An Algorithm for Solving Phase‐Lag Nonlinear Mixed Integral Equation With Discontinuous Generalized Kernel - Al-Bugami, Abeer M.; Abdou, M. A. (Advances in Mathematical Physics, 2025)
In this work, a nonlinear fractional integrodifferential equation (NFIo‐DE) with discontinuous generalized kernel in position and time is explored in space L 2 ( Ω ) × C [0, T ], T < 1, with respect to the phase‐lag time. Here, Ω is the domain of integration with respect to position, Ω ∈ (−1, 1), while T is the time. The integral equation (IE) under discussion has a general kernel. The equation can be transformed into a Volterra‐Fredholm integral equation (V‐FIE) of mixed type by applying the integration technique and using initial conditions. The study examines the presence of a singular solution, the convergence of the solution, and the stability of the error under specific circumstances. A Fredholm integral equation (FIE) with a general singular kernel could be obtained by applying a particular technique to separate the variables. This method was shown to relate the time variables to the IE’s kernel in terms of the existence of a unique solution. Using the Toeplitz matrix method (TMM) it was possible to obtain an algebraic system that was studied in terms of the existence of a unique solution and its convergence. Finally, some numerical results are calculated, when the kernel takes a general form of logarithmic kernel, Carleman function, and Cauchy kernel. Furthermore, the error estimate is calculated for each case.