Numerical solution methods for semi-li̇near parabolic equations with periodic boundary conditions, singular perturbation and delay arguments

This study presents numerical solution methods for singularly perturbated semilinear parabolic type equations with periodic boundary conditions and time delay arguments. The stability of the problem was investigated by using the energy inequality and evaluated with respect to the L_2 norm. Then, with the help of interpolation quadrature rules with exponential basis functions and the remainder term in integral form, difference schemes with exponential coefficients were constructed in the uniform network. The stability of the solution of the discrete problem was analyzed with the help of discrete energy inequalities, and the error rate was determined. The theoretical results were validated through numerical examples, for which computer codes were employed to provide computational support for the problem under study.