Numerical simulation, existence and uniqueness for solving nonlinear mixed partial integro-differential equations with discontinuous kernels

This study describes a new effective technique for solving mixed partial integro-differential equations that are nonlinear with discontinuous kernels (NMPI-DEs). We have used two well-known different numerical techniques, the toeplitz matrix technique (TMT), and the product Nystrom technique (PNT). We have outlined the characteristics of TMT and PNT in both cases, as well as the significance of each approach for characterizing and demystifying the problems' complexity. These methods have used to convert a system of nonlinear algebraic equations has been derived from the nonlinear Fredholm integral equation (NFIE). Banach's fixed point theory is employed to investigate the existence and uniqueness of the solution to the nonlinear mixed integral problem. Compared to other approaches, these strategies have shown excellent results in the first instance of being utilized to solve this kind of complex problem. Lastly, a comparison of the two distinct approaches is shown using several cases by using tables and figures. The Maple software has been utilized to compute and obtain all of the numerical results.