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Loewy Decomposition of Linear Differential Equations

  • Book
  • © 2012

Overview

Authors:
  1. Fritz Schwarz

    1. Institute SCAI, Fraunhofer Gesellschaft, Sankt Augustin, Germany

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  • Most advanced and most complete text on closed form solutions of linear partial differential equations
  • Provides more than 50 worked out examples and exercises including solutions
  • The results described in the book may be applied for determining Lie symmetries of nonlinear differential equations
  • Includes supplementary material: sn.pub/extras

Part of the book series: Texts & Monographs in Symbolic Computation (TEXTSMONOGR)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. Loewy’s Results for Ordinary Differential Equations

    • Fritz Schwarz
    Pages 1-20

  3. Rings of Partial Differential Operators

    • Fritz Schwarz
    Pages 21-59

  4. Equations with Finite-Dimensional Solution Space

    • Fritz Schwarz
    Pages 61-79

  5. Decomposition of Second-Order Operators

    • Fritz Schwarz
    Pages 81-90

  6. Solving Second-Order Equations

    • Fritz Schwarz
    Pages 91-118

  7. Decomposition of Third-Order Operators

    • Fritz Schwarz
    Pages 119-147

  8. Solving Homogeneous Third-Order Equations

    • Fritz Schwarz
    Pages 149-175

  9. Summary and Conclusions

    • Fritz Schwarz
    Pages 177-179

  10. Back Matter

    Pages 181-230
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • Institute SCAI, Fraunhofer Gesellschaft, Sankt Augustin, Germany

    Fritz Schwarz

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