Loewy Decomposition of
Linear Differential Equations
1
Fritz Schwarz
Fraunhofer Gesellschaft
Institut SCAI
53754 Sankt Augustin
Germany
1
Fritz Schwarz (Fraunhofer-SCAI)
Or: How to solve them in closed form
Loewy Decomposition
JKU Linz 2015
1 / 11
Introduction
General remarks
Background Knowledge
Basic facts about linear ordinary differential equations.
Edward L. Ince, Ordinary Differential Equations, Dover 1956;
Chapters V, VI, VII, VIII and IX, p. 114-222.
Erich Kamke, Differentialgleichungen Lösungsmethoden und Lösungen,
Akademische Verlagsgesellschaft Leipzig, 1967;
Part A, no. 16-22, p. 69-109; no. 24-27, p. 116-140; Part C, no. 2-5, p.
396-541.
Special References
Loewy Decomposition of Linear Differential Equations, Springer, 2014;
quoted as LD.
Bulletin of Mathematical Sciences 3, page 19-71 (2013)
http://link.springer.com/article/10.1007/s13373-012-0026-7
Algorithmic Lie Theory, CRC Press, 2007, Chapter 2; quoted as ALT.
Fritz Schwarz (Fraunhofer-SCAI)
Loewy Decomposition
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2 / 11
Introduction
General remarks
Partition of Differential Equations
Ordinary vs. partial equation
Linear vs. nonlinear equations
Equations of order 1,2,3,...
Subject of this Lecture
Linear ordinary equations of order 2 and 3
Linear partial equations of order 2 and 3 in the plane
Determining general solutions in closed form
Fritz Schwarz (Fraunhofer-SCAI)
Loewy Decomposition
JKU Linz 2015
3 / 11
Introduction
General remarks
Schedule of Lectures
Introduction
Basic Knowledge on Linear ODE’s
Solving Second-Order Linear ODE’s
Solving Third-Order Linear ODE’s
Rings of Partial Differential Operators: Basic Results
Rings of Partial Differential Operators: Decomposition of Ideals
Decomposing Second-Order Operators in the Plane
Decomposing Third-Order Operators with Leading Derivative
∂xxx , ∂xxy or ∂xyy
Solving Homogeneous Second-Order Linear PDE’s
Solving Inhomogeneous Second-Order Linear PDE’s
Solving Third-Order Linear PDE’s with Leading Derivative ∂xxx ,
∂xxy or ∂xyy
Solving Riccati Equations
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Loewy Decomposition
JKU Linz 2015
4 / 11
Introduction
Examples
Solving 2nd Order Linear ODE’s
Example 1
3 y0 + 4 y = 0
y 00 − x
x2
General solution : y = C1 x2 + C2 x2 log x
Example 2
y 00 − xy 0 + 2y = 0
General solution : y = C1
Fritz Schwarz (Fraunhofer-SCAI)
(x2
− 1) + C2
Loewy Decomposition
(x2
Z
− 1)
exp 12 x2
dx
(x2 − 1)2
JKU Linz 2015
5 / 11
Introduction
Examples
Solving 3rd Order Linear ODE’s
Example 3
1 + 2 y 00 −
6 + 6 + 4 y 0
y 000 + x +
1 x
x + 1 x x2
8
8 +8+
− x+
1 x x2 + 4 y = 0
x3
General solution : y = C1 x2 + C2 x2 log x + C3 x2 (log x)2
Fritz Schwarz (Fraunhofer-SCAI)
Loewy Decomposition
JKU Linz 2015
6 / 11
Introduction
Examples
Example 4
y 000 −
4x 0
4
y=0
1y +
x− 2
x − 12
General solution:
y = C1 x + C2
e2x
+ C3
1 2x
4e
Z
x exp (−4x)dx
x2 − x + 14
− 14 x
Fritz Schwarz (Fraunhofer-SCAI)
Z
Loewy Decomposition
exp (−2x)dx x2 − x + 41
JKU Linz 2015
7 / 11
Introduction
Examples
Solving Linear PDE’s
Example 5
2z = 0
zxx − zyy − x
x
General solution : z = F (x + y) − xF 0 (x + y) + G(x − y) − xG0 (x − y)
Example 6
zxxx + xzxxy + 2zxx + 2(x + 1)zxy + zx + (x + 2)zy = 0
General solution : z = F (y − 12 ) + G(y)e−x
x
Z
+ H(ȳ + 12 x2 )e−x dx
1
ȳ=y− 2 x2
Fritz Schwarz (Fraunhofer-SCAI)
Loewy Decomposition
JKU Linz 2015
8 / 11
Introduction
Solutions and how to find them
What is a solution?
Answer:
Finite expression in closed form in well defined function field.
Examples: Elementary or liouvillian extension of the base field.
Closed Forms: What They are and Why We Care, Jonatan M. Borwein, Richard E.
Crandall, Notices of the AMS 60, page 50-65, 2013.
Excluded:
Numerical or graphical solutions.
Infinite series expansions.
General Solution: Contains maximal number of undetermined elements.
For linear ode’s: Maximal number of constants.
For linear pde’s: Maximal number of undetermined functions and
its arguments.
Fritz Schwarz (Fraunhofer-SCAI)
Loewy Decomposition
JKU Linz 2015
9 / 11
Introduction
Solutions and how to find them
How are solutions obtained?
Excluded:
Trial and error methods.
Data base like collections of solved examples, e.g. Kamke:
y00 − x2 y0 − (x + 1)2 y = 0
Z
y = exp 13 x3 + x C1 + C2 exp −
2.109
1
3
2.57
− 2x dx
xy00 − xy0 − y = x(x + 1)ex
y = (x2 − xlogx − 1)ex + C1 xex + C2 xex
Z
dx
x2 ex
Morris-Brown, Diff. Equations, S. 136,369
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Loewy Decomposition
JKU Linz 2015
10 / 11
Introduction
Final Goal
Final Goal
Algorithms that guarantee to find solutions of particular type.
Failure of algorithm proves non-existence of solution of particular
type.
Prove that decision procedure for existence of solution of a certain
type does not exist.
Fritz Schwarz (Fraunhofer-SCAI)
Loewy Decomposition
JKU Linz 2015
11 / 11