Fractional Differential Equations

Hydrodynamic and hydromagnetic stability

The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi-step DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

https://hal.archives-ouvertes.fr/hal-00430504/document

A multi-step differential transform method and application to non-chaotic or chaotic systems

A new Rabotnov fractional‐exponential function‐based fractional derivative for diffusion equation under external force

We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.

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Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets

Differential transform method for solving the linear and nonlinear Klein–Gordon equation

In this paper, we propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schrodinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.

Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations

This paper applies He's variational iteration method for solving nonlinear singular boundary value problems. The solution process is illustrated and various physically relevant results are obtained. Comparison of the obtained results with exact solutions shows that the method used is an effective and highly promising method for treating various classes of both linear and nonlinear singular boundary value problems.

K. Aruna

Papers

Differential transform method for solving the linear and nonlinear Klein–Gordon equation

Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations

He's variational iteration method for treating nonlinear singular boundary value problems

Solution of singular two-point boundary value problems using differential transformation method

Differential transform method for solving linear and non-linear systems of partial differential equations