Książka - szczegóły

CONTENTS
INTRODUCTION............................................................................................................................... 3
Chapter I. ALGEBRAIC PROPERTIES OF SOLUTIONS OF ABSTRACT DIFFERENTIAL
EQUATIONS
§ 1. Ordinary abstract differential equations
1. Taylor’s formula for an abstract derivative.......................................................................... 4
2 π-solutions................................................................................................................................. 5
§ 2. Fundamental system of solving operations in linear spaces and algebras
1. Operational independence and solving operations.......................................................... 8
2. One linear differential equation of the first order................................................................. 9
3. A system of linear differential equations of the first order............................................... 11
4. Linear differential equations of order n.............................................................................. 15
5. Partial derivatives..................................................................................................................... 18
6. Linear partial differential equations...................................................................................... 20
7. Wroński's fundamentality criteria in algebras................................................................. 24
8. Examples................................................................................................................................ 25
§ 3. Universal spaces of analytic elements
1. Introduction............................................................................................................................. 26
2. The space $C_N(ℬ)$........................................................................................................... 27
3. Multiplications, superposition and convolution of elements
of $C_N(ℬ)$.................................................................................................................................. 29
4. The space $C_N^m(ℬ)$ of analytic functions of many multipliers................................. 32
6. Examples.................................................................................................................................. 33
Chapter II. ANALYTIC PROPERTIES OF SOLUTIONS OF ABSTRACT DIFFERENTIAL
EQUATIONS
§ 4. Existence, uniqueness and continuity of solutions
1. Regular operations in $K_Z$-linear spaces....................................................................... 35
2. The well-defined problem of solution of an abstract differential equation.................... 37
3. Examples................................................................................................................................... 41
§ 5. Analytic elements
1. Introduction.............................................................................................................................. 43
§ 6. The separation of variables
1. The separation of variables.................................................................................................. 46
2. Examples................................................................................................................................. 49
§ 7. Summation theorem
1. The Kojima-Schur and the Toeplitz theorems................................................................. 52
2. Euler’s theorems..................................................................................................................... 64
3. Newton’s interpolation formulas........................................................................................ 55
4. Examples................................................................................................................................. 59
REFERENCES............................................................................................................................ 61