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E-raamat: Handbook of Differential Equations

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(Brown University, Providence, Rhode Island, USA), (Rensselaer Polytechnic Institute, NY, USA)
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Through the previous three editions, Handbook of Differential Equations has proven an invaluable reference for anyone working within the field of mathematics, including academics, students, scientists, and professional engineers.

The book is a compilation of methods for solving and approximating differential equations. These include the most widely applicable methods for solving and approximating differential equations, as well as numerous methods. Topics include methods for ordinary differential equations, partial differential equations, stochastic differential equations, and systems of such equations.

Included for nearly every method are:











The types of equations to which the method is applicable The idea behind the method The procedure for carrying out the method At least one simple example of the method Any cautions that should be exercised Notes for more advanced users

The fourth edition includes corrections, many supplied by readers, as well as many new methods and techniques. These new and corrected entries make necessary improvements in this edition.
Preface xiii
Introduction xv
How to Use This Book xvii
I.A Definitions and Concepts
1 Definition of Terms
1(11)
2 Alternative Theorems
12(2)
3 Bifurcation Theory
14(5)
4 Chaos in Dynamical Systems
19(5)
5 Classification of Partial Differential Equations
24(5)
6 Compatible Systems
29(3)
7 Conservation Laws
32(2)
8 Differential Equations - Diagrams
34(3)
9 Differential Equations - Symbols
37(1)
10 Differential Resultants
38(2)
11 Existence and Uniqueness Theorems
40(3)
12 Fixed Point Existence Theorems
43(2)
13 Hamilton-Jacobi Theory
45(3)
14 Infinite Order Differential Equations
48(2)
15 Integrability of Systems
50(4)
16 Inverse Problems
54(1)
17 Limit Cycles
55(3)
18 PDEs & Natural Boundary Conditions
58(2)
19 Normal Forms: Near-Identity Transformations
60(4)
20 q-Differential Equations
64(2)
21 Quaternionic Differential Equations
66(1)
22 Self-Adjoint Eigenfunction Problems
67(5)
23 Stability Theorems
72(2)
24 Stochastic Differential Equations
74(2)
25 Sturm-Liouville Theory
76(4)
26 Variational Equations
80(4)
27 Web Resources
84(2)
28 Well-Posed Differential Equations
86(2)
29 Wronskians & Fundamental Solutions
88(3)
30 Zeros of Solutions
91(4)
I.B Transformations
31 Canonical Forms
95(2)
32 Canonical Transformations
97(2)
33 Darboux Transformation
99(2)
34 An Involutory Transformation
101(2)
35 Liouville Transformation - 1
103(1)
36 Liouville Transformation - 2
104(1)
37 Changing Linear ODEs to a First Order System
105(2)
38 Transformations of Second Order Linear ODEs - 1
107(3)
39 Transformations of Second Order Linear ODEs - 2
110(2)
40 Transforming an ODE to an Integral Equation
112(1)
41 Miscellaneous ODE Transformations
113(4)
42 Transforming PDEs Generically
117(3)
43 Transformations of PDEs
120(4)
44 Transforming a PDE to a First Order System
124(2)
45 Prufer Transformation
126(1)
46 Modified Prufer Transformation
127(2)
II Exact Analytical Methods
47 Introduction to Exact Analytical Methods
129(1)
48 Look-Up Technique
130(19)
48.1 Ordinary Differential Equations
131(5)
48.2 Partial Differential Equations
136(5)
48.3 Systems of Differential Equations
141(2)
48.4 Hamiltonians Representing Differential Equations
143(2)
48.5 The Laplacian in Different Coordinate Systems
145(1)
48.6 Parametrized Differential Equations at Specific Values
146(3)
49 Look-Up ODE Forms
149(2)
II.A Exact Methods for ODEs
50 Use of the Adjoint Equation
151(3)
51 An Nth Order Equation
154(1)
52 Autonomous Equations - Independent Variable Missing
155(2)
53 Bernoulli Equation
157(3)
54 Clairaut's Equation
160(1)
55 Constant Coefficient Linear ODEs
161(1)
56 Contact Transformation
162(3)
57 Delay Equations
165(5)
58 Dependent Variable Missing
170(1)
