Exact and Approximate to Exact Methods for Solution Linear Boundary Value Problems Using Laplace Transform.

It is a new method, a mixture of numerical and exact methods, each of which has a role to obtain the solution. It is known that the Laplace transforms method gives a closed-form for initial value problems but in the present study, we were able to use it with the aid of high-accurate numerical methods to solve linear boundary value problems. The novelty of the present method that it is converted Linear Boundary Value Problems to initial Value Problems using accurate numerical methods and then uses Laplace transforms method to find approximate to the exact solution. Approximate to Exact Method (AEM) is an algorithm, with a very strong accuracy that approaches the exact solution because it is a mixture of numerical methods of very high accuracy with a closed-form method. The uniqueness, convergence, and stability of the new technique are verified and tested by comparisons with a fourth-order accurate finite difference (FOFDM) solution.