Abstract and Applied Analysis

Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order

Veyis Turut and Nuran Güzel

Full-text: Open access

Abstract

Two tecHniques were implemented, the Adomian decomposition method (ADM) and 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 has been solved and converted to power series by Adomian decomposition method (ADM), then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 746401, 12 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448861

Digital Object Identifier
doi:10.1155/2013/746401

Mathematical Reviews number (MathSciNet)
MR3035303

Zentralblatt MATH identifier
1275.65088

Citation

Turut, Veyis; Güzel, Nuran. Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 746401, 12 pages. doi:10.1155/2013/746401. https://projecteuclid.org/euclid.aaa/1393448861


Export citation

References

  • J. H. He, “Nonlinear oscillation with fractional derivative and its applications,” in Proceedings of International Conference on Vibrating Engineering, pp. 288–291, Dalian, China, 1998.
  • J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science and Technology, vol. 15, no. 2, pp. 86–90, 1999.
  • Y. Luchko and R. Gorenflo, The Initial Value Problem for Some Fractional Differential Equations with the Caputo Derivative, Series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, Berlin, Germany, 1998.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • S. Momani, “Non-perturbative analytical solutions of the space- and time-fractional Burgers equations,” Chaos, Solitons & Fractals, vol. 28, no. 4, pp. 930–937, 2006.
  • Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 27–34, 2006.
  • S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006.
  • S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248–1255, 2007.
  • Z. M. Odibat and S. Momani, “Approximate solutions for boundary value problems of time-fractional wave equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 767–774, 2006.
  • G. Domairry and N. Nadim, “Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation,” International Communications in Heat and Mass Transfer, vol. 35, no. 1, pp. 93–102, 2008.
  • G. Domairry, M. Ahangari, and M. Jamshidi, “Exact and analytical solution for nonlinear dispersive $K(m,p)$ equations using homotopy perturbation method,” Physics Letters A, vol. 368, no. 3-4, pp. 266–270, 2007.
  • G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.
  • G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
  • S. Momani, “An explicit and numerical solutions of the fractional KdV equation,” Mathematics and Computers in Simulation, vol. 70, no. 2, pp. 110–118, 2005.
  • Ph. Guillaume and A. Huard, “Multivariate Padé approximation,” Journal of Computational and Applied Mathematics, vol. 121, no. 1-2, pp. 197–219, 2000.
  • V. Turut, E. Çelik, and M. Yiğider, “Multivariate Padé approximation for solving partial differential equations (PDE),” International Journal for Numerical Methods in Fluids, vol. 66, no. 9, pp. 1159–1173, 2011.
  • V. Turut and N. Güzel, “Comparing numerical methods for solving time-fractional reaction-diffusion equations,” ISRN Mathematical Analysis, vol. 2012, Article ID 737206, 28 pages, 2012.
  • V. Turut, “Application of Multivariate padé approximation for partial differential equations,” Journal of Life Sciences, vol. 2, no. 1, pp. 17–28, 2012.
  • V. Turut and N. Güzel, “On solving partial differential eqauations of fractional order by using the variational iteration method and multivariate padé approximation,” European Journal of Pure and Applied Mathematics. Accepted.
  • S. Momani and R. Qaralleh, “Numerical approximations and Padé approximants for a fractional population growth model,” Applied Mathematical Modelling, vol. 31, no. 9, pp. 1907–1914, 2007.
  • S. Momani and N. Shawagfeh, “Decomposition method for solving fractional Riccati differential equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1083–1092, 2006.
  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  • M. Caputo, “Linear models of dissipation whose Q is almost frequency independent. Part II,” Journal of the Royal Astronomical Society, vol. 13, no. 5, pp. 529–539, 1967.
  • Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations of fractional order,” Applied Mathematical Modelling, vol. 32, no. 1, pp. 28–39, 2008.
  • A. Cuyt and L. Wuytack, Nonlinear Methods in Numerical Analysis, vol. 136 of North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam, The Netherlands, 1987.