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

In this paper, a numerical investigation of a class of parabolic Volterra integro-differential equations (parabolic-VIDEs) is conducted. The approach focuses on semi-discretizing parabolic-VIDEs by utilizing a second-order compact finite difference method O(delta t(2)) 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 L-2 and L-infinity-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.