Explicit Solutions for Nonlinear Partial Differential Equations Using Bezier Functions

This paper presents a methodology for generating solutions of non linear partial differential equations through Bezier functions. These functions define corresponding Bezier surfaces using a bipolynomial Bernstein basis function. The solution, or essentially the coefficients, is identified through design optimization. The set up is direct, elegantly simple, and involves minimizing the error in the residuals of the differential equations over the domain. No domain discretization is necessary. The procedure is not problem dependent and is adaptive through the selection of the order of the Bezier functions. Two examples: (1) the laminar flow over a flat plate; and (2) displacement of an ionic polymer-metal composite membrane are solved. Alternate solution to these problems is referenced in the paper.