The Shooting Method in the Analysis of Two-Point Boundary-Value Problems (Adolfo J. Rumbos, Pomona College)

Abstract:

Two-point boundary-value problems (BVPs) appear frequently in applied mathematics.  When looking for solutions of boundary-value problems for some partial differential equations (PDEs) in mathematical physics, two-point BVPs come up as a result of applying the method of separation of variables, for instance. In the case of linear PDEs, the resulting two-point BVPs fall into a class of problems known as Sturm-Liouville eigenvalue problems.

This presentation deals with the use of the shooting method to prove existence of solutions of two-point BVPs.  The shooting method is a numerical technique used to estimate solutions of two-point BVPs once a solution is known to exist.  In this talk we illustrate how the shooting method can be used to prove existence of eigenvalues of linear Sturm-Liouville problems.  We also show how the shooting method can be applied to prove existence and uniqueness of solutions for some nonlinear, two-point BVPs, and existence of eigenvalues for some nonlinear eigenvalue problems.

The presentation describes research conducted with collaborators Vaidehi Srinivasan (Pomona College class of 2027) and Gavin Zhao (Pomona College class of 2029) in the summer of 2025 with the support of the Summer Undergraduate Research Program at Pomona College.