Jamshad Ahmad

An Efficient Algorithm for Some Highly Nonlinear Fractional PDEs in Mathematical Physics

In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and subsequently Reduced Differential Transform Method (RDTM) is applied on the transformed system of linear and nonlinear time-fractional PDEs. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs and hence can be extended to other complex problems of diversified nonlinear nature.

Analytic solutions of the Helmholtz and Laplace equations by using local fractional derivative operators

Abstract In this paper we develop analytical solutions for the Helmholtz and Laplace equations involving local fractional derivative operators. We implement the local fractional decomposition method (LFDM) for finding the exact solutions. The iteration procedure is based upon the local fractional derivative sense. The numerical results, whichwe present in this paper, show that the methodology used provides an efficient and simple tool for solving fractal phenomena arising in mathematical physics and engineering. Several illustrative examples are also provided.

Differential Transform Method with Complex Transforms to Some Nonlinear Fractional Problems in Mathematical Physics

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.