Numerical Approximation for a Nonlocal and Nonlinear Reaction–Diffusion Problem with Robin Boundary Conditions

Tudor Barbu (), Ana-Maria Moşneagu and Gabriela Tănase
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Tudor Barbu: Institute of Computer Science, Romanian Academy–Iaşi Branch, Str. Teodor Codrescu, nr. 2, 700481 Iaşi, Romania
Ana-Maria Moşneagu: Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Bvd. Carol I, No. 11, 700506 Iaşi, Romania
Gabriela Tănase: Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Bvd. Carol I, No. 11, 700506 Iaşi, Romania

Mathematics, 2025, vol. 13, issue 21, 1-18

Abstract: In this paper we consider a reaction–diffusion model with nonlocal diffusion and a nonlinear reaction term analogous to a local reaction–diffusion problem with Robin boundary conditions. Firstly, we investigate the existence of solutions in a two-dimensional spatial domain. Then we attach a semi-implicit numerical scheme by using finite differences in order to approximate the solution. We use the iterative Newton method to numerically solve the resulting implicit problem. Based on theoretical results we generate an adaptive mesh in time that ensures the stability of the corresponding numerical scheme. Numerical experiments that illustrate the effectiveness of the theoretical results are provided.

Keywords: adaptive time-step method; integro-differential equations; phase transitions; nonlocal diffusion; finite difference scheme; newton iterative method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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