Error Analysis and Stability of Numerical Schemes for Initial Value problems “IVP’s”

Most of initial value problems are natural phenomena written in the language of mathematics. Solving these initial value problems is one of the most challenging fields in mathematics, because of the mathematicians’ continuous desire of exactness. This work focuses mainly on developing algorithms and programs to construct higher order Taylor’s methods for approximating the solution of first order initial value problems, systems of first order initial value problems and higher order initial value problems. Moreover, it concentrates on studying error and stability of numerical methods for solving initial value problems. For this purpose, we developed programs to find the error amplification functions of Taylor’s and Runge-Kutta methods and to plot boundaries of stability regions for these methods and other methods. We concluded that with the programs we developed, higher order Taylor’s methods could be a good choice for approximating solutions of a wide range of initial value problems.