A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation

作者：Bashier E B M; Patidar K C^*

来源：Applied Mathematics and Computation, 2011, 217(9): 4728-4739.

DOI：10.1016/j.amc.2010.11.028

• Delay parabolic partial differential equation
• Singular perturbations
• Fitted operator finite difference methods
• Stability
• Convergence

We design a robust fitted operator finite difference method for the numerical solution of a singularly perturbed delay parabolic partial differential equation. This method is unconditionally stable and is convergent with order O(k + h(2)), where k and h are respectively the time and space step-sizes, which is better than the one obtained by Ansari et al. [A. R. Ansari, S. A. Bakr, G. I. Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205 (2007) 552-566] where they have used a fitted mesh finite difference method. Their method was of the order O(N(t)(-1) + N(x)(-2)ln(2)N(x)), where N(t) and N(x) denote the total number of sub-intervals in the time and space directions. The performance of our method is illustrated through some numerical experiments. We also compare our results with those obtained by a standard finite difference method as well as other works seen in the literature. In addition, we provide a novel proof for the bounds on partial derivatives of the solution of the continuous problem.

• 出版日期2011-1-1