Differential Equations and Their Applications: An Introduction to Applied Mathematics

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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.
 

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This book is not very organized. It is hard to find different sections of chapters because there are no breaks in between. It explains things VERY well if you already know what is going on, but before I understood differential equations a little bit (I used Schaum's) it was rather difficult. It has some fascinating examples of how differential equations are applied and is over all a good book, it is just somewhat difficult to learn from. 

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
269
34 Applications of linear algebra to differential equations
281
35 The theory of determinants
287
36 Solutions of simultaneous linear equations
300
37 Linear transformations
310
38 The eigenvalueeigenvector method of finding solutions
323
39 Complex roots
331
310 Equal roots
335

19 Exact equations and why we cannot solve very many differential equations
52
110 The existenceuniqueness theorem Picard iteration
58
111 Finding roots of equations by iteration
71
1111 Newtons method
77
112 Difference equations and how to compute the interest due on your student loans
81
113 Numerical approximations Eulers method
86
1131 Error analysis for Eulers method
90
114 The three term Taylor series method
97
115 An improved Euler method
99
116 The RungeKutta method
102
117 What to do in practice
106
Secondorder linear differential equations
117
22 Linear equations with constant coefficients
128
221 Complex roots
131
222 Equal roots reduction of order
135
23 The nonhomogeneous equation
141
24 The method of variation of parameters
143
25 The method of judicious guessing
147
26 Mechanical vibrations
155
261 The Tacoma Bridge disaster
163
262 Electrical networks
165
27 A model for the detection of diabetes
168
28 Series solutions
175
257 Singular points Euler equations
188
282 Regular singular points the method of Frobenius
193
283 Equal roots and roots differing by an integer
209
29 The method of Laplace transforms
215
210 Some useful properties of Laplace transforms
223
211 Differential equations with discontinuous righthand sides
228
212 The Dirac delta function
233
213 The convolution integral Consider the initialvalue problem
241
214 The method of elimination for systems
247
215 Higherorder equations
249
Systems of differential equations
254
32 Vector spaces
263
311 Fundamental matrix solutions eAl
345
312 The nonhomogeneous equation variation of parameters
350
313 Solving systems by Laplace transforms
358
Qualitative theory of differential equations
362
42 Stability of linear systems
368
43 Stability of equilibrium solutions
375
44 The phaseplane
384
45 Mathematical theories of war
388
452 Lanchesters combat models and the battle of Iwo Jima
395
46 Qualitative properties of orbits
404
47 Phase portraits of linear systems
408
48 Long time behavior of solutions the PoincareBendixson Theorem
418
49 Introduction to bifurcation theory
427
410 Predatorprey problems or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I
433
411 The principle of competitive exclusion in population biology
441
412 The Threshold Theorem of epidemiology
448
413 A model for the spread of gonorrhea
455
Separation of variables and Fourier series
466
52 Introduction to partial differential equations
471
53 The heat equation separation of variables
473
54 Fourier series
477
55 Even and odd functions
483
56 Return to the heat equation
488
57 The wave equation
493
58 Laplaces equation
498
SturmLiouville boundary value problems
504
62 Inner product spaces
505
63 Orthogonal bases Hermitian operators
516
64 SturmLiouville theory
523
Some simple facts concerning functions of several variables
535
Sequences and series
537
C Programs
539
Answers to oddnumbered exercises
547
Index
565
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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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