Name: William Ted Mahavier
School: Nicholls State University
A Numerical Method for Solving Singular DE's
A numerical method is developed for solving singular
differential equations using steepest descent based on
weighted Sobolev gradients. The
method is demonstrated on a variety of first and second
order problems, including linear constrained,
unconstrained, and partially constrained first order problems,
a nonlinear first order problem with irregular singularity,
and two second order variational problems.
The method is an extension of steepest descent in Sobolev
spaces which is a variation of descent based on the Euclidean
gradient. The differential equation is cast as a least-squares
problem yielding a functional representing the equation.
A weighted Sobolev space for the problem is chosen
where the weights are based on the functional. The
gradients associated with the functional take into account both
the weights and the boundary conditions for the given equation.
Results are presented which demonstrate the improvements
obtained by computing based on weighted Sobolev gradients
rather than computing based on either unweighted Sobolev
gradients or on the Euclidean gradient.