Schwarz F. Loewy Decomposition of Linear Differential Equations

Springer-Verlag, 2012. — 238 p. — (Texts & Monographs in Symbolic Computation). — ISBN10: 3709112850.The central subject of the book is the generalization of Loewy's decomposition - originally introduced by him for linear ordinary differential equations - to linear partial differential equations. Equations for a single function in two independent variables of order two or three are comprehensively discussed. A complete list of possible solution types is given. Various ad hoc results available in the literature are obtained algorithmically. The border of decidability for generating a Loewy decomposition are explicitly stated. The methods applied may be generalized in an obvious way to equations of higher order, in more variables or systems of such equations.Loewy’s Results for Ordinary Differential Equations
Rings of Partial Differential Operators
Equations with Finite-Dimensional Solution Space
Decomposition of Second-Order Operators
Solving Second-Order Equations
Decomposition of Third-Order Operators
Solving Homogeneous Third-Order Equations
Summary and Conclusions