Weakly Nonlocal Boundary Value Problems with Application to Geology

TL;DR

This paper investigates the existence of solutions for a weakly nonlocal Sturm-Liouville boundary value problem modeling groundwater flow, using analytical methods like the implicit function theorem to handle small perturbations.

Contribution

It introduces an analytical approach to establish solution existence for nonlocal boundary conditions in Sturm-Liouville problems with small parameters.

Findings

Existence of solutions established for small perturbation parameter .

Application of implicit function theorem to nonlocal boundary conditions.

Framework applicable to groundwater flow models with nonlocal effects.

Abstract

In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: \begin{equation*} x''(t)+\lambda x(t)=h(t)+\varepsilon f(x(t)),\hspace{.1in}t\in(0,\pi) \end{equation*} subject to non-local boundary conditions \begin{equation*} x(0)=h_1+\varepsilon\eta_1(x)\text{ and } x(\pi)=h_2+\varepsilon\eta_2(x). \end{equation*} In this paper, we study the existence of solutions to the above Sturm-Liouville problem under the assumption that $\varepsilon$ is a small parameter. Our method will be analytical, utilizing the implicit function theorem and its generalizations.