Direct linearization method for nonlinear PDE’s and the related kernel RBFs

The standard methodology handling nonlinear PDE’s involves the two steps: numerical discretization to get a set of nonlinear algebraic equations, and then the application of the Newton iterative linearization technique or its variants to solve the nonlinear algebraic systems. Here we present an alternative strategy called direct linearization method (DLM). The DLM discretization algebraic equations of nonlinear PDE’s is simply linear rather than nonlinear. The basic idea behind the DLM is that we see a nonlinear term as a new independent systematic variable and transfer a nonlinear PDE into a linear PDE with more than one independent variable. It is stressed that the DLM strategy can be applied combining any existing numerical discretization techniques. The resulting linear discretization equations can be either over-posed or well-posed. In particular, we also discuss how to create proper radial basis functions in conjunction with the DLM.