Methods of Applied Mathematics with a MATLAB Overview / Edition 1

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"Methods of Applied Mathematics with a MATLAB Overview" is devoted to the applications of Fourier analysis; topics encompass the classical applications in partial differential equations and boundary value problems as well as the concepts and methods associated to the Laplace, Fourier, and discrete transform. Transform inversion problems are covered, along with the necessary background in complex variables and Sturm-Liouville theory. The final chapter provides an examination of wavelets, short-time Fourier analysis, and geometrically-based transforms. The use of the computer program MATLAB has been integrated throughout the text, with particular emphasis on the program's numerical and graphing capabilities. Key features:
• Clear, consistent exposition with a strong focus on mathematical fundamentals and useful techniques
• Extensive examples, illustrations, and modern applications
• A variety of problems of wide-ranging difficulty, many with solutions
• Numerous MATLAB exercises and routines, with an introduction to MATLAB provided in an appendix
• Comprehensive references and index Requiring only familiarity with calculus and linear algebra and some introductory acquaintance with differential equations and vector calculus, this work can be used flexibly, either for a one-semester survey or an in-depth, year-long course. With its broad scope and careful pedagogy, this treatment will serve students in pure and applied mathematics, physical sciences, and engineering.

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Editorial Reviews

From the Publisher

"Overall, this textbook has an attractive format with lots of figures, programs, and formulas, and it presents, in a very traditional way, a body of material that is fundamental in applied mathematics, science, and engineering. It would make an excellent textbook for courses focused around Fourier analysis and applications to differential equations." —SIAM Review

"The aim of this book is to provide an introduction to a number of methods of applied mathematics, especially those arising from Fourier analysis. Classical problems of [a] mathematical physics nature are discussed throughout the book. Those problems represent the pretext or the basis for the elaboration of mathematical models. The book presents numerical schemes and analytical results as well.... This book is excellent material for students of applied mathematics and engineers, covering the theory of Fourier analysis with interesting applications and numerical examples in MATLAB." —Zentralblatt MATH

"This book is devoted to the application of Fourier analysis. The author mixed in a remarkable way theoretical results and applications illustrating the results. Flexibility of presentation (increasing and decreasing level of rigor, accessibility) is a key feature.

...The book contains extensive examples, presented in an intuitive way with high quality figures (some of them quite spectacular), useful MATLAB codes. MATLAB exercises and routines are well integrated within the text, and a concise introduction into MATLAB is given in an appendix. The emphasis is on the program's numerical and graphical capabilities and its applications, not on its syntax.... Applications are modern and up to date.... Comprehensive references are attached to each chapter.

Intended audience: especially students in pure and applied mathematics, physics and computer science, but also useful to applied mathematicians, engineers and computer scientists interested in applications of Fourier analysis." —Mathematica

"The topics covered are useful both in traditional continuum mechanics and mathematical physics areas, as well as in applied mathematics domains such as control and communications. The book provides in a clear and distinct manner the fundamental concepts and techniques of this area and a wide variety of problems in which these methods are useful. Both the theoretical and the computational aspects are emphasized.

...Davis's book has many novel features being quite different from most other textbooks on applied mathematics. Besides a nice treatment of all these also contains a very deep treatment of MATLAB implementations of methods and algorithms. Mainly it has a clear and consistent exposition with a strong focus on mathematical fundamentals and useful techniques. It has numerous extensive examples, illustrations, comments, and a very modern graphical presentation of results. A variety of problems of wide-ranging difficulty together with their solutions are presented. Plenty of MATLAB routines and exercises are also provided.

...The book has style. Every theorem and mathematical result has a wonderful appealing comment. While the author does not purposely go out of his way to be rigorous and very technical, he illustrates all these results by numerical examples implemented and solved in the MATLAB environment.

