# zbMATH — the first resource for mathematics

A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula. (English) Zbl 1148.65099
For the approximate solution of fractional partial differential equations, the authors suggest to use a method based on a (generalized) Taylor expansion of the solution. As in the classical case one can then try to compute the coefficients of the expansion, and hence the exact solution, by suitable recurrence relations based on the differential equation. This path seems to be viable only in the case of a sufficiently simple equation.
In particular, the method requires that the solution can be expanded in a series of a special form, and in a typical application it is by no means clear whether such an expansion is possible. Moreover it seems (see, e.g., Example 5.2) that the computed approximate solution does not depend on any boundary conditions. Obviously the exact solution does depend on the boundary conditions, and this discrepancy raises strong concerns about the correctness of the approach.

##### MSC:
 65R20 Numerical methods for integral equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 45K05 Integro-partial differential equations 45G10 Other nonlinear integral equations 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Full Text:
##### References:
 [1] Al-Khaled, K.; Momani, S., An approximate solution for a fractional diffusion-wave equation using the decomposition method, Appl. math. comput., 165, 2, 473-483, (2005) · Zbl 1071.65135 [2] Baeumer, B.; Meerschaert, M.M., Stochastic solutions for fractional Cauchy problems, Fractional calculus appl. anal., 4, 481-500, (2001) · Zbl 1057.35102 [3] B. Baeumer, M.M. Meerschaert, M. Kovacs, Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bull. Math. Biol., 2007, to appear, Preprint available at $$\langle$$http://www.stt.msu.edu/ mcubed/BMBseed.pdf$$\rangle$$. · Zbl 1296.92195 [4] Bildik, N.; Konuralp, A.; Bek, F.; Kucukarslan, S., Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Appl. math. comput., 172, 551-567, (2006) · Zbl 1088.65085 [5] Caputo, M., Linear models of dissipation whose Q is almost frequency independent. part II, J. roy. austral. soc., 13, 529-539, (1967) [6] del Castillo-Negrete, D.; Carreras, B.A.; Lynch, V.E., Front dynamics in reaction – diffusion systems with levy flights: a fractional diffusion approach, Phys. rev. lett., 91, 1, 018302, (2003) [7] I.H. Hassan, Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos Solitons Fract., in press, doi:10.1016/j.chaos.2006.06.040. · Zbl 1152.65474 [8] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. mech. engrg., 167, 57-68, (1998) · Zbl 0942.76077 [9] Lynch, V.E.; Carreras, B.A.; del-Castillo-Negrete, D.; Ferriera-Mejias, K.M.; Hicks, H.R., Numerical methods for the solution of partial differential equations of fractional order, J. comput. phys., 192, 406-421, (2003) · Zbl 1047.76075 [10] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space – time fractional diffusion equation, Fractional calculus appl. anal., 4, 2, 153-192, (2001) · Zbl 1054.35156 [11] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. math., 56, 80-90, (2006) · Zbl 1086.65087 [12] Momani, S., Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Appl. math. comput., 165, 2, 459-472, (2005) · Zbl 1070.65105 [13] Momani, S., Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. math. comput., 170, 2, 1126-1134, (2005) · Zbl 1103.65335 [14] Momani, S., An explicit and numerical solutions of the fractional $$K \operatorname{d} V$$ equation, Math. comput. simulation, 70, 2, 110-118, (2005) · Zbl 1119.65394 [15] Momani, S., Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos solitons fract., 28, 4, 930-937, (2006) · Zbl 1099.35118 [16] Momani, S.; Odibat, Z., Analytical solution of a time-fractional navier – stokes equation by Adomian decomposition method, Appl. math. comput., 177, 488-494, (2006) · Zbl 1096.65131 [17] Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A,, 355, 271-279, (2006) · Zbl 1378.76084 [18] S. Momani, Z. Odibat, Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl., in press, doi:10.1016/j.camwa.2006.12.037. · Zbl 1141.65398 [19] Momani, S.; Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. lett. A, 365, 345-350, (2007) · Zbl 1203.65212 [20] S. Momani, Z. Odibat, Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. Lett. A, in press, doi:10.1016/j.physleta.2007.05.083. · Zbl 1209.35066 [21] Odibat, Z.; Momani, S., Approximate solutions for boundary value problems of time-fractional wave equation, Appl. math. comput., 181, 1351-1358, (2006) · Zbl 1110.65068 [22] Z. Odibat, S. Momani, Numerical methods for nonlinear partial differential equations of fractional order, Appl. Math. Modelling, in press, doi:10.1016/j.apm.2006.10.025. · Zbl 1133.65116 [23] Odibat, Z.; Momani, S., A reliable treatment of homotopy perturbation method for klein – gordon equations, Phys. lett. A, 365, 351-357, (2007) · Zbl 1203.65213 [24] Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., in press, doi:10.1016/j.aml.2007.02.022. · Zbl 1132.35302 [25] Odibat, Z.; Shawagfeh, N., Generalized Taylor’s formula, Appl. math. comput., 186, 286-293, (2007) · Zbl 1122.26006 [26] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 [27] Schneider, W.R.; Wyess, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004 [28] Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Baeumer, B., Fractal mobile/immobile solute transport, Water resour. res., 39, 10, 1296, (2003) [29] Tadjeran, C.; Meerschaert, M.M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. comput. phys., 220, 813-823, (2007) · Zbl 1113.65124 [30] Zhang, Y.; Benson, D.A.; Meerschaert, M.M.; Scheffler, H.P., On using random walks to solve the space-fractional advection dispersion equations, J. statist. phys., 123, 1, 89-110, (2006) · Zbl 1092.82038 [31] Zhou, J.K., Differential transformation and its applications for electrical circuits, (1986), Huazhong University Press Wuhan China, (in Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.