Finite Difference Scheme for Two-Dimensional Poisson Equation with the Multiple Integral Boundary Condition

This article investigates the numerical solution of the two-dimensional Poisson equation defined over a rectangular domain subject to a double integral nonlocal boundary condition. We propose a finite difference scheme by discretizing the integral term using the two-dimensional trapezoidal rule. The main difficulty of this problem is that, in the non-classical case, we cannot use the method of separation of variables and decompose the problem into one-dimensional problems. Our approach involves reducing the integral boundary condition from the complete domain to the interior points and strategically partitioning the computational domain into the boundary and interior points. We propose a method that allows us to find a solution by solving the Poisson equation with classical boundary conditions, and using the solutions found to construct a solution to a problem with a nonlocal integral condition. This method requires solving a linear system whose dimension is much smaller than the original. Under certain conditions on the kernel, the proposed method is correct.