Numerical discussion of phase-lag mixed integral equations with general discontinuous kernels

In this research, the effect of phase-lag time on an integro-differential equation in position and time is studied in L2Ω×C0,T,T<1, space. Here, Ω is the domain of integration concerning position, while T is the time. The integral equation to be discussed has a general nucleus that is anomalous in position. Using the integration technique, with the help of initial conditions, it is possible to transform the equation into a Volterra–Fredholm integral equation (V-FIE) of mixed type. Under certain conditions, the existence of a single solution, the convergence of the solution, and the stability of the error are studied. After using a specific method to separate the variables, it was possible to obtain a Fredholm integral equation (FIE) with a general singular kernel. This integral equation has coefficients that vary in time and phase-lag time. It was proven that this method linked the time variables to the kernel of the integral equation in terms of the existence of a single solution. Using the Toeplitz matrix method, which is the best method for solving anomalous integral equations (because it converts anomalous integrals into ordinary integrals that are easy to solve), it was possible to obtain an algebraic system that was studied in terms of the existence of a unique solution as well as its convergence. The integral error equation for this method was also investigated. Finally, some numerical results are calculated when the kernel takes a general form of the logarithmic kernel, Carleman function, and Cauchy kernel. In addition, the error estimate for each case is computed.