Analytical and Numerical Treatment of an Integro Partial Differential Equation in Position and Time with an Anomalous Position Kernel

This work presents a second-order Integro-Partial Differential Equation (Io-PDE) with respect to time and space, incorporating a generalized anomalous spatial kernel k(|x−y|), −1 ≤ x, y ≤ 1. From this general kernel, one can derive many special kernels, such as the logarithmic and Carleman types, the Cauchy kernel, and the strongly anomalous kernel. Other special cases can also be derived from the proposed generalized kernel. A delayed-phase formulation is also extracted as a specific instance. By imposing initial conditions, the Io-PDE is reformulated as a Mixed Integral Equation (MIE) defined in both space and time domains. We establish the uniqueness and existence of a solution and demonstrate the convergence properties. A separation of variables approach is then applied, leading to a System of Fredholm Integral Equations (SFIEs) characterized by singular spatial kernels and time-dependent coefficients. The Toeplitz Matrix Method (TMM), known for its robustness in handling anomalous equations, is utilized to numerically solve these SFIEs. This method simplifies complex anomalous integrals into standard numerical forms. Numerical experiments are conducted using logarithmic and Carleman kernels, and associated error metrics are evaluated.