A compact finite-difference and Haar wavelets collocation technique for parabolic volterra integro-differential equations

Abstract

In this paper, a numerical investigation of a class of parabolic Volterra integrodifferential equations (parabolic-VIDEs) is conducted. The approach focuses on
semi-discretizing parabolic-VIDEs by utilizing a second-order compact finite difference method O(δt2
) for the time variable and approximating the integral term
using the trapezoidal rule. Further, the spatial derivative is approximated by Haar
wavelet method. This hybrid methodology leverages the strengths of both techniques to solve the equations more efficiently and accurately. The error analysis,
using L2 and L∞-norms, shows low computational costs. Numerical experiments
show that second-order accuracy in both time and space, respectively. Stability and
convergence of the proposed method are given through Sobolve space. Some numerical examples are tested for the effectiveness and accuracy of the proposed method.
Furthermore, the presented numerical approach is compared with standard finite
difference numerical methods and the results are reported in a table.