Overview

There are many approaches to the solution of linear and nonlinear boundary-value problems, and they range from completely analytical to completely numerical. Of these, the following deserve attention:

1. • Direct integration (exact solution):

2. • Separation of variables

3. • Similarity solutions

4. • Fourier and Laplace transformations

5. • Approximate solutions

6. • Perturbation

7. • Power series

8. • Probability schemes

9. • Method of weighted residuals (MWR)

10. • Finite difference techniques

11. • Ritz method

12. • Finite element method

For a few problems, it is possible to obtain an exact solution by direct integration of the governing differential equation. This is accomplished, sometimes, by an obvious separation of variables or by applying a transformation that makes the variables separable and leads to a similarity solution. Occasionally, a Fourier or Laplace transformation of the differential equation leads to an exact solution. However, the number of problems with exact solutions is severely limited, and most of these have already been solved.

Because regular and singular perturbation methods are primarily applicable when the nonlinear terms in the equation are small in relation to the linear terms, their usefulness is limited. The power-series method is powerful and has been employed with some success, but because the method requires generation of a coefficient for each term in the series, it is relatively tedious. It is also difficult, if not impossible, to demonstrate that the series converges.