Homotopy analysis method for solving Abel differential equation of fractional order

In this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.     

General fractional variational problem depending on indefinite integrals

In this report, we consider two kind of general fractional variational problem depending on indefinite integrals include unconstrained problem and isoperimetric problem. These problems can have multiple dependent variables, multiorder fractional derivatives, multiorder integral derivatives and boundary conditions. For both problems, we obtain the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Also, we apply the Rayleigh-Ritz method for solving the unconstrained general fractional variational problem depending on indefinite integrals. By this method, the given problem is reduced to the problem for solving a system of algebraic equations using shifted Legendre polynomials basis functions. An approximate solution for this problem is obtained by solving the system. We discuss the analytic convergence of this method and finally by some examples will be showing the accurately and applicability for this technique.     

Solitary pattern solutions for fractional Zakharov-Kuznetsov equations with fully nonlinear dispersion

The fractional Zakharov-Kuznetsov equations are increasingly used in modeling various kinds of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. This has led to a significant interest in the study of these equations. In this work, solitary pattern solutions of fractional Zakharov-Kuznetsov equations are investigated by means of the homotopy perturbation method with consideration of Jumarie’s derivatives. The effects of fractional derivatives for the systems under consideration are discussed. Numerical results and a comparison with exact solutions are presented.     

Analytical modelling of fractional advection–dispersion equation defined in a bounded space domain

The fractional advection–dispersion equation (FADE) known as its non-local dispersion, is used in groundwater hydrology and has been proven to be a reliable tool to model the transport of passive tracers carried by fluid flow in a porous media. In this paper, compact structures of FADE are investigated by means of the homotopy perturbation method with consideration of a promising scheme to calculate nonlinear terms. The problems are formulated in the Jumarie sense. Analytical and numerical results are presented.     

On dual Bernstein polynomials and stochastic fractional integro-differential equations

In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro-differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.     

Solving a fourth‐order fractional diffusion‐wave equation in a bounded domain by decomposition method

In this article, the Adomian decomposition method has been used to obtain solutions of fourth‐order fractional diffusion‐wave equation defined in a bounded space domain. The fractional derivative is described in the Caputo sense. Convergence of the method has been discussed with some illustrative examples. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

This study presents a robust modification of Chebyshev ? ‐weighted Crank–Nicolson method for analyzing the sub‐diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub‐fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of sub‐diffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.     

Reanalysis of an open problem associated with the fractional Schrödinger equation

It was recently shown that there are some difficulties in the solution method proposed by Laskin for obtaining the eigenvalues and eigenfunctions of the one-dimensional time-independent fractional Schrödinger equation with an infinite potential well encountered in quantum mechanics. In fact, this problem is still open. We propose a new fractional approach that allows overcoming the limitations of some previously introduced strategies. In deriving the solution, we use a method based on the eigenfunction of the Weyl fractional derivative. We obtain a solution suitable for computations in a closed form in terms of Mittag–Leffler functions and fractional trigonometric functions. It is a simple extension of the results previously obtained by Laskin et al.