2025, Vol. 6, Issue 1, Part D

This work presents a novel hybrid method for numerically solving the second-kind Fredholm-Volterra as well as Volterra-Friedholm integral equations in one and two dimensions. The proposed hybrid approach makes use of a multi-linear Legendre wavelet, where a multi-linear Legendre wavelet series can be used to approximate the unknown function in the equations. The newly developed numerical method is exercised on a series of linear and non-linear reference test models, including ones that have both discrete and non-discrete solutions. Comparative analyses were carried out against other existing numerical methods, such as the Monte Carlo method, the rational Haar function method, the operational matrix with the impulse mass function method, and the Bernstein matrices method. The results demonstrate that the operational and direct method of the arrangement line P illustrates the applicability and validity of the proposed method in solving multiple physical and engineering problems.

The numerical method created performs well under discontinuities and without needing the solutions to be non-differentiable, which is a restriction on many other numerical methods available. Further, the approximate and exact solutions comparison using L1 parameters provides another reasonable check of the newly developed numerical method's accuracy, flexibility and robustness.