Explicit Solutions for Linear Partial Differential Equations Using Bezier Functions

Solutions in basic polynomial form are obtained for linear partial differential equations through the use of Bezier functions. The procedure is a direct extension of a similar technique employed for nonlinear boundary value problems defined by systems of ordinary differential equations. The Bezier functions define Bezier surfaces that are generated using a bipolynomial Bernstein basis function. The solution is identified through a standard design optimization technique. The set up is direct 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. Three examples: (1) the Poisson equation; (2) the one dimensional heat equation; and (3) the slender two-dimensional cantilever beam are solved. The Bezier solutions compare excellently with the analytical solutions.