Loewy Decomposition of Linear Differential Equations

1. Front Matter

   Pages i-xv

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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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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.

• Computer Algebra Software
• Differential Algebra
• Partial Differential Equations

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)

• Institute SCAI, Fraunhofer Gesellschaft, Sankt Augustin, Germany

  Fritz Schwarz

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