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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.
List of contents
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.
Summary
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.
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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)
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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)
