# Solving Nonlinear Boundary Value Problems Using the Homotopy Analysis Method.

#### Abstract

Analytical solutions of differential equations are very important for all researchers from different discipline. Obtaining such solutions is difficult in most cases, especially if the differential equation is nonlinear. One of the mostly used methods are the series methods, where the solution is represented as an infinite series. Different methods are available to evaluate the terms of this series. These methods include the well-known Taylor series method, the Adomian decomposition method, the Homotopy iteration method, and the Homotopy analysis method.

In this thesis we give a survey of the different series methods available to solve initial and boundary value problems. The methods to be presented are the Taylor series method, the Adomina decomposition method, and the Homotopy analysis method. The main features of each method will be presented and the error analysis will be discussed as well. For the Homotopy analysis method, the error is controlled by introducing the parameter known as h, then the error is controlled by monitoring the value of the solution at a specific point for different values of h. This produces what is known as the h curve. The mathematical foundation of this method is not very well established, and the method will not work at all times. The error for the Taylor series and the Adomian decomposition method is controlled by adding more terms to the series solution which might be costly and difficult to calculate especially if the differential equation is nonlinear.

In this study we will show that the error can be controlled by other means. A modified Taylor series method has been developed and will be discussed. The method is based on controlling the error through different choices of the point of expansion.

The mathematical foundation of the method and application of the method to differential equations with singularities and eigenvalue problems will be presented.