A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations

DOI10.1016/j.apm.2009.06.025zbMath1185.65139OpenAlexW1965027360MaRDI QIDQ967844

Hang Xu, Shaher Momani, Zaid M. Odibat

Publication date: 2 May 2010

Published in: Applied Mathematical Modelling (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.apm.2009.06.025

zbMATH Keywords

homotopy analysis methodfractional differential equationcaputo fractional derivative

Mathematics Subject Classification ID

Fractional derivatives and integrals (26A33) Numerical methods for ordinary differential equations (65L99) Fractional ordinary differential equations (34A08)

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Cites Work

• Unnamed Item
• The multistage homotopy-perturbation method: a powerful scheme for handling the Lorenz system
• Series solutions of non-linear Riccati differential equations with fractional order
• Homotopy perturbation method for nonlinear partial differential equations of fractional order
• Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation
• Homotopy analysis method for fractional IVPs
• Application of generalized differential transform method to multi-order fractional differential equations
• Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method
• Variational iteration method for solving nonlinear boundary value problems
• A second-order accurate numerical method for the two-dimensional fractional diffusion equation
• Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations
• A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula
• Numerical approach to differential equations of fractional order
• A kind of approximation solution technique which does not depend upon small parameters. II: An application in fluid mechanics
• On the homotopy analysis method for nonlinear problems.
• Numerical methods for the solution of partial differential equations of fractional order.
• Numerical solutions for systems of fractional differential equations by the decomposition method
• Application of variational iteration method to nonlinear differential equations of fractional order
• An approximate solution technique not depending on small parameters: A special example
• Numerical comparison of methods for solving linear differential equations of fractional order
• Series solutions of nano boundary layer flows by means of the homotopy analysis method
• A generalized differential transform method for linear partial differential equations of fractional order
• Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method
• Numerical methods for nonlinear partial differential equations of fractional order
• Non-perturbative analytical solutions of the space- and time-fractional Burgers equations
• Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method
• Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order
• An explicit and numerical solutions of the fractional KdV equation
• Finite difference approximations for two-sided space-fractional partial differential equations

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