A fully discrete spectral method for coupled parabolic systems with nonlocal terms

Articles in Press

Document Type : Research Article

Authors

M. Hatami

Davoud Rostami

Department of Mathematics, Imam Khomeini International University, Qazvin, Iran

10.22067/ijnao.2026.98401.1864

Abstract

This paper presents a comprehensive analysis of a coupled system of parabolic equations with nonlocal Kirchhoff-type terms. The model involves standard time derivatives and spatial fractional Laplacian operators. We first establish the well-posedness of the weak formulation in fractional Sobolev spaces using Schaefer’s fixed point theorem and detailed energy estimates. Then, we develop a novel fully discrete numerical scheme that combines a spectral collocation method based on Lucas polynomials for spatial discretization with an implicit Euler method for temporal discretization. We provide a rigorous a priori error analysis, proving exponential convergence in space (for analytic solutions) and first-order convergence in time. A complete stability analysis of the implicit Euler scheme for the coupled system is presented, showing unconditional stability. Extensive numerical experiments validate the theoretical results, demonstrating the efficiency, accuracy, and robustness of the proposed method. The numerical scheme handles the nonlinear nonlocal terms effectively and shows excellent performance for long-time simulations.

Keywords

Parabolic equations

Kirchhoff-type nonlocal term

Lucas polynomials

Spectral collocation

Implicit Euler method

Error analysis

Unconditional stability

Subjects

• Numerical methods for differential equations and its applications

---