Some Analytic Methods for Solving Higher Order Nonlinear Boundary and Initial Value Problems

Abstract

Nonlinear phenomenon plays vital role in engineering and applied sciences. Higher order nonlinear boundary value and initial value problems are known to describe a large variety of phenomena. We are mainly concerned with finding the analytic solution of boundary value and initial value problems of the type T ( u ) = 0, subject to some boundary and initial conditions, where T is any differential or integral operator. We study and develop some analytic methods for solving higher order nonlinear boundary value and initial value problems. We propose several modifications in variation of parameters method for finding the approximate solution of several higher order nonlinear boundary value, initial value, parametric boundary value, higher dimensional and system of nonlinear Volterra integro differential problems. We apply the exp-function method to construct generalized soliton, travelling wave and periodic solution of several higher order nonlinear partial differential equations. We apply the exp-function method for finding the exact solution of some higher order nonlinear boundary value problems. All the suggested techniques are compared with well known classical techniques and are found quite efficient, well suited and reliable for the solution of higher order nonlinear boundary and initial value problems of diversified nature.