A first-order fractional–steps–type method to approximate a nonlinear reaction–diffusion equation with homogeneous Cauchy–Neumann boundary conditions
Gabriela Tănase
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\ge 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 ${\mathtt{GTanase-\; alg-frac\_rd\_CN-bc}}$, 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.