On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions

Abstract

The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both θ(t,x) and φ(t,x). Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, f1, f2, w1, w2, α, ξ, θ0, α0, φ0, and ξ0, we prove the well-posedness of a solution θ,α,φ,ξ, which is θ(t,x),α(t,x)∈Wp1,2(Q)×Wp1,2(Σ), φ(t,x),ξ(t,x)∈Wν1,2(Q)×Wp1,2(Σ), ν=min{q,μ}. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future.

Keywords

• boundary value problems for nonlinear parabolic PDE
• dynamic boundary conditions
• fractional step method
• convergence of numerical scheme
• numerical algorithm
• phase changes