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El. knyga: Loewy Decomposition of Linear Differential Equations

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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.

Recenzijos

From the reviews:

This monograph pretends to describe the start point for developing a systematic way for solving linear partial differential equations (PDEs) based on the Loewys 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)

1 Loewy's Results for Ordinary Differential Equations
1(20)
1.1 Basic Facts for Linear ODE's
1(2)
1.2 Factorization and Loewy Decomposition
3(8)
1.3 Solving Linear Homogeneous Ode's
11(5)
1.4 Solving Second-Order Inhomogeneous Ode's
16(4)
1.5 Exercises
20(1)
2 Rings of Partial Differential Operators
21(40)
2.1 Basic Differential Algebra
21(2)
2.2 Janet Bases of Ideals and Modules
23(2)
2.3 General Properties of Ideals and Modules
25(5)
2.4 Differential Type Zero Ideals in Q(x, y)[ ∂x, ∂y]
30(2)
2.5 Differential Type Zero Modules over Q(x, y)[ ∂x,∂y]
32(2)
2.6 Laplace Divisors Lxm (L) and Lyn (L)
34(11)
2.7 The Ideals Jxxx and Jxxy
45(2)
2.8 Lattice Structure of Ideals in Q(x, y)[ ∂x, ∂y]
47(12)
2.9 Exercises
59(2)
3 Equations with Finite-Dimensional Solution Space
61(20)
3.1 Equations of Differential Type Zero
61(2)
3.2 Loewy Decomposition of Modules M(0,2)
63(3)
3.3 Loewy Decomposition of Ideals J(0,2) and J(0,3)
66(7)
3.4 Solving Homogeneous Equations
73(2)
3.5 Solving Inhomogeneous Equations
75(3)
3.6 Exercises
78(3)
4 Decomposition of Second-Order Operators
81(10)
4.1 Operators with Leading Derivative ∂xx
81(5)
4.2 Operators with Leading Derivative ∂xy
86(4)
4.3 Exercises
90(1)
5 Solving Second-Order Equations
91(28)
5.1 Solving Homogeneous Equations
91(8)
5.2 Solving Inhomogeneous Second Order Equations
99(8)
5.3 Solving Equations Corresponding to the Ideals Jxxx and Jxxy
107(5)
5.4 Transformation Theory of Second Order Linear PDE's
112(5)
5.5 Exercises
117(2)
6 Decomposition of Third-Order Operators
119(30)
6.1 Operators with Leading Derivative ∂xxx
119(11)
6.2 Operators with Leading Derivative ∂xxy
130(9)
6.3 Operators with Leading Derivative ∂xyy
139(8)
6.4 Exercises
147(2)
7 Solving Homogeneous Third-Order Equations
149(28)
7.1 Equations with Leading Derivative Zxxx
149(7)
7.2 Equations with Leading Derivative Zxxy
156(9)
7.3 Equations with Leading Derivative Zxyy
165(7)
7.4 Transformation Theory of Third-Order Linear PDE's
172(3)
7.5 Exercises
175(2)
8 Summary and Conclusions
177(4)
A Solutions to the Exercises
181(24)
B Solving Riccati Equations
205(8)
B.1 Ordinary Riccati Equations
205(1)
B.2 Partial Riccati Equations
206(2)
B.3 Partial Riccati-Like Systems
208(1)
B.4 First Integrals of Differential Equations
209(3)
B.5 Exercises
212(1)
C The Method of Laplace
213(4)
C.1 Exercises
215(2)
D Equations with Lie Symmetries
217(4)
D.1 Exercises
219(2)
E ALLTYPES in the Web
221(2)
List of Notation 223(2)
References 225(4)
Index 229
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