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 solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. (English) Zbl 1234.35300
Summary: Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2] and [0,2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko’s theorem [Y. Luchko and R. Gorenflo, Acta Math. Vietnam. 24, No. 2, 207–233 (1999; Zbl 0931.44003)], we propose some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations.
35R11Fractional partial differential equations
33E12Mittag-Leffler functions and generalizations
[1]Agrawal, O. P.: Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear dynam. 29, 145-155 (2002) · Zbl 1009.65085 · doi:10.1023/A:1016539022492
[2]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 · doi:10.1016/j.jmaa.2007.06.023
[3]Gafiychuk, V. V.; Datsko, B. Yo.: Pattern formation in a fractional reaction diffusion system, Physica A 365, 300-306 (2006)
[4]Daftardar-Gejji, V.; Bhalekar, S.: Boundary value problems for multi-term fractional differential equations, J. math. Anal. appl. 345, 754-765 (2008) · Zbl 1151.26004 · doi:10.1016/j.jmaa.2008.04.065
[5]Golbabai, A.; Sayevand, K.: Analytical modelling of fractional advection-dispersion equation defined in a bounded space domain, Math. comput. Modelling 53, 1708-1718 (2011) · Zbl 1219.76035 · doi:10.1016/j.mcm.2010.12.046
[6]Gorenflo, R.; Mainardi, F.: Random walk models for space-fractional diffusion processes, Fract. calc. Appl. anal. 1, 167-191 (1998) · Zbl 0946.60039
[7]Gorenflo, R.; Mainardi, F.: Signalling problem and Dirichlet-Neumann map for time-fractional diffusion-wave equation, Matimyas mat. 21, 109-118 (1998) · Zbl 0932.35021
[8], Applications of fractional calculus in physics (2000)
[9]Ilić, M.; Liu, F.; Turner, I.; Anh, V.: Numerical approximation of a fractional-in-space diffusion equation, Fract. calc. Appl. anal. 8, No. 3, 323-341 (2005) · Zbl 1126.26009
[10]Kelly, J. F.; Mcgough, R. J.: Analytical time-domain greens functions for power-law media, J. acoust. Soc. am. 124, 2861-2872 (2008)
[11]Liu, F.; Anh, V.; Turner, I.: Numerical solution of the space Fokker-Planck equation, J. comput. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[12]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 · doi:10.1016/j.amc.2006.08.162
[13]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
[14]Y. Luchko, Maximum principle for the fractional differential equations and its application, in: Proceedings of FDA’10, the 4th IFAC Workshop Fractional Differentiation and Applications, Badajoz, Spain, October 18-22, 2010.
[15]Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. math. Anal. appl. 374, 538-548 (2011) · Zbl 1202.35339 · doi:10.1016/j.jmaa.2010.08.048
[16]Kilbas, A. A.; Srivastava, N. H. M.; Trujillo, J. J.: Theory and applications of fractional differential equation, (2006)
[17]Magin, R.; Ortigueira, M. D.; Podlubny, I.; Trujillo, J.: On the fractional signals and systems, Signal process. 91, 350-371 (2011) · Zbl 1203.94041 · doi:10.1016/j.sigpro.2010.08.003
[18]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
[19]Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997)
[20]Mainardi, F.; Paradisi, P.: Fractional diffusive waves, J. comput. Acoust. 9, 1417-1436 (2001)
[21]Meerschaert, M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equation, J. comput. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[22]Meerschaert, M.; Nane, E.; Vellaisamy, P.: Distributed-order fractional diffusions on bounded domains, J. math. Anal. appl. 379, 216-228 (2011) · Zbl 1222.35204 · doi:10.1016/j.jmaa.2010.12.056
