Handbook of Differential Equations, Volume 1

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Gulf Professional Publishing, 1998 - 801 pagine
This book compiles the most widely applicable methods for solving and approximating differential equations. as well as numerous examples showing the methods use. Topics include ordinary differential equations, symplectic integration of differential equations, and the use of wavelets when numerically solving differential equations.

  • For nearly every technique, the book provides:
    • 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
    • References to the literature for more discussion or more examples, including pointers to electronic resources, such as URLs
 

Sommario

A Definitions and Concepts 1 Definition of Terms
1
Alternative Theorems
13
Bifurcation Theory
16
A Caveat for Partial Differential Equations
24
Chaos in Dynamical Systems
26
50
28
Classification of Partial Differential Equations
33
Compatible Systems
39
132
509
134
516
Monges Method
522
138
528
Boundary Layer Method
541
Functional Iteration
542
Multiple Scales
549
Regular Perturbation
553

Conservation Laws
43
Differential Resultants
46
Existence and Uniqueness Theorems
49
Fixed Point Existence Theorems
54
HamiltonJacobi Theory
56
Integrability of Systems
60
43
62
Internet Resources
66
Inverse Problems
69
Limit Cycles
72
Natural Boundary Conditions for a PDE
76
NearIdentity Transformations
78
Random Differential Equations
83
SelfAdjoint Eigenfunction Problems
86
Stability Theorems
92
SturmLiouville Theory
94
Classification of SturmLiouville Problems
97
Variational Equations
99
Well Posed Differential Equations
104
Wronskians and Fundamental Solutions
108
Zeros of Solutions
111
I
115
29
121
31
127
34
133
Transformation of an ODE to an Integral Equation
143
Reduction of PDEs to a First Order System
149
Transformations of Partial Differential Equations
156
2
163
Finite Element Method
174
51
175
Hybrid Computer Methods Invariant Imbedding
176
Multigrid Methods
177
5
188
II
205
ComputerAided Solution
218
Constant Coefficient Linear Equations
225
55
236
58
243
Equidimensionalinx Equations
250
61
256
64
263
67
272
69
279
71
285
73
297
75
303
77
309
79
319
80
326
83
344
85
352
88
360
91
370
96
381
Exact Methods for PDEs
387
99
396
101
403
104
411
108
419
Separation of Variables
443
114
450
Approximate Analytical Methods
463
Equation Splitting
471
The Phase Plane
479
The Tangent Field
485
Integral Methods
493
129
499
Strained Coordinates 144
556
Picard Iteration
560
Reversion Method
562
Singular Solutions
566
SolitonType Solutions
567
Stochastic Limit Theorems
569
Taylor Series Solutions
572
Eigenvalue Approximation
575
RayleighRitz
578
WKB Method
581
Concepts 154 Introduction to Numerical Methods
587
Definition of Terms for Numerical Methods
589
Integrating Stochastic Equations Symplectic Integration
591
Available Software
592
Rectilinear Grid 157 2 Two Dimensions Rectilinear Grid
599
Irregular Grid
601
Triangular Grid
602
Ux
605
184
606
Finite Difference Methodology
607
159
611
and for polar coordinates
612
Grid Generation Richardson Extrapolation
615
ODE Approximations
621
dependence
624
Courant Criterion
625
Von Neumann Test
626
Testing Differential Equation Routines
627
B Numerical Methods for ODEs 165 Analytic Continuation
631
Box Method
634
Shooting Method
638
Continuation Method
641
Continued Fractions
644
Cosine Method
647
Differential Algebraic Equations
650
EigenvalueEigenfunction Problems
657
Eulers Forward Method
659
542
705
Use of Wavelets
710
Weighted Residual Methods
712
Numerical Methods for PDEs 186 Boundary Element Method
717
549
721
Domain Decomposition 717 721
724
Finite Differences
729
560
733
MonteCarlo Method
737
Relaxation
739
Method of Characteristics
742
Finite Differences 742
746
Lattice Gas Dynamics
750
Method of Lines
752
Explicit Method
756
Implicit Method
760
MonteCarlo Method
765
Pseudospectral Method
771
569
780
765
782
575
783
578
784
589
785
592
786
599
789
605
792
611
793
615
797
626
798
659
799
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Informazioni sull'autore (1998)

Dr. Daniel Zwillinger is a Senior Principal Systems Engineer for the Raytheon Company. He was a systems requirements "book boss” for the Cobra Judy Replacement (CJR) ship and was a requirements and test lead for tracking on the Ungraded Early Warning Radars (UEWR). He has improved the Zumwalt destroyer's software accreditation process and he was test lead on an Active Electronically Scanned Array (AESA) radar. Dan is a subject matter expert (SME) in Design for Six Sigma (DFSS) and is a DFSS SME in Test Optimization, Critical Chain Program Management, and Voice of the Customer. He is currently leading a project creating Trust in Autonomous Systems. At Raytheon, he twice won the President's award for best Six Sigma project of the year: on converting planning packages to work packages for the Patriot missile, and for revising Raytheon's timecard system. He has managed the Six Sigma white belt training program. Prior to Raytheon, Dan worked at Sandia Labs, JPL, Exxon, MITRE, IDA, BBN, and The Mathworks (where he developed an early version of their Statistics Toolbox).For ten years, Zwillinger was owner and president of Aztec Corporation. As a small business, Aztec won several Small Business Innovation Research (SBIR) contracts. The company also created several software packages for publishing companies. Prior to Aztec, Zwillinger was a college professor at Rensselaer Polytechnic Institute in the department of mathematics.Dan has written several books on mathematics on the topics of differential equations, integration, statistics, and general mathematics. He is editor-in-chief of the Chemical Rubber Company's (CRC's) "Standard Mathematical Tables and Formulae”, and is on the editorial board for CRC's "Handbook of Chemistry and Physics”. Zwillinger holds a bachelor's degree in mathematics from the Massachusetts Institute of Technology (MIT). He earned his doctorate in applied mathematics from the California Institute of Technology (Caltech). Zwillinger is a certified Raytheon Six Sigma Expert and an ASQ certified Six Sigma Black Belt. He also holds a pilot's license.

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