On the numerical approximation of the solution to a nonlinear second-order anisotropic reaction-diffusion model. Applications in medical image processing

In the present paper, we study a second-order nonlinear anisotropic reaction-diffusion problem, endowed with a cubic nonlinearity reaction term and inhomogeneous Neumann boundary conditions. We investigate both the local problem (with classical Laplacian diffusion) and its nonlocal counterpart (with convolution-type diffusion). The well-posedness of the local problem is established in the class $ W_p^{1, 2}(Q) $, $ p\ge 2 $, using the Leray-Schauder principle combined with a priori estimates and $ L^p $ theory. For the nonlocal problem, we prove the existence of solutions in the space $ C([0, \bar t], C^\infty(\Omega)) $. A first-order fractional-steps scheme is developed to numerically approximate the unique solution of the nonlinear problem, and the convergence of this scheme is rigorously established. Three explicit numerical approximation schemes based on finite-difference methods are constructed, including an adaptive time-stepping approach that ensures numerical stability. Experimental results in 2D are presented to illustrate the effectiveness of the proposed methods, demonstrating their applicability to medical image processing.