×

zbMATH — the first resource for mathematics

Multivariate Padé approximation for solving nonlinear partial differential equations of fractional order. (English) Zbl 1275.65088
Summary: Two techniques are implemented, the Adomian decomposition method (ADM) and the multivariate Padé approximation (MPA), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation is solved and converted to a power series by the Adomian decomposition method (ADM), then the power series solution of the fractional differential equation is put into multivariate Padé series. Finally, numerical results are compared and presented in tables and figures.

MSC:
65N99 Numerical methods for partial differential equations, boundary value problems
35C10 Series solutions to PDEs
35R11 Fractional partial differential equations
41A21 Padé approximation
35G20 Nonlinear higher-order PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] He, J. H., Nonlinear oscillation with fractional derivative and its applications, Proceedings of International Conference on Vibrating Engineering
[2] He, J. H., Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science and Technology, 15, 2, 86-90, (1999)
[3] Luchko, Y.; Gorenflo, R., The Initial Value Problem for Some Fractional Differential Equations with the Caputo Derivative. The Initial Value Problem for Some Fractional Differential Equations with the Caputo Derivative, Series A08-98, Fachbreich Mathematik und Informatik, (1998), Berlin, Germany: Freic Universitat Berlin, Berlin, Germany
[4] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198, xxiv+340, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[5] Momani, S., Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos, Solitons & Fractals, 28, 4, 930-937, (2006) · Zbl 1099.35118
[6] Odibat, Z. M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 27-34, (2006) · Zbl 1401.65087
[7] Momani, S.; Odibat, Z., Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Applied Mathematics and Computation, 177, 2, 488-494, (2006) · Zbl 1096.65131
[8] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, 31, 5, 1248-1255, (2007) · Zbl 1137.65450
[9] Odibat, Z. M.; Momani, S., Approximate solutions for boundary value problems of time-fractional wave equation, Applied Mathematics and Computation, 181, 1, 767-774, (2006) · Zbl 1148.65100
[10] Domairry, G.; Nadim, N., Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation, International Communications in Heat and Mass Transfer, 35, 1, 93-102, (2008)
[11] Domairry, G.; Ahangari, M.; Jamshidi, M., Exact and analytical solution for nonlinear dispersive \(K(m, p)\) equations using homotopy perturbation method, Physics Letters A, 368, 3-4, 266-270, (2007) · Zbl 1209.65108
[12] Adomian, G., A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135, 2, 501-544, (1988) · Zbl 0671.34053
[13] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method. Solving Frontier Problems of Physics: The Decomposition Method, Fundamental Theories of Physics, 60, xiv+352, (1994), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0802.65122
[14] Momani, S., An explicit and numerical solutions of the fractional KdV equation, Mathematics and Computers in Simulation, 70, 2, 110-118, (2005) · Zbl 1119.65394
[15] Guillaume, Ph.; Huard, A., Multivariate Padé approximation, Journal of Computational and Applied Mathematics, 121, 1-2, 197-219, (2000) · Zbl 1090.41505
[16] Turut, V.; Çelik, E.; Yiğider, M., Multivariate Padé approximation for solving partial differential equations (PDE), International Journal for Numerical Methods in Fluids, 66, 9, 1159-1173, (2011) · Zbl 1219.65116
[17] Turut, V.; Güzel, N., Comparing numerical methods for solving time-fractional reaction-diffusion equations, ISRN Mathematical Analysis, 2012, (2012) · Zbl 1250.65128
[18] Turut, V., Application of Multivariate padé approximation for partial differential equations, Batman University Journal of Life Sciences, 2, 1, 17-28, (2012)
[19] Turut, V.; Güzel, N., On solving partial differential eqauations of fractional order by using the variational iteration method and multivariate padé approximation · Zbl 1413.65401
[20] Momani, S.; Qaralleh, R., Numerical approximations and Padé approximants for a fractional population growth model, Applied Mathematical Modelling, 31, 9, 1907-1914, (2007) · Zbl 1167.45300
[21] Momani, S.; Shawagfeh, N., Decomposition method for solving fractional Riccati differential equations, Applied Mathematics and Computation, 182, 2, 1083-1092, (2006) · Zbl 1107.65121
[22] Oldham, K. B.; Spanier, J., The Fractional Calculus, xiii+234, (1974), New York, NY, USA: Academic Press, New York, NY, USA
[23] Caputo, M., Linear models of dissipation whose Q is almost frequency independent. Part II, Journal of the Royal Astronomical Society, 13, 5, 529-539, (1967)
[24] Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling, 32, 1, 28-39, (2008) · Zbl 1133.65116
[25] Cuyt, A.; Wuytack, L., Nonlinear Methods in Numerical Analysis. Nonlinear Methods in Numerical Analysis, North-Holland Mathematics Studies, 136, x+278, (1987), Amsterdam, The Netherlands: North-Holland Publishing, Amsterdam, The Netherlands
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.