Solution of Differential Equations by Means of One-parameter Groups (Chapman & Hall CRC Research Notes in Mathematics Series)

Cover......Page 1
ISBN 0-273-08506-9......Page 2
Title: Solution ofdifferential equations by means of one -parameter groups......Page 3
Preface......Page 4
Acknowledgement......Page 6
Dedication......Page 7
Contents......Page 8
1.1 ORDINARY DIFFERENTIAL EQUATIONS......Page 10
1.2 PARTIAL DIFFERENTIAL EQUATIONS......Page 15
PROBLEMS......Page 18
2. 1 ONE—PARAMETER TRANSFORMATION GROUPS......Page 21
2.2 LIE SERIES AND THE COMMUTATION THEOREM......Page 27
PROBLEMS......Page 30
3.1 FIRST ORDER EQUATION y\' + p(x)y = g(x)......Page 35
3.2 SECOND ORDER HOMOGENEOUS EQUATION y\" + p(x) y = 0......Page 39
3.3 THIRD ORDER HOMOGENEOUS EQUATION y\" + p(x)y\' + q(x)y =0......Page 42
3.4 FOURTH ORDER SELF-ADJOINT EQUATION y\" + [p(x)y\']\' + q(x) y = 0......Page 45
PROBLEMS......Page 48
4 First order differential equations......Page 57
4.1 INFINITESIMAL VERSIONS OF y\' AND y\' = F(x,y) AND THE FUNDAMENTAL PROBLEM......Page 58
4.2 INTEGRATING FACTORS AND CANONICAL COORDINATES FOR y\' = F(x,y)......Page 60
4.3 THE ALTERNATIVE PROBLEM......Page 64
4.4 THE FUNDAMENTAL PROBLEM AND SINGULAR SOLUTIONS OF y\' = F(x,y)......Page 67
4.5 INVARIANCE OF THE ASSOCIATED FIRST ORDER PARTIAL DIFFERENTIAL EQUATION......Page 68
4.6 LIE\'S PROBLEM AND AREA PRESERVING GROUPS......Page 71
PROBLEMS......Page 75
5.1 INFINITESIMAL VERSIONS OF y\" AND y\" = F(x,y,y\')......Page 86
5.2 EXAMPLES OF THE DETERMINATION OF ((x,y) AND ri(x,y)......Page 88
5.3 DETERMINATION OF THE MOST GENERAL DIFFERENTIAL EQUATION INVARIANT UNDER A GIVEN GROUP......Page 92
5.4 APPLICATIONS......Page 96
PROBLEMS......Page 101
6 Linear partial differential equations......Page 106
6.1 FORMULAE FOR PARTIAL DERiVATIVES......Page 108
6.2 Classical Groups for the Diffusion Equation......Page 110
6.3 SIMPLE EXAMPLES FOR THE DIFFUSiON EQUATION......Page 112
6.4 MOVING BOUNDARY PROBLEMS......Page 114
6.5 FOKKER-PLANCK EQUATION......Page 118
6.6 EXAMPLES FOR tHE FOKKER—PLANCK EQUATION......Page 125
6.7 NON-CLASSICAL GROUPS FOR THE DIFFUSION EQUATION......Page 129
PROBLEMS......Page 132
7 Non-linear partial differential equations......Page 144
7.1 FORMULAE FOR PARTIAL DERIVATIVES......Page 145
7.2 Classical Groups for Non-Linear Diffusion......Page 149
7.3 NON-CLASSICAL GROUPS FOR NON-LINEAR DIFFIJSI ON......Page 155
7.4 TRANSFORMATIONS OF THE NON-LINEAR DIFFUSION EQUATiON......Page 157
PROBLEMS......Page 159
References......Page 168