[23]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. rep. 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[24]Metzler, R.; Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. phys. A: math. Gen. 37, 161-208 (2004) · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01
[25]Park, Y. S.; Baik, J. Jin.: Analytical solution of the advection-diffusion equation for a ground-level finite area source, Atmos. environ. 42, 9063-9069 (2008)
[26]Podlubny, I.: Fractional differential equations, (1999)
[27]Povstenko, Y. Z.: Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation, Int. J. Solids struct. 44, 2324-2348 (2009) · Zbl 1121.74022 · doi:10.1016/j.ijsolstr.2006.07.008
[28]Povstenko, Y. Z.: Signaling problem for time-fractional diffusion-wave equation in a half-plane, Fract. calc. Appl. anal. 11, 329-352 (2008) · Zbl 1157.45004
[29]Povstenko, Y. Z.: Fundamental solution to three dimensional diffusion-wave equation and associated diffusive stresses, Chaos solitons fractals 36, 961-972 (2008) · Zbl 1131.74022 · doi:10.1016/j.chaos.2006.07.031
[30]Povstenko, Y. Z.: Analysis of fundamental solutions to fractional diffusion-wave equation in polar coordinates, Sci. issues Ján długosz univ. Czestochowa,math·14,97-104(2009)
[31]Povstenko, Y. Z.: Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry, Nonlinear dynam. 55, 593-605 (2010) · Zbl 1189.35169 · doi:10.1007/s11071-009-9566-0
[32]Y.Z. Povstenko, Theories of thermoelasticity based on space-time-fractional Cattaneo-type equations, in: Proceedings of FDA’10, the 4th IFAC Workshop Fractional Differentiation and Its Applications, Badajoz, Spain, October 18-20, 2010.
[33]Qi, H.; Jiang, X.: Solutions of the space-time fractional Cattaneo diffusion equation, Physica A 390, 1876-1883 (2011) · Zbl 1225.35253 · doi:10.1016/j.physa.2011.02.010
[34]Roop, J. P.: Computational aspects of FEM approximation of fractional advection dispersion equation on boundary domains in R2, J. comput. Appl. math. 193, No. 1, 243-268 (2006) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[35]Shakhmurov, V. B.; Shahmurova, A.: Nonlinear abstract boundary value problems modelling atmospheric dispersion of pollutants, Nonlinear anal. Real world appl. 11, 932-951 (2010) · Zbl 1186.35234 · doi:10.1016/j.nonrwa.2009.01.037
[36]Shen, S.; Liu, F.; Anh, V.: Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order, J. appl. Math. comput. 28, 147-164 (2008) · Zbl 1157.65520 · doi:10.1007/s12190-008-0084-x
[37]Shen, S.; Liu, F.; Anh, V.; Turner, I.: The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation, IMA J. Appl. math. 73, 850-872 (2008) · Zbl 1179.37073 · doi:10.1093/imamat/hxn033
[38]Shen, S.; Liu, F.; Anh, V.: Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numer. algorithms 56, No. 3, 383-403 (2011) · Zbl 1214.65046 · doi:10.1007/s11075-010-9393-x
[39]Stojanovic, M.: Numerical method for solving diffusion-wave phenomena, J. comput. Appl. math. 235, No. 10, 3121-3137 (2011) · Zbl 1213.65133 · doi:10.1016/j.cam.2010.12.010
[40]Treeby, B. E.; Cox, B. T.: Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian, J. acoust. Soc. am. 127, 2741-2748 (2010)
[41]Uchaikin, V. V.: Method of fractional derivatives, (2008)
[42]Wang, K.; Wang, H.: A fast characteristic finite difference method for fractional advection-diffusion equations, Adv. water resour. 34, 810-816 (2011)
[43]Yang, Q.; Liu, F.; Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. math. Model. 34, 200-218 (2010) · Zbl 1185.65200 · doi:10.1016/j.apm.2009.04.006
[44]Zhang, Y.; Benson, D. A.; Reeves, D. M.: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications, Adv. water resour. 32, 561-581 (2009)
[45]Zhou, Y.; Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal. Real world appl. 11, 4465-4475 (2010)