This Ebook is designed for science and engineering students taking a course in numerical methods of differential equations. Most of the material in this Ebook has its origin based on lecture courses given to advanced and early postgraduate students. This Ebook covers linear difference equations, linear multistep methods, Runge Kutta methods and finite difference methods for elliptic, parabolic and hyperbolic equations. As a course in numerical analysis it contains a variety of finite difference schemes and efficient algorithms implemented in mathematics. The mathematical modulae attached to each chapter with solutions of practical examples should help readers to understand the text and apply the methods. It is expected that the readers will find theorems with proofs and applications interesting and informative.