Toeplitz Matrix Method and Nonlinear Volterra–Fredholm Integral Equation With Hilbert Kernel

This work emphasizes the investigation of the solution to the nonlinear Volterra–Fredholm integral equation (NV-FIE) and the necessary conditions for a unique solution. The first step is to convert the NV-FIE into a system of nonlinear Fredholm integral equations (NFIEs) using the splitting of the time interval. Analytical and semianalytical approaches are unable to solve this kind of singular integral equation due to the cumulative increase in error. While the Toeplitz matrix method (TMM) is considered one of the best methods to solve singular integral equations, its importance lies in the fact that it addresses singularity and provides simple, direct integrals. Therefore, in this study, the TMM is employed on the MIE to obtain an algebraic system. Finally, a numerical example is discussed as an application, and the error is calculated. One of the most prominent results of this study is the flexibility and efficiency of TMM in solving integral equations when the kernel takes the Hilbert type.