A Simple Direct Matrix Approach for Computing Jacobians of Nonlinear Discretized Integro-differential Equations

In this article, we present a simple direct matrix method for analytically computing the Jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integrodifferential equations. This method is based on a formulation of the discretized equations in vector form using only matrix-vector products and component-wise operations. By applying simple matrixbased differentiation rules, we are able to calculate the analytical Jacobian of the discretized equations directly in matrix form in a manner reminiscent of computing derivatives in single-variable calculus. After presenting the method, we briefly discuss its connection to the Newton-Kantorovich method (also known as quasilinearization) for numerically solving nonlinear integro-differential equations and apply it to a few example problems.