An efficient Tau approach for solving a class of third-kind Volterra integral equations with proportional delays

Abstract

This work presents a new algorithm for solving a kind of Volterra delay integral equations of the third kind (VDIEs). Using the Tau method and generalized polynomial bases, our developed method is a robust approach for solving these equations. Specifically, we employ simple matrix operations to enhance the Tau approach. The underlying strategy leverages orthogonal polynomial bases to change the original equation into a matrix-vector form. Such a transformation makes the third-kind VDIEs easier to handle by turning them into a set of algebraic equations. Importantly, this method exhibits good stability, reduces memory usage, and is computationally cost-effective. The paper details the algorithm’s formulation and shows its capability to provide approximate polynomial solutions. We perform a thorough error estimation to check the method’s accuracy. To demonstrate its practical effectiveness, we use several numerical examples. The obtained results highlight the performance of the method and prove its alignment with theoretical error predictions. Furthermore, a comparative analysis with analytical solutions and alternative methods reaffirms the efficiency of the developed approach.