Extension of Solutions to some Nonlinear Differential Equations

Abstract

Differential equations are necessary in scientific modeling of physical problems, newlinewhich find relevance in almost every sphere of human endeavor from Agricultural newlineSciences, Engineering, Medical Sciences, Physical Sciences to Social Sciences. newlineThe subject Differential Equations is a well established part of mathematics and newlineits systematic development goes back to the early days of the development of Calculus. newlineMany recent advances in mathematics, paralleled by a renewed and flourishing newlineinteraction between mathematics, the sciences, and engineering, have again shown that newlinemany phenomena in the applied sciences, modeled by differential equations will yield newlinesome mathematical explanation of these phenomena. newlineFor finding exact solutions of nonlinear Initial Value Problems (IVPs) is a goal newlinefor mathematicians, engineers, and scientists, and it plays an important role in real world newlineapplications. In recent years, first and second order nonlinear IVPs were considered by newlinemany authors. For instance, Adomian Decomposition Method (ADM) is used to solve newlinenonlinear differential equations such as Dung-Vanderpole equation, solved nonlinear newlineIVPs by the Laplace Adomian decomposition method (LADM), obtained approximate newlinesolutions by the method of differential transforms (DTM), and the variation iteration newlinemethods (VIM) were used by many authors. newline2. OBJECTIVES : newlineThe main objectives are carried out as follows : newline1. To study the existence and uniqueness of solution of nonlinear differential equations newlineand its extension of solutions. newlineiii newline2. To discuss nonlinear ordinary differential equations for their different behavior of the newlinesolutions. newline3. To analyze some applications of nonlinear ordinary differential equations studied in newlinethe present work to some concrete problem of the other areas of mathematics. newline4. To develop the method for existence and uniqueness of solutions by using fixed points newlinetheorem. newline5. To study the existence and uniqueness of solutions of nonlinear ordinary differential newlineequations in various aspects. newline3. BRIEF DESCRIPTION OF THE THESIS