59 Differentiation Method
171(1)
60 Differential Equations with Discontinuities
172(2)
61 Eigenfunction Expansions
174(5)
62 Equidimensional-in-x Equations
179(2)
63 Equidimensional-in-y Equations
181(2)
61 Euler Equations
183(2)
65 Exact First Order Equations
185(2)
66 Exact Second Order Equations
187(2)
67 Exact Nth Order Equations
189(1)
68 Factoring Equations
190(1)
69 Factoring/Composing Operators
191(5)
70 Factorization Method
196(2)
71 Fokker Planck Equation
198(4)
72 Fractional Differential Equations
202(2)
73 Free Boundary Problems
204(3)
71 Generating Functions
207(2)
75 Green's Functions
209(9)
76 ODEs with Homogeneous Functions
218(2)
77 Hypergeometric Equation
220(4)
78 Method of Images
224(3)
79 Integrable Combinations
227(1)
80 Integrating Factors
228(2)
81 Interchanging Dependent and Independent Variables
230(2)
82 Integral Representation: Laplace's Method
232(4)
83 Integral Transforms: Finite Intervals
236(3)
84 Integral Transforms: Infinite Intervals
239(6)
85 Lagrange's Equation
245(2)
86 Lie Algebra Technique
247(4)
87 Lie Groups: ODEs
251(10)
88 Non-normal Operators
261(2)
89 Operational Calculus
263(3)
90 Pfaffian Differential Equations
266(3)
91 Quasilinear Second Order ODEs
269(3)
92 Quasipolynomial ODEs
272(3)
93 Reduction of Order
275(2)
94 Resolvent Method for Matrix ODEs
277(2)
95 Riccati Equation - Matrices
279(2)
96 Riccati Equation - Scalars
281(2)
97 Scale-Invariant Equations
283(2)
98 Separable Equations
285(1)
99 Series Solution
286(4)
100 Equations Solvable for x
290(1)
101 Equations Solvable for y
291(2)
102 Superposition
293(1)
103 Undetermined Coefficients
294(2)
104 Variation of Parameters
296(2)
105 Vector ODEs
298(5)
II.B Exact Methods for PDEs
106 Backlund Transformations
303(2)
107 Cagniard-de Hoop Method
305(3)
108 Method of Characteristics
308(4)
109 Characteristic Strip Equations
312(2)
110 Conformal Mappings
314(3)
111 Method of Descent
317(1)
112 Diagonalizable Linear Systems of PDEs
318(2)
113 Duhamel's Principle
320(2)
114 Exact Partial Differential Equations
322(1)
115 Fokas Method/Unified Transform
323(8)
116 Hodograph Transformation
331(2)
117 Inverse Scattering
333(3)
118 Jacobi's Method
336(1)
119 Legendre Transformation
337(3)
120 Lie Groups: PDEs
340(5)
121 Many Consistent PDEs
345(1)
122 Poisson Formula
346(2)
123 Resolvent Method for PDEs
348(3)
124 Riemann's Method
351(4)
125 Separation of Variables
355(4)
126 Separable Equations: Stackel Matrix
359(2)
127 Similarity Methods
361(3)
128 Exact Solutions to the Wave Equation
364(2)
129 Wiener-Hopf Technique
366(5)
III Approximate Analytical Methods
130 Introduction to Approximate Analysis
371(1)
131 Adomian Decomposition Method
372(3)
132 Chaplygin's Method
375(2)
133 Collocation
377(1)
134 Constrained Functions
378(2)
135 Differential Constraints
380(1)
136 Dominant Balance
381(2)
137 Equation Splitting
383(1)
138 Floquet Theory
384(3)
139 Graphical Analysis: The Phase Plane
387(5)
140 Graphical Analysis: Poincare Map
392(2)
141 Graphical Analysis: Tangent Field
394(2)
142 Harmonic Balance
396(2)
143 Homogenization
398(3)
144 Integral Methods
401(2)
145 Interval Analysis
403(2)
146 Least Squares Method
405(2)
147 Equivalent Linearization and Nonlinearization
407(3)
148 Lyapunov Functional
410(2)
149 Maximum Principles
412(4)
150 McGarvey Iteration Technique
416(2)
151 Moment Equations: Closure
418(2)
152 Moment Equations: Ito Calculus
420(2)
153 Monge's Method
422(2)
154 Newton's Method
424(2)
155 Pade Approximants
426(3)
156 Paratnetrix Method
429(1)
157 Perturbation Method: Averaging
430(2)
158 Perturbation Method: Boundary Layers
432(6)
159 Perturbation Method: Functional Iteration
438(4)
160 Perturbation Method: Multiple Scales
442(3)
161 Perturbation Method: Regular Perturbation
445(2)
162 Perturbation Method: Renormalization Group
447(5)
163 Perturbation Method: Strained Coordinates
452(2)
164 Picard Iteration
454(2)
165 Reversion Method
456(1)
166 Singular Solutions
457(3)
167 Soliton-Type Solutions
460(2)
168 Stochastic Limit Theorems
462(2)
169 Structured Guessing
464(2)
