Introduction to Nonlinear Differential and Integral EquationsCourier 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. |
Table des matières
Nonlinear Operators | 1 |
Some Particular Differential Equations | 7 |
Integration of the Riccati Equation | 63 |
Solution of the Riccati Equation by Means of Continued Frac | 70 |
The Generalized Riccati Equation | 76 |
AN INTRODUCTION TO SECOND ORDER EQUATIONSTHE | 95 |
Solution of the Problem of Growth of Two Conflicting Populations | 102 |
A Generalization of Volterras Problem | 109 |
The Solution of y6y2 by the Method of Continuous Analytic Continuation | 256 |
The Numerical Evaluation of the First Painlevé Transcendent | 258 |
The Numerical Evaluation of the Second Painlevé 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 |
The Hereditary Factor in the Problem of Growth | 112 |
Curves of Pursuit | 113 |
Linear Pursuit | 115 |
Pursuit When the Path of the Pursued Is a Circle | 119 |
Conditions of Capture | 123 |
General Pursuit Curves | 125 |
ELLIPTIC INTEGRALS ELLIPTIC FUNCTIONS AND THETA 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 Formulas | 150 |
Expansions of the Elliptic Functions in Powers of u | 152 |
The Poles of the Elliptic Functions | 153 |
The Zeta Elliptic Function of Jacobi | 155 |
The Elliptic Functions of Weierstrass | 156 |
THETA FUNCTIONS | 157 |
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 |
A Equations Solved by Elliptic Functions B Equations in Which Critical Points Are Fixed Points C The General ized Riccati Equation of Second Orde... | 188 |
Existence Theorems | 189 |
The Problem of the Pendulum | 192 |
y6y² | 194 |
The Solution of y6y2 as a Laurent Series | 197 |
The Solution of y6y2 as a Taylors Series | 200 |
y6y21282 | 202 |
yAy+By³_ | 207 |
Solution of the General Elliptic Equation | 209 |
SECOND ORDER DIFFERENTIAL EQUATIONS OF POLYNOMIAL CLASS Page 1 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 Painlevé Transcendent__ | 229 |
The Boutroux Transformation of the first Painlevé Equation | 232 |
Definition of a New Transcendental Function | 234 |
Determination of the Parameter h | 236 |
Generalization of Sv | 237 |
The Second Painlevé Transcendent | 239 |
The Boutroux Transformation of the Second Painlevé Equation | 242 |
Methods of Analytic Continuation | 245 |
CONTINUOUS ANALYTIC CONTINUATION 1 Introduction | 247 |
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 |
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 | 309 |
A Preliminary Example | 311 |
The Stability Theorem | 317 |
An Application of the Stability Theorem | 322 |
Limit Cycles | 331 |
Some Further Comments About Limit Cycles | 336 |
Periodic SolutionsThe Homogeneous Polynomial Case | 339 |
Periodic SolutionsThe General Quadratic Equation | 343 |
Topological ConsiderationsPoincarés IndexBendixons Theorem | 351 |
SOME PARTICULAR EQUATIONS Page 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 Generalized Equation of Blasius | 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 uxС Kxt ut dt | 429 |
The Equation of Bratu | 432 |
The Nonlinear Convolution Theorem | 434 |
PROBLEMS FROM THE CALCULUS OF VARIATIONS 1 Introduction | 439 |
The Euler Condition | 441 |
The Euler Condition in the Isoperimetric Case | 445 |
The Euler Condition for a Double Integral | 450 |
The Problem of the Minimal Surface | 452 |
Hamiltons PrincipleThe Principle of Least Action | 456 |
The Canonical Equations of Hamilton | 461 |
THE 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 Painlevé Transcendent | 501 |
BIBLIOGRAPHY | 545 |
559 | |
Autres éditions - Tout afficher
Introduction to Nonlinear Differential and Integral Equations Harold Thayer Davis Affichage du livre entier - 1960 |
Introduction to Nonlinear Differential and Integral Equations Harold Thayer Davis Affichage d'extraits - 1962 |
Introduction to Nonlinear Differential and Integral Equations Harold Thayer Davis Affichage d'extraits - 1962 |