A first-order fractional–steps–type method to approximate a nonlinear reaction–diffusion equation with homogeneous Cauchy–Neumann boundary conditions

Abstract

In this present paper, we consider a nonlinear reaction-diffusion problem (1), endowed with a cubic nonlinear reaction term and homogeneous Cauchy-Neumann boundary conditions. We will approach the proposed nonlinear parabolic problem in the spirit of Hadamard's well-posedness conditions (see [26, p. 46]). Practically, we start our study by investigating the solvability of such a problem in the class W 1,2 p (Q), p ≥ 2. The second goal is to develop an iterative splitting scheme, corresponding to the nonlinear reaction-diffusion problem in question. Results about the convergence of the numerical scheme and error estimation are established, too. On the basis of the proposed numerical scheme, we formulate a conceptual algorithm GTanase-alg-frac rd CN-bc, which represents a delicate challenge for our future works, in order to approximate the solution of the nonlinear parabolic problem (1). The benefit of such a method aims at simplifying the process of numerical computations due to its decoupling feature.• t ∈ (0, T ], x = (x 1 , . . . , x n ) ∈ Ω; • p d , p r , p s and p t are positive values, representing the diffusion, reaction, source, and thermal transfer parameters, respectively; • V (s, y), (s, y) ∈ Q, is the unknown function (the order parameter in Q, for example);

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

“…where φ ε − stands for the left-hand limit of φ ε . Detailed discussions with respect to the advantage of ( 23)-(25) can be found in the works [3,4,15,17,18].…”

“…• K 1 s, y, θ(s, y) and K 2 s, y, φ(s, y) are the mobility functions (attached to the solution θ(s, y), φ(s, y), (s, y) ∈ Q, of (1) 1 and (1) 2 , respectively; see [2] for more details); • f 1 (t, x) ∈ L p (Q) and f 2 (t, x) ∈ L q (Q) are given functions (see [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16] for more details).…”