Transformation Techniques to Find the Analytical and Numerical Solutions of Ordinary Differential Equations (ODES)

Abstract

The analytical solutions are very rare in the field of differential equation. However, in Physics and engineering sciences it is always preferred to find analytical solution. Analytical solutions are exact and error free solutions supported with a logical proof. The motivation is to find the analytical solution of second order ordinary differential equations (ODEs). In this thesis we have presented some methods to find the analytical solutions of different forms of differential equations (DEs). The chapter wise summary of this thesis is as under.

The first chapter presents brief introduction to the problem of study, its objectives and methodology used to achieve the goal.

In the second chapter we have presented a comprehensive form of literature review related to the studying problem understudy. The current literature is discussed in detail and results are summarized to draw conclusions and further directions.

The third chapter is preliminaries, showing notions and basic definitions which are associated with this study.

In chapter four, we have presented our main results with their proofs for solving ODEs by using a transformation technique. We have implemented the results in the form of solution methods to provide general solutions to practical examples. We have also solved some problems to explain the solution procedure and its implementation. The numerical verification and comparison of each problem is also included in this chapter. In chapter five, we have proposed an idea to solve second order differential equation with polynomial coefficients by using Laplace transformation. The operator form of solution is presented and discussed in this chapter.