On a Semilinear Parabolic Problem with Four-Point Boundary Conditions

Marián Slodička

doi:10.3390/math9050468

Abstract

This paper studies a semilinear parabolic equation in 1D along with nonlocal boundary conditions. The value at each boundary point is associated with the value at an interior point of the domain, which is known as a four-point boundary condition. First, the solvability of a steady-state problem is addressed and a constructive algorithm for finding a solution is proposed. Combining this schema with the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem is shown using the semigroup theory in C-spaces. Numerical experiments support the theoretical algorithms.

Highlights

Mathematical description frequently leads to an appropriate partial differential equation (PDE)
The problem in this paper describes a transient semilinear heat equation in ( a, b) with two controllers located at the interior points c, d, where a < c < d < b
The values of α, β in (3) are the solution of (12). In this way we can see that the solvability of the nonlocal problem (3) is linked to the solvability of the classical Dirichlet boundary-value problems (BVPs) (4)–(8) and the solvability of the algebraic system (12)

Summary

Introduction

The heat equation is usually accompanied by one of the following three classical BCs. Robin/Newton when a linear combination of the temperature and the normal component of the flux is known at the boundary. Using the method of energy inequalities, a priori estimates for the corresponding differential and finite-difference problems were obtained in a weighted L2 norms This proof technique has been generalized to multi-point BCs for linear parabolic problems in [14]. That is why we first developed a new solution method for the steady-state differential problem This is based on the principle of linear superposition. We designed a numerical scheme for approximation of the solution to a semilinear parabolic equation accompanied with the four-point BCs (1), which is based. We carried out some numerical experiments to support our results

Linear Steady-State Case

Resolvent Estimate

Parabolic Problem

Steady-State Example

Transient Problem

Conclusions