ABSTRACT
Because of this, approximate solutions of Equation 4.1 are often desired. There are three basic categories of approximation methods:
The operator L is approximated by an operator Lˆ whose domain and range are the same as L, and an exact solution uˆ is obtained for the approximate equation
ˆ ˆLu f= (4.2)
For example, the differential equation for the vibrations of a beam with a slowly varying cross-section might be approximated by the differential equation for a beam with a uniform cross-section. Obviously when the operator is approximated, uˆ not the exact solution. Improvements to the approximation are often obtained using asymptotic expansions. If, when nondimensionalized, the problem formulation involves a small nondimensional parameter, often denoted by ε, the solution may be expanded into an asymptotic expansion in terms of linearly independent functions of ε.
