Differential Equations and Their Applications: An Introduction to Applied Mathematics

Front Cover
Springer Science & Business Media, Dec 5, 1992 - Mathematics - 578 pages
Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as weil as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high Ievel of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. T AM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Seiences ( AMS) series, which will focus on advanced textbooks and research Ievel monographs. Preface to the Fourth Edition There are two major changes in the Fourth Edition of Differential Equations and Their Applications. The first concerns the computer programs in this text. In keeping with recent trends in computer science, we have replaced all the APL programs with Pascal and C programs. The Pascal programs appear in the text in place ofthe APL programs, where they are followed by the Fortran programs, while the C programs appear in Appendix C.
 

Contents

Firstorder differential equations
12 Firstorder linear differential equations
13 The Van Meegeren art forgeries
5
14 Separable equations
14
15 Population models
20
16 The spread of technological innovations
33
17 An atomic waste disposal problem
40
18 The dynamics of tumor growth mixing problems and orthogonal trajectories
46
33 Dimension of a vector space
273
34 Applications of linear algebra to differential equations
285
35 The theory of determinants
291
36 Solutions of simultaneous linear equations
304
37 Linear transformations
314
38 The eigenvalueeigenvector method of finding solutions
327
39 Complex roots
335
310 Equal roots
339

19 Exact equations and why we cannot solve very many differential equations
52
110 The existenceuniqueness theorem Picard iteration
61
111 Finding roots of equations by iteration
75
1111 Newtons method
81
112 Difference equations and how to compute the interest due on your student loans
85
113 Numerical approximations Eulers method
90
1131 Error analysis for Eulers method
94
114 The three term Taylor series method
101
115 An improved Euler method
103
116 The RungeKutta method
106
117 What to do in practice
110
Secondorder linear differential equations
121
22 Linear equations with constant coefficients
132
221 Complex roots
135
222 Equal roots reduction of order
139
23 The nonhomogeneous equation
145
24 The method of variation of parameters
147
25 The method of judicious guessing
151
26 Mechanical vibrations
159
261 The Tacoma Bridge disaster
167
262 Electrical networks
169
27 A model for the detection of diabetes
172
28 Series solutions
179
257 Singular points Euler equations
192
282 Regular singular points the method of Frobenius
197
283 Equal roots and roots differing by an integer
213
29 The method of Laplace transforms
219
210 Some useful properties of Laplace transforms
227
211 Differential equations with discontinuous righthand sides
232
212 The Dirac delta function
237
213 The convolution integral Consider the initialvalue problem
245
214 The method of elimination for systems
251
215 Higherorder equations
253
Systems of differential equations
258
32 Vector spaces
267
311 Fundamental matrix solutions eAl
349
312 The nonhomogeneous equation variation of parameters
354
313 Solving systems by Laplace transforms
362
Qualitative theory of differential equations
366
42 Stability of linear systems
372
43 Stability of equilibrium solutions
379
44 The phaseplane
388
45 Mathematical theories of war
392
452 Lanchesters combat models and the battle of Iwo Jima
399
46 Qualitative properties of orbits
408
47 Phase portraits of linear systems
412
48 Long time behavior of solutions the PoincareBendixson Theorem
422
49 Introduction to bifurcation theory
431
410 Predatorprey problems or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I
437
411 The principle of competitive exclusion in population biology
445
412 The Threshold Theorem of epidemiology
452
413 A model for the spread of gonorrhea
459
Separation of variables and Fourier series
470
52 Introduction to partial differential equations
475
53 The heat equation separation of variables
477
54 Fourier series
481
55 Even and odd functions
487
56 Return to the heat equation
492
57 The wave equation
497
58 Laplaces equation
502
SturmLiouville boundary value problems
508
62 Inner product spaces
509
63 Orthogonal bases Hermitian operators
520
64 SturmLiouville theory
527
Some simple facts concerning functions of several variables
539
Sequences and series
541
C Programs
543
Answers to oddnumbered exercises
551
Index
569
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About the author (1992)

Martin Braun is Professor of Differential Equations at Queen's College, The City University of New York, USA.

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