 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 · doi:10.1007/s00440-003-0309-8
[19] Beghin, L.; Orsingher, E.: The telegraph process stopped at stable-distributed times connection with the fractional telegraph equation, Fract. calc. Appl. anal. 2, 187-204 (2003) · Zbl 1083.60039
[20] Momani, S.: Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. math. Comput. 170, 1126-1134 (2005) · Zbl 1103.65335 · doi:10.1016/j.amc.2005.01.009
[21] Dimovski, I. H.: Convolution calculus, (1982) · Zbl 0517.44012
[22] Luchko, Y.; Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives, Acta math. Vietnam 24, 207-233 (1999) · Zbl 0931.44003
[23] Daftardar-Gejji, V.; Jafari, H.: Boundary value problems for fractional diffusion -- wave equation [J], Aust. J. Math. anal. Appl. 3, No. 1 (2006) · Zbl 1093.35041
[24] Agrawal, O. P.: Solution for a fractional diffusion -- wave equation defined in a bounded domain, J. nonlinear dynam. 29, 145-155 (2002) · Zbl 1009.65085 · doi:10.1023/A:1016539022492
[25] Group, Computational Mathematics: Partial differential equation, (1979)