Analytic solutions of a class of nonlinear partial differential equations

Abstract

An approach is presented for computing the adjoint operator vector of a class of nonlinear (that is, partial-nonlinear) operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author. A unified theory is then given to solve a class of nonlinear (partial-nonlinear and including all linear) and non-homogeneous differential equations with a mathematical mechanization method. In other words, a transformation is constructed by homogenization and triangulation, which reduces the original system to a simpler diagonal system. Applications are given to solve some elasticity equations.

Arbitrariness of the general solution of the partial differential equations and its applications

“…By using the formal theory we can conclude some properties of the system without solving them. And under the "AC=BD" model for mathematics mechanization presented by Zhang [25,26], we can present an approach to reduce the original overdetermined system to a well-determined one.…”

“…Then, by considering the differential representations, we improve the Diophantine equation of the number of arbitrary functions, moreover we get more accurate arbitrariness than the results gotten by Seiler for some differential equations. Next, under the "AC=BD" model for mathematics mechanization presented by Zhang [25,26] and based on the above resuslts we present an approach to reduce an overdetermined system to a well-determined one, i.e. for an overdetermined system Au = f , we can construct a transformation u = Cv in order to reduce the original system to Dv = g, which is well determined and has the same arbitrariness with Au = f .…”