Reproducing kernel method for the numerical solution of the Brinkman–Forchheimer momentum equation

We consider two efficient methods for the solution of the Brinkman–Forchheimer momentum equation with boundary conditions on the square. Physically, this model describes the flow of fully developed forced convection in a porous-saturated rectangular duct. After first demonstrating the existence and symmetry properties of a solution, we apply the reproducing kernel method in order to solve the Brinkman–Forchheimer momentum equation. We then demonstrate the applicability of the method by considering several specific numerical examples, which allow us to understand the variation of the physical solutions as one changes any of the several model parameters. The numerical results demonstrate the utility of the reproducing kernel method for solving nonlinear elliptic partial differential equations on compact domains.