Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Abstract

Keywords:Boundary value problems for nonlinear parabolic PDE Stability and convergence of numerical method Thermodynamics Heat transferThe paper studies the existence, uniqueness, regularity and the approximation of solutions to the reaction-diffusion equation endowed with a general nonlinear regular potential and Cauchy-Neumann boundary conditions. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation.

Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

“…, the linear parabolic boundary value problem formulated in (15) has a unique solution, that is (see (14) (3) and (7)), allows us to conclude that…”

“…Let us show that the nonlinear operator H(u, λ) defined by (14) satisfies the following Properties A and B.…”

“…and then considering the difference H(u n , λ n ) − H(u n , λ), we obtain from relations (14) and 15that…”

“…In addition, the nonlinear problems of type (1) 1 , occur in the phase-field transition system (e.g., [6]) where the phase function v(t, x) describes the transition between the solid and liquid phases in the solidification process of a material occupying a region Ω. For more general assumptions and with various types of boundary conditions, Equation (5) has been numerically investigated (e.g., [6][7][8][9][10][11][12][13][14][15][16][17]). The error analysis for the implicit backward Euler approximation is presented in [16], and computations with several different higher-order time-stepping schemes are used in [11].…”

“…The Leray-Schauder principle (see [1,4,[11][12][13][14][15][19][20][21] and reference therein); • The L p -theory of linear and quasi-linear parabolic equations; • Green's first identity…”