The Variational Iteration Transform Method for Solving the Time-Fractional Fornberg–Whitham Equation and Comparison with Decomposition Transform Method

In this article, modified techniques, namely the variational iteration transform and Shehu decomposition method, are implemented to achieve an approximate analytical solution for the time-fractional Fornberg–Whitham equation. A comparison is made between the results of the variational iteration transform method and the Shehu decomposition method. The solution procedure reveals that the variational iteration transform method and Shehu decomposition method is effective, reliable and straightforward. The variational iteration transform methods solve non-linear problems without using Adomian’s polynomials and He’s polynomials, which is a clear advantage over the decomposition technique. The solutions achieved are compared with the corresponding exact result to show the efficiency and accuracy of the existing methods in solving a wide variety of linear and non-linear problems arising in various science areas.

Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems.

A Friendly Iterative Technique for Solving Nonlinear Integro-Differential and Systems of Nonlinear Integro-Differential Equations

In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential and systems of nonlinear integro-differential equations (IDEs). The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, it does not require discretization, linearization or any restrictive assumption of any form in providing analytical or approximate solution to linear and nonlinear equations. Also, this technique does not require calculating Adomian’s polynomials, Lagrange’s multiplier values or equating the terms of equal powers of the impeding parameter which need more computational procedures and time. These advantages make it reliable and its efficiency is demonstrated with numerical examples.

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.

A Simple Modification of Homotopy Perturbation Method for the Solution of Blasius Equation in Semi-Infinite Domains

A simple modification of the homotopy perturbation method is proposed for the solution of the Blasius equation with two different boundary conditions. Padé approximate is used to deal with the boundary condition at infinity. The results obtained from the analytical method are compared to Howarth’s numerical solution and fifth order Runge-Kutta Fehlberg method indicating a very good agreement. The proposed method is a simple and reliable modification of homotopy perturbation method, which does not require the existence of a small parameter, linearization of the equation, or computation of Adomian’s polynomials.

A Comparison between Adomian’s Polynomials and He’s Polynomials for Nonlinear Functional Equations

We will compare the standard Adomian decomposition method and the homotopy perturbation method applied to obtain the solution of nonlinear functional equations. We prove analytically that the two methods are equivalent for solving nonlinear functional equations. In Ghorbani (2009), Ghorbani presented a new definition which he called as He’s polynomials. In this paper, we also show that He’s polynomials are only the Adomian polynomials.