Introduction to Nonlinear Differential and Integral Equations

Couverture
Courier Corporation, 1 janv. 1962 - 566 pages

Within recent years interest in nonlinear equations has grown enormously. They are extremely important as basic equations in many areas of mathematical physics, and they have received renewed attention because of progress in their solution by machines.
This volume undertakes a definition of the field, indicating advances that have been made up through 1960. The author's position is that while the advent of machines has resulted in much new knowledge, one should not disregard analytical methods, since the solution of nonlinear equations possesses singularities which only the analytical method (as based upon the work of Poincare, Liapounoff, Painleve and Goursatl can discover.
After a general survey of the problem presented by nonlinear equations, the author discusses the differential equation of the first order, following this by chapters on the Riccati equation (as a bridge between linear and nonlinear equations) and existence theorems, with special reference to Cauchy's method. Second order equations are introduced via Volterra's problem and the problem of pursuit, and succeeding chapters cover elliptic integrals and functions and theta functions; differential equations of the second order; and second order differential equations of the polynomial class, with special reference to Painleve transcendents. The technique of continuous analytical continuation is shown, while phenomena of the phase plane are studied as an introduction to nonlinear mechanics. Nonlinear 111echanics is then discussed, with various classical equations like Van der Pol's equations, Emden's equation, and the Duffing problem. The remaining chapters are concerned with nonlinear integral equations, problems from the calculus of variations, and numerical integration of nonlinear equations. Throughout the book the results of distinguished analysis of the past and modern machine computations are both taken into account. Despite the thoroughness of its coverage this is a very fine introduction to this important area of mathematics, and it can easily be followed by the mathematically sophisticated reader who knows very little about nonlinear equations.

 

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Table des matières

INTRODUCTION e 1 N onlinear Operators
1
Nonlinear Equations
3
The Origin of Nonlinear Equations
9
The Problem of Nonlinear Equations
12
Systems of Nonlinear Equations
16
Nonlinear Partial Differential Equations
18
TRE DIFFERENTIAL EQUATION OF FIRST ORDER 1 Introduction
25
Isoclines and Curvature
29
The Boutroux Transformation of the first Painleve Equation
232
Definition of a New Transcendental Function
234
Determination of the Parameter A
236
Generalization of Sp
237
The Second Painleve Transcendent
239
The Boutroux Transformation of the Second Painleve Equation
242
Methods of Analytic Continuation
245
CONTINUOUS ANALYTIC CONTINUATIGN 1 Introduction
247

