Hybrid solution of two-point linear boundary value problems

Abstract

We discuss a general approach to solve linear two points boundary value problems (BV) for ordinary differential equations of second and higher order. The combination of symbolic and numeric methods in a hybrid calculation allows us to derive solutions for boundary value problems in a symbolic and numeric representation. The combination of symbolic and numeric calculations simplifies not only the set up of iteration formulas which allow us to numerically represent the solution but also offers a way to standardize calculations and deliver a symbolic approximation of the solution. We use the properties of distributions and their approximations to set up interpolation formulas which are efficient and precise in the representation of solutions. In our examples we compare the exact results for our test examples with the numerical approximations to demonstrate that the solutions have an absolute error of about 10-12. This order of accuracy is rarely reached by traditional numerical approaches, like sweep and shooting methods, but is within the limit of accuracy if we combine numerical methods with symbolic ones. © 2009 Springer Berlin Heidelberg.