170 Taylor Series Solutions
466(3)
171 Variational Method: Eigenvalue Approximation
469(5)
172 Variational Method: Rayleigh Ritz
474(1)
173 WKB Method
474(3)
IV.A Numerical Methods: Concepts
174 Introduction to Numerical Methods
477(2)
175 Terms for Numerical Methods
479(2)
176 Finite Difference Formulas
481(5)
176.1 One Dimension: Rectilinear Grid
481(1)
176.2 Two Dimensions: Rectilinear Grid
482(1)
176.3 Two Dimensions: Irregular Grid
483(1)
176.1 Two Dimensions: Triangular Grid
483(1)
176.5 Numerical Schemes for the ODE: y' = f (x,y)
484(1)
176.6 Explicit Numerical Schemes for the PDE: aux + ut = 0
484(1)
176.7 Implicit Numerical Schemes for the PDE: aux +ut = S(x,1)
485(1)
176.8 Numerical Schemes for the PDE: F(u)x + ut = 0
485(1)
176.9 Numerical Schemes for the PDE: ux = utt
485(1)
177 Finite Difference Methodology
486(3)
178 Grid Generation
489(3)
179 Richardson Extrapolation
492(2)
180 Stability: ODE Approximations
494(3)
181 Stability: Courant Criterion
497(2)
182 Stability: Von Neumann Test
499(2)
183 Testing Differential Equation Routines
501(2)
IV.B Numerical Methods for ODEs
184 Analytic Continuation
503(2)
185 Boundary Value Problems: Box Method
505(2)
186 Boundary Value Problems: Shooting Method
507(3)
187 Continuation Method
510(2)
188 Continued Fractions
512(1)
189 Cosine Method
513(3)
190 Differential Algebraic Equations
516(4)
191 Eigenvalue/Eigenfunction Problems
520(2)
192 Euler's Forward Method
522(2)
193 Finite Element Method
524(7)
194 Hybrid Computer Methods
531(2)
195 Invariant Imbedding
533(3)
196 Multigrid Methods
536(2)
197 Neural Networks & Optimization
538(1)
198 Nonstandard Finite Difference Schemes
539(1)
199 ODEs with Highly Oscillatory Terms
540(3)
200 Parallel Computer Methods
543(2)
201 Predictor-Corrector Methods
545(2)
202 Probabilistic Methods
547(2)
203 Quantum Computing
549(1)
204 Runge-Kutta Methods
550(5)
205 Stiff Equations
555(3)
206 Integrating Stochastic Equations
558(3)
207 Symplectic Integration
561(3)
208 System Linearization via Koopman
564(1)
209 Using Wavelets
565(2)
210 Weighted Residual Methods
567(4)
IV.C Numerical Methods for PDEs
211 Boundary Element Method
571(3)
212 Differential Quadrature
574(2)
213 Domain Decomposition
576(3)
214 Elliptic Equations: Finite Differences
579(4)
215 Elliptic Equations: Monte-Carlo Method
583(4)
216 Elliptic Equations: Relaxation
587(2)
217 Hyperbolic Equations: Method of Characteristics
589(2)
218 Hyperbolic Equations: Finite Differences
591(3)
219 Lattice Gas Dynamics
594(2)
220 Method of Lines
596(2)
221 Parabolic Equations: Explicit Method
598(3)
222 Parabolic Equations: Implicit Method
601(3)
223 Parabolic Equations: Monte-Carlo Method
604(5)
224 Pseudospectral Method
609(4)
V Computer Languages and Systems
225 Computer Languages and Packages
613(2)
226 Julia Programming Language
615(2)
227 Maple Computer Algebra System
617(4)
228 Mathematica Computer Algebra System
621(3)
229 MATLAB Programming Language
624(4)
230 Octave Programming Language
628(3)
231 Python Programming Language
631(1)
232 R Programming Language
632(3)
233 Sage Computer Algebra System
635(4)
Mathematical Nomenclature 639(60)
Named Differential Equations 699(4)
Index 703
Daniel Zwillinger has more than 35 years of proven technical expertise in numerous areas of engineering and the physical sciences. He earned a Ph.D. in applied mathematics from the California Institute of Technology. He is the Editor of CRC Standard Mathematical Tables and Formulas, 33rd edition and also Table of Integrals, Series, and Products, Gradshteyn and Ryzhik. He serves as the Series Editor on the CRC Series of Advances in Applied Mathematics.

Vladimir A. Dobrushkin is a Professor at the Division of Applied Mathematics, Brown University. He holds a Ph.D. in Applied mathematics and Dr.Sc. in mechanical engineering. He is the author of three books for CRC Press, including Applied Differential Equations: The Primary Course, Applied Differential Equations with Boundary Value Problems, and Methods in Algorithmic Analysis.
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