General solutions of the Laplace equation

Abstract

With the purpose of concisely and effectively obtaining the general or exact solutions of partial differential equations (PDEs), we put forward some universal Z1transformations in present paper. Not only many linear equations can be solved, but also analytical solutions of some nonlinear equations can be obtained by utilizing this method, and many solutions contain arbitrary functions. Taking as the typical case, we gain the general solution of Laplace equation for the first time. During the solving process, we find that the form of the general solution of some PDEs is not unique. On the basis of the practical cases, we also find that general solutions of some first-order linear PDEs obtained by the characteristic equation method are incomplete.

Keywords

• Transformation method
• General solution
• Exact solution
• Problem of definite solution
• Laplace equation