A Novel Approach for solving nonlinear differential equations via Global Parametric Linear Treatment (GPLT

In this paper, we propose a novel approach for solving nonlinear differential equations, especially non-smooth ones. We call our approach Global Parametric Linear Treatment (GPLT). For smooth mappings, GPLT is linear part of Taylor expansion where the specified point which is the center of the expansion is considered to be a varying point. The approach in global sense converges to the original nonlinear mappings. So based on GPLT, instead of original nonlinear differential equation, we have a parametric linear differential equation which has a parametric analytic solution that makes the analysis more straightforward and easier. One of the main advantages of our approach is that the approach can be extended for non-smooth mappings which do not have Taylor expansion by a novel definition of Global Weak Differentiation in the sense of space. Finally, a numerical example is used to show the effectiveness of the proposed approach in solving nonlinear non-smooth differential equations.