Numerical Solution of a Class of Nonlinear Delay Partial Differential Equations Using a Linear Compact Difference Scheme

Abstract

       In this paper, we propose numerical solution of a certain class of nonlinear DPDEs via a linear compact difference scheme. Of course, and we know very well that in technology, physics or biology as well as other scientific fields like time lags which may be modelled with nonlinear parabolic PDEs in the context of continuum mechanics or space confinements are to describe those systems. not characteristic were the strong nonlinearity and delay terms that often lead to numericenarios UNssnasosty an large precisionestabiy losses in numericalthe loss of high precision. To address these issues, we propose a linear compact finite difference scheme that is able to provide high-order spatial accuracy and icely conserve the time delay term. By discretization of the spatial derivative terms with compact stencil and the linearization of the nonlinear terms through suitable transformations, an efficient and readily implementable algorithm is developed. The scheme is analyzed for stability and convergence by rigorous mathematical method, guaranteeing that the present numerical solution can be obtained with high accuracy. Some typical examples including benchmark problems and nonlinear with complicated time delay structure are examined to demonstrate the accuracy, efficiency, validity, and practicality of the proposed method. Numerical experiments show improved error estimates for the proposed method compared to classical FD schemes and stability with respect to different discretization parameters and initial conditions. The results emphasize that the linear compact scheme is an efficient and useful tool for simulating nonlinear delay PDEs arisen in science and engineering. Extension of the method to multi-dimensions and exploration into adaptive meshes as a means for improving computational efficiency and quality of the solutions can be features that may be included in future developments.