a numerical method of elliptic–parabolic integro-differential equations arising in applied problems

This presentation introduces a hybrid numerical method for solving an elliptic-parabolic integro-differential equation in two spatial dimensions. such equations model physical phenomena with memory effects, such as heat conduction in materials with memory. the proposed method combines the gregory quadrature formula, the finite difference method for spatial discretization, and a multi-step method adamsmoulton or runge-kutta) for time discretization. the stability and convergence of the method are discussed analytically, and its effciency is demonstrated through a numerical example.