Various Numerical Methods for Singularly Perturbed Boundary Value Problems

As Science & technology develop, many practical problems, such as the mathematical boundary layer theory or approximation of solution of various problems described by differential equations involving large or small parameters have become increasingly complex and therefore require the use of asymptotic methods. However, the theory of asymptotic analysis for differential operators has mainly been developed for regular perturbations where the perturbations are subordinate to the unperturbed operator. In some problems, the perturbations occur over a very narrow region across which the dependent variable undergoes very rapid changes. These narrow regions are frequently adjacent to the boundaries of the domain of interest because a small parameter multiplies the highest derivative. Consequently, they are usually referred to as boundary layers in fluid mechanics, edge layers in solid mechanics, skin layers in electrical applications, shock layers in fluid and solid mechanics, transition points in quantum mechanics, WKB problems, the modeling of steady and unsteady viscous flow problems with large Reynolds numbers and convective heat transport problems with large peclet numbers.