Solving Volterra–Hammerstein integral equations with time-varying delays using the local RBF-collocation method

Mathematical models with time delays offer profound insights into systems influenced by their historical states. In particular, delay integral equations have found wide application in fields such as population dynamics, epidemiology, and economics due to their ability to model memory-dependent systems effectively. This paper introduces a numerical method for solving a specific type of non-vanishing delay Volterra–Hammerstein integral equations with difference kernels. The considered equations feature time-dependent delay arguments and may also involve kernels with weak singularities. The proposed method uses a modified discrete collocation technique based on radial basis functions (RBFs), relying on local data within the neighborhood of a given point rather than all nodes, to estimate the solution. Therefore, it is more stable and uses less computer memory compared to its global counterpart. This enables the approach to run easily on typical computer systems, with no demand for advanced hardware. Since the approach avoids cell structures to obtain the numerical solution, it can be included in meshless schemes. The study also includes a detailed error estimate, which shows the high convergence performance achieved with selected RBFs. We also present several numerical experiments to highlight the effectiveness and computational advantage of the method.