PhD Defence: Symbolic Solutions of First-Order Algebraic Differential Equations

Symbolic Solutions of First-Order Algebraic Differential Equations

Differential equations have been intensively studied for a long time. Various exact solution methods have been proposed for specific cases. Nevertheless, there is no general algorithm for finding explicit exact solutions. The main aim of this thesis is to develop and investigate new methods for computing explicit exact solutions of algebraic differential equations. For this purpose, the differential problem is transformed into an algebraic geometric one by considering the differential equation to be an algebraic equation. Such an equation defines an algebraic variety and hence, tools from algebraic geometry can be applied. In particular, parametrizations of algebraic varieties are intrinsically used to solve the problem and prove properties of the obtained solutions. A general idea for solving first-order autonomous algebraic differential equations is presented.

The main results of the thesis are applications of this general idea to ordinary and partial differential equations. The idea is introduced for first-order autonomous algebraic ordinary differential equations. The presented method is a generalization of an existing algorithm for computing rational solutions. It admits an extension to the computation of radical solutions. Moreover, it allows a further generalization to higher-order algebraic ordinary differential equations.

A second focus lies on the application of the general idea to partial differential equations in arbitrary many variables. The presented method reduces the problem to another one for which solution methods exist. Various well-known differential equations are solved by this method. Furthermore, classes of differential equations with rational, radical or algebraic solutions are presented. With the help of linear transformations a solution method for certain non-autonomous differential equations is achieved.

The procedures are constructed in such a way that the obtained solutions thereof satisfy certain requirements. It is shown that algebraic solutions of ordinary differential equations are general solutions. Rational solutions of partial differential equations are proven to be proper and complete.