A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations

• Published in: Mathematical modelling and analysis Vol. 28; no. 3; pp. 469 - 486
• Main Authors: Xie, Ruifeng, Zhang, Jian, Niu, Jing, Li, Wen, Yao, Guangming
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 04.09.2023
• Subjects: Analysis; Approximation; Boundary value problems; collocation method; Collocation methods; Convergence; delay parabolic equation; Differential equations; Exact solutions; Kernels; Mathematical analysis; Mathematical functions; Methods; numerical solution; Parabolic differential equations; Partial differential equations; reproducing kernel method; Singular perturbation; collocation method; numerical solution; delay parabolic equation; reproducing kernel method
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2023.16852

In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution  to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.