A note on the existence of positive solutions of singular initial-value problem for second order differential equations

We are interested in the existence of positive solutions to initial-value problems for second-order nonlinear singular differential equations. Existence of solutions is proven under conditions which are directly applicable and considerably weaker than previously known conditions

Analyzing “Homotopy Perturbation Method for Solving Fourth-Order Boundary Value Problem”

We analyze a previous paper by S. T. Mohyud-Din and M. A. Noor (2007) and show the mistakes in it. Then, we demonstrate a more efficient method for solving fourth-order boundary value problems

A Singular Initial-Value Problem for Second-Order Differential Equations

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions

A Homotopy-Analysis Approach for Nonlinear Wave-Like Equations with Variable Coefficients

We are interested in the approximate analytical solutions of the wave-like nonlinear equations with variable coefficients. We use a wave operator, which provides a convenient way of controlling all initial and boundary conditions. The proposed choice of the auxiliary operator helps to find the approximate series solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method

Comments on “Homotopy Perturbation Method for Solving Reaction-Diffusion Equations”

The paper entitled “Homotopy perturbation method for solving reaction diffussion equation” contains some mistakes and misinterpretations along with a false conclusion. Applying the homotopy perturbation method (HPM) in an incorrect manner, the authors have drawn the false conclusion that this approach is efficient for reaction-diffusion type of equation. We show that HPM in the proposed form is not efficient in most cases, and hence, we will introduce the correct form of HPM

Homotopy Perturbation Method for Solving Wave-Like Nonlinear Equations with Initial-Boundary Conditions

The homotopy perturbation method is employed to obtain approximate analytical solutions of the wave-like nonlinear equations with initial-boundary conditions. An efficient way of choosing the auxiliary operator is presented. The results demonstrate reliability and efficiency of the method

An Analytic Structure of the Real Spectrum of Multiparameter Operator System

This article is devoted to the geometry and analytical structure of the spectrum of self-adjoins multiparameter operators