Convergence analysis of the homogeneous second order difference method for a singularly perturbed Volterra delay-integro-differential equation

• Published in: Chaos, solitons and fractals solitons and fractals, 2021-09, Vol.150, p.111100, Article 111100
• Main Authors: Yapman, Ömer, Amiraliyev, Gabil M.
• Format: Article
• Language: English
• Subjects: Finite difference method; Singular perturbation; Uniform convergence; Volterra delay-integro-differential equation
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2021.111100

•A new approach for designing the computational method has been considered.•This new approach results in a local truncation error containing much lower derivatives of exact solution compared to classical methods.•The method allows to weaken or get rid of the smoothness of the data functions, a determining factor for convergence analysis.•The previous works related to Volterra integro-differential equation were only concerned with regular cases.•Problems involving boundary layers have a solution with bad behaviours in applications encountered in different fields of engineering. A linear Volterra delay-integro-differential equation with a singular perturbation parameter ε is considered. The problem is discretized using exponentially fitted schemes on the Shishkin type meshes. It is proved that the numerical approximations generated by this method are O(N−2lnN) convergent in the discrete maximum norm, where N is the mesh parameter. Numerical results show a good agreement with the theoretical analysis.