On theRF-pair operations for the exact solution of some classes of nonlinear Volterra integral equations

P Darania,M Hadizadeh

doi:10.1155/mpe/2006/97020

Abstract

We study the exact solution of some classes of nonlinear integral equations by series of some invertible transformations andRF-pair operations. We show that this method applies to several classes of nonlinear Volterra integral equations as well and give some useful invertible transformations for converting these equations into differential equations of Emden-Fowler type. As a consequence, we analyze the effect of the proposed operations on the exact solution of the transformed equation in order to find the exact solution of the original equation. Some applications of the method are also given. This approach is effective to find a great number of new integrable equations, which thus far, could not be integrated using the classical methods.

Highlights

Many problems in mathematical physics, contact problems in the theory of elasticity, and mixed boundary value problems are transformed into nonlinear Volterra integral equations
Due to the fact that this equation appears in many branches of physics and engineering, for example, in stellar dynamics, quantum mechanics, and fluid dynamics, there is great interest in finding both exact or closed form approximate solutions
By means of appropriate transformations of RF-pair operations, some classes of nonlinear integral equations are transformed into some solvable differential equations of Emden-Fowler class

Summary

Introduction

Many problems in mathematical physics, contact problems in the theory of elasticity, and mixed boundary value problems are transformed into nonlinear Volterra integral equations (see [2, 8, 12]). The theory and application of integral equations is an important subject within applied mathematics. Note that, closed form solutions of Emden-Fowler equations are very scarce and the search for the conservation which can in many respects shed light on the physical mechanism of the dynamical problem has its full vindication. By means of appropriate transformations of RF-pair operations, some classes of nonlinear integral equations are transformed into some solvable differential equations of Emden-Fowler class. This approach is effective to find a great number of new integrable equations which far, could not be integrated using the classical methods

Preliminaries

Some applications and further comments

Conclusion