---

As the most complete text on closed form solutions of linear partial differential equations, this book´s coverage of the generalization of Loewy´s decomposition includes more than fifty worked out examples and exercises in addition to their solutions.

---

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 third-order equations.- Summary and conclusions.- Solutions to the exercises.- Solving Riccati equations.- The method of Laplace.- Equations with Lie symmetries.

---

From the reviews:

"This monograph pretends to describe the start point for developing a systematic way for solving linear partial differential equations (PDE´s) based on the Loewy´s decomposition method, working in an proper ring of differential operators and including algorithmic alternatives for several problems considered in classic literature. ... this monograph is truly a guide book for the problem of decomposing differential operators, written in a very clear and objective language, and providing the necessary tools towards more general problems." (Ana Rita Martins, Zentralblatt MATH, Vol. 1261, 2013)

---