New Iterative Method: An Application for Solving Fractional Physical Differential Equations

Abstract

The new iterative method with a powerful algorithm is developed for the solution of linear and nonlinear ordinary and partial differential equations of fractional order as well. The analysis is accompanied by numerical examples where this method, in solving them, is used without linearization or small perturbation which con firm the power, accuracy, and simplicity of the given method compared with some of the other methods.

Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method

“…Comparing the above results with the obtained results by the iterative method which is introduced in [24], affirms the simplicity and accuracy of the given method.…”

“…In Figure 1, approximate solution of the new modified recursion scheme and the iterative method [24] for α = 0.98 and the exact solution have been plotted. …”

“…Since most fractional differential equations do not have exact analytic solutions, approximate and numerical techniques, are used extensively, such as homotopy analysis method [10,16,31], homotopy perturbation method [20,21], variational iteration method [15,17,18,19], Chebyshev spectral method [12,13], c 2014 Diogenes Co., Sofia pp. 382-400 , DOI: 10.2478/s13540-014-0176-2 new iterative method [9,22,23,24], orthogonal polynomial method [33,36], Oldham-Spanier L1 method [32], Grunwald-Letnikov method [33,14], fractional Adams method [11], and several other methods [33,27,28,46].…”

An analytical treatment to fractional Fornberg–Whitham equation

“…To describe the idea of the NIM, consider the following general functional equation [10,[16][17][18][19][20]:…”

New iterative method for solving gas dynamic equation

“…The science arises from the studies of gas flows, often around or within physical bodies, especially at speeds comparable to the speed of sound or beyond, and sometimes with a significant change in gas and objects temperatures [1]. Recently, [2][3][4][5][6] summarized various NIM algorithms for various nonlinear equations including fractional differential equation. Though much achievement has been achieved, application of the NIM to Gas dynamic equation has not yet been dealt with.…”