Journal ArticleDOI
Solving nonlinear fractional partial differential equations using the homotopy analysis method
TLDR
In this paper, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations (FPDE) with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives, and the results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.Abstract:
In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m,n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010read more
Citations
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Journal ArticleDOI
The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions
Rahmat Ellahi,Rahmat Ellahi +1 more
TL;DR: In this article, the authors examined the magnetohydrodynamic flow of non-Newtonian nanofluid in a pipe and derived explicit analytical expressions for the velocity field, the temperature distribution and nano concentration.
Journal ArticleDOI
A new jacobi operational matrix: an application for solving fractional differential equations
TL;DR: In this paper, the shifted Jacobi operational matrix (JOM) of fractional derivatives was derived and applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs).
Journal ArticleDOI
A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations
Ali H. Bhrawy,Mahmoud A. Zaky +1 more
TL;DR: This paper proposes and analyzes an efficient operational formulation of spectral tau method for multi-term time-space fractional differential equation with Dirichlet boundary conditions using shifted Jacobi operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives.
Book
Numerical Methods for Partial Differential Equations
TL;DR: In this paper, the authors present an extension of Matrix Stability Analysis to the case of Parabolic Equations (see Section 2.1.1). But they do not discuss the use of the concept of a simple explicit method.
Journal ArticleDOI
An approximation algorithm for the solution of the nonlinear lane-emden type equations arising in astrophysics using hermite functions collocation method
TL;DR: A collocation method for solving some well-known classes of Lane–Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain based on a Hermite function collocation (HFC) method is proposed.
References
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Book
An Introduction to the Fractional Calculus and Fractional Differential Equations
Kenneth S. Miller,Bertram Ross +1 more
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Book
Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Book
Solitons, Nonlinear Evolution Equations and Inverse Scattering
M. A. Ablowitz,Peter A. Clarkson +1 more
TL;DR: In this article, the authors bring together several aspects of soliton theory currently only available in research papers, including inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multidimensional space, and the ∂ method.
Book
Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
TL;DR: In this article, the authors present a method for computing fractional derivatives of the Fractional Calculus using the Laplace Transform Method and the Fourier Transformer Transform of fractional Derivatives.