Monotone Method for Nonlinear First-order Hyperbolic Initial-boundary Value Problems of Moving Boundary

Abstract

Moving boundary problems arise in many important applications to biology and chemistry. Comparing to the fixed boundary problem, moving boundary problem is more reasonable. To the best of our knowledge, there’s few results on the moving boundary for nonlinear first-order hyperbolic initial-boundary value problems. In the present paper, we mainly clarify the problem and show the existence and uniqueness of the solution for such kind of problems. We take a classical transform to straighten the moving boundary and develop a monotone approximation, based on upper and lower solutions technique, for solving a class of first-order hyperbolic initial-boundary value problems of moving boundary. Such an approximation results in the existence and uniqueness of the solution for the problem. The idea behind such a method is to replace the actual solution in all the nonlinear and nonlocal terms with some previous guess for the solution, then solve the resulting linear model to obtain a new guess for the solution. Iteration of such a procedure yields the solution of the original problem upon passage to the limit. A novelty of such a technique is that an explicit solution representation for each of these iterates is obtained, and hence an efficient numerical scheme can be developed. The key step is a comparison principle between consecutive guesses.