A Fully Discretized Scheme Based on High‐Precision Galerkin–Jacobi Approach for Solving Vanishing–Nonvanishing Delayed Volterra Integral Equations

In this research study, a fully discretized scheme based on the Jacobi–Galerkin approach to solve the vanishing–nonvanishing delayed Volterra integral equations has been proposed. Difficulty of this research study is because of the terms of the vanishing and nonvanishing delays. Therefore, one can apply some associated transformations on the variables of the original equation and obtain its numerical solution efficiently by using the Jacobi–Galerkin approach in fully discretized form (i.e., using high accurate Gauss–Legendre quadrature rule). The convergence analysis is discussed in detail by using the important lemmas, which proves that the error of the approximated solution will be decayed very fast in both of the infinity and weighted space norms, respectively, with the condition of sufficiently smooth solutions. Otherwise, as seen in the last example, the rate of convergence will be algebraic instead of exponential. Some test problems are presented to prove the theoretical investigations experimentally and the efficiency of the approach. Superior results can be observed in the case of errors with respect to a recently proposed approach in the literature.