The Integrating Factor
32
The Homogeneous Case
36
Ax + By+E dx Cx+Dy + F dy
38
Special Cases for Which the Integrating Factor Is Known
45
Some Particular Differential Equations
49
Singular Solutions
53
TRE RICCATI EQUATION 1 The Riccati Equation
57
Relationship Between the Riccati Equation and the Linear Dif ferential Equation of Second Order
59
The Motion of a Body Which Falls in a Medium Where the Re sistance Varies as the Square of the Velocity of the Body
60
The CrossSection Theorem for the Riccati Equation
62
Integration of the Riccati Equation
63
Solution of the Original Riccati Equation
65
Solution of the Riccati Equation by Means of Continued Frac tions
70
TARLES
71
The Character of the Singularities of the Riccati Equation
73
Abels Equation
74
The Generalized Riccati Equation
76
EXISTENCE TREOREMS 1 Introduction
79
The Method of Successive Approximations
83
An Example and Critique of the Method of Successive Approxi mations
87
The CauchyLipschitz Method
88
Introduction
95
The Logistic Curve
96
The Problem of Growth in Two Populations Conflicting With One Another
99
Solution of the Problem of Growth of Two Conflicting Populations
102
A Generalization of Volterras Problem
109
The Hereditary Factor in the Problem of Growth
112
Curves of Pursuit
113
Linear Pursuit
115
Conditions of Capture
123
General Pursuit Curves
125
ELLIPTIC INTEGRALS ELLIFTIP FUNCTIONS AND TRETA FUNCTIONS 1 Introduction
129
Elliptic Integrals
131
Expansions of the Complete Elliptic Integrals of First and Second Kinds
133
Expansions of the Elliptic Integrals of First and Second Kinds
135
Differential Equations Satisfied by the Complete Elliptic In grals
137
Gausss Limit
144
ELLIPTIC FUNCTIONS 8 The Elliptic Functions of Jacobi
145
Derivatives and Integrals of the Elliptic Functions
147
Addition Theorems
148
DoubleAngle and HalfAngle Formula?
150
Expansions of the Elliptic Functions in Powers ofu
152
The Poles of the Elliptic Functions
153
The Zeta Elliptic Function of Jacobi
155
The Elliptic Functions of Weierstrass
156
Theta Functions
162
The Differential Equation of the Theta Functions
165
Representation of the Jacobi Elliptic Functions as Fourier Series
167
The Elliptic Modular Functions
169
Solution of the Quintic Equation by Modular Functions
172
Tables of the Elliptic Functions
175
DIFFERENTIAL EQUATIONS OF SECOND ORDER 1 Introduction
179
Classification of Nonlinear Differential Equations of Second Order
182
in Which Critical Points Are Fixed Points C The General ized Riccati Equation of Second Order D Equations Having Periodic Solutions E Miscellane...
188
Existence Theorems
189
The problem of the Pendulum
192
6
194
as a Laurent Series
197
The Solution of y 6y2 as a Taylors Series
200
ff6Jigi
202
y Ay+By
207
Solution of the General Elliptic Equation
209
Introduction
213
Applications of the Linear Fractional Transformation
218
Transformations of the Independent Variable
221
Equations With Fixed Critical Points and Movable Poles
225
The First Painleve Transcendent _
229
The Method of Curvature
248
Analytic Continuation
250
The Method of Continuous Analytic Continuation
251
An Elementary Example Illustrating the Method of Continuous Analytic Continuation
254
The Solution of y 6y1 by the Method of Continuous Analytic Continuation
256
The Numerical Evaluation of the First Painleve Transcendent
258
The Numerical Evaluation of the Second Painleve Transcendent
260
The Analytic Continuation of the Van der Pol Equation
261
The Analytic Continuation of Volterras Equation
262
The Technique of Continuous Analytic Continuation Around a Singular Point
263
THE PHASE PLANE AND ITS PHENOMENA 1 Introduction
267
The Phase Plane and Limit Cycles
268
Phase Curves and Forcing Functions
273
Nonperiodic Solutions in a Closed Area
284
The Pendulum Problem as a Fourier Series
291
Periodic Solutions
297
Additional Aspects of PeriodicityFloquets Theory
300
Periodicity as a Phenomenon of the Phase Plane
303
NONLINEAR MECHANICS 1 Introduction 3O
309
A Preliminary Example
317
An Application of the Stability Theorem
322
Limit Cycles
331
Some Further Comments About Limit Cycles
336
ise
339
Periodic SolutionsThe General Quadratic Equation
343
Topological ConsiderationsPoincares IndexBendixons Theorem
351
SOME PARTICULAR EQUATIONS 1 Introduction
357
The Equation of Van der Pol
358
An Analytical Approximation to the Solution of the Van der Pol Equation
364
Stellar Pulsation as a LimitCycle Phenomenon
368
Emdens Equation
371
The Differential Equation of Isothermal Gas Spheres
377
Equations of Emden Type
381
The Duffing Problem
386
Nonlinear ResonanceThe Jump Phenomenon
395
The Generalixed Equation of Blasiua
400
Miscellaneous Examples
405
NONLINEAR INTEGRAL EQUATIONS 1 Introduction
413
An Existence Theorem for Nonlinear Integral Equations of Vol terra Type
415
The IntegroDifferential Problem of Volterra
417
An Existence Theorem for Nonlinear Integral Equations of Fredholm Type
424
A Particular Example
426
The Equation u2Xf Kxt ut dt
429
The Nonlinear Convolution Theorem
434
PRORLEMS FROM TRE CALCULUS OP VARIATIONS 1 Introduction
439
The Euler Condition
445
The Euler Condition fora Double Integral
450
The Problem of the Minimal Surface
452
Hamiltons PrincipleThe Principle of Least Action
456
The Canonical Equations of Hamilton
461
TRE NUMERICAL INTEGRATION OF NONLINEAR EQUATIONS 1 Introduction
467
The Calculus of Finite Differences
468
Differences and Derivatives
470
Integration Formulas
473
An Illustrative Example
478
The AdamsBashforth Method
481
The RungeKutta Method
482
The Milne Method
486
Application to Differential Equations of Higher Order and to Systems of Equations
488
Types of Equations With Fixed Critical Points
495
Elements of the Linear Fractional Transformation
499
Coefficients of the Expansion of the First lainleve Transcendent
501
Coefficients of the Expansion of the Second Painleve Transcendent
503
BIRLIOGRAPRY
545
INDEX or NAMES
559
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