Modified Homotopy Perturbation Method for Solving Fractional Differential Equations

Abstract

The modified homotopy perturbation method is extended to derive the exact solutions for linear (nonlinear) ordinary (partial) differential equations of fractional order in fluid mechanics. The fractional derivatives are taken in the Caputo sense. This work will present a numerical comparison between the considered method and some other methods through solving various fractional differential equations in applied fields. The obtained results reveal that this method is very effective and simple, accelerates the rapid convergence of the series solution, and reduces the size of work to only one iteration.

Local fractional analytical methods for solving wave equations with local fractional derivative

“…Many problems of applied mathematics, physics, biology, and engineering are formulated by means of fractional differential equations with arbitrary orders. [1][2][3][4][5][6][7] We mention but few, fractional diffusion and wave, 8 fractional telegraph, 9 fractional evolution, 10 fractional KdV, 11 fractional Klein-Gordon, 12 and fractional Fisher equations. 13 The fractional differential equations were considered in the sense of the Caputo derivative and the Riemann-Liouville derivative.…”

Approximate Analytical Solution for Non-Linear Reaction Diffusion Equations in Modeling of a Bacterial and Fungal Biofilter

“…A.D.Dorado et.al., [23] represents the nonlinear partial dierential equation in the modeling of bacteria and fungi biolter. In this work, the analytical expressions have been derived from the concentration of the Pollution concentration in the biolm phase using Homotopy perturbation method (HPM) [24,25,26,27]. Also, the obtained analytical results are compared with numerical solution with the help of MATLAB software [28] and the error percentage is noted.…”

New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science

“…Nonlinear evolution equation are often used to describe complex aspects of various models arising in the field of nonlinear sciences such as mathematical physics, chemical physics, chemistry, biological sciences etc. Attention from different researchers has been paid to this area in searching for new solutions to the different class of NLEEs where various powerful method are formulated such as the generalized and improved   / GG  -expansion method [1], the Jacobi elliptic-function method [2], the modified simple equation method [3,4], the sine-Gordon expansion method [5][6][7], the extended tanh method [8], the improved Bernoulli sub-equation function method [9], the rational sine-cosine method [10], the RicattiBernoulli sub-ODE method [11], the Homotopy perturbation method [12] and so on. However, in this work we aim at investigating solution of the ill-posed Boussinesq equation [13] by using the modified exp…”