Solving the elliptic Monge–Ampère equation by Kansaʹs method

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation, which originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing and others. The numerical solution of the elliptic Monge–Ampère equation has been a subject of increasing interest recently. In this paper, Kansaʹs method (with multiquadric basis functions) is introduced and used to solve numerically the Monge–Ampère equation. Kansaʹs method is a meshfree method which uses the combination of some radial basis functions (RBFs) to approximate the solution of the partial differential equation. We prove the classical consistency of the Kansaʹs method for the elliptic Monge–Ampère equation. Finally, we also present some numerical experiment to demonstrate the effectiveness of Kansaʹs method for the elliptic Monge–Ampère equation.