...All in all, I greatly enjoyed reviewing this book, and I recommend it without any hesitation as a textbook for advanced graduate or master's level courses in engineering." —Studies in Informatics and Control

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Product Details

  • ISBN-13: 9780817643317
  • Publisher: Birkhauser Verlag
  • Publication date: 11/3/2003
  • Series: Applied and Numerical Harmonic Analysis Series
  • Edition description: 2004
  • Edition number: 1
  • Pages: 721
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.56 (d)

Table of Contents

Preface v
1 Introduction 1
1.1 An Overview 1
1.2 Topics by Chapter 2
1.3 Applying Mathematics 3
References 3
2 Fourier Series 5
2.1 Introduction 5
2.2 Inner Products and Fourier Expansions 14
2.3 Convergence of Fourier Series 24
2.4 Pointwise and Uniform Convergence of Fourier Series 37
2.5 Gibb's Phenomenon and Summation Methods 43
2.6 Summation Methods 46
2.7 Fourier Series Properties 51
2.8 Periodic Solutions of Differential Equations 64
2.9 Impedance Methods and Periodic Solutions 69
2.10 Power Spectrum and Parseval's Theorem 70
References 78
3 Elementary Boundary Value Problems 79
3.1 Introduction 79
3.2 The One-Dimensional Diffusion Equation 80
3.3 The Wave Equation 86
3.4 The Potential Equation 91
3.5 Discrete Models of Boundary Value Problems 95
3.6 Separation of Variables 103
3.7 Half-Range Expansions and Symmetries 120
3.8 Some Matters of Detail 147
References 160
4 Sturm-Liouville Theory and Boundary Value Problems 163
4.1 Further Boundary Value Problems 163
4.2 Selfadjoint Eigenvalue Problems 170
4.3 Sturm-Liouville Problems 180
4.4 Power Series and Singular Sturm-Liouville Problems 199
4.5 Cylindrical Problems and Bessel's Equation 211
4.6 Multidimensional Problems and Forced Systems 225
4.7 Finite Differences and Numerical Methods 250
4.8 Variational Models and Finite Element Methods 258
4.9 Computational Finite Element Methods 272
References 277
5 Functions of a Complex Variable 279
5.1 Complex Variables and Analytic Functions 279
5.2 Domains of Definition of Complex Functions 287
5.3 Integrals and Cauchy's Theorem 294
5.4 Cauchy's Integral Formula, Taylor Series, and Residues 301
5.5 Complex Variables and Fluid Flows 316
5.6 Conformal Mappings and the Principle of the Argument 327
References 343
6 Laplace Transforms 345
6.1 Introduction 345
6.2 Definitions of the Laplace Transform 346
6.3 Mechanical Properties of Laplace Transforms 350
6.4 Elementary Transforms and Fourier Series Calculations 358
6.5 Elementary Applications to Differential Equations 368
6.6 Convolutions, Impulse Responses, and Weighting Patterns 376
6.7 Vector Differential Equations 391
6.8 Impedance Methods 404
References 409
7 Fourier Transforms 411
7.1 Introduction 411
7.2 Basic Fourier Transforms 417
7.3 Formal Properties of Fourier Transforms 423
7.4 Convolutions and Parseval's Theorem 433
7.5 Comments on the Inversion Theorem 447
7.6 Fourier Inversion by Contour Integration 458
7.7 The Laplace Transform Inversion Integral 478
7.8 An Introduction to Generalized Functions 493
7.9 Fourier Transforms, Differential Equations and Circuits 504
7.10 Transform Solutions of Boundary Value Problems 513
7.11 Band-limited Functions and Communications 529
References 541
8 Discrete Variable Transforms 543
8.1 Some Discrete Variable Models 543
8.2 Z-Transforms 551
8.3 Z-Transform Properties 554
8.4 z-Transform Inversion Integral 562
8.5 Discrete Fourier Transforms 577
8.6 Discrete Fourier Transform Properties 581
8.7 Some Applications of Discrete Transform Methods 597
8.8 Finite and Fast Fourier Transforms 610
8.9 Finite Fourier Properties 613
8.10 Fast Finite Transform Algorithm 620
8.11 Computing The FFT 626
References 633
9 Additional Topics 635
9.1 Local Waveform Analysis 635
9.2 Uncertainty Principle 636
9.3 Short-Time Fourier Transforms 641
9.4 Function Shifts and Scalings 645
9.5 Orthonormal Shifts 647
9.6 Multi-Resolution Analysis and Wavelets 648
9.7 On Wavelet Applications 661
9.8 Two-Sided Transforms 663
9.9 Walsh Functions 668
9.10 Geometrically Based Transforms 672
References 683
A Linear Algebra Overview 685
A.1 Vector spaces 685
A.2 Linear Mappings 687
A.3 Inner Products 688
A.4 Linear Functionals and Dual Spaces 689
A.5 Canonical Forms 689
References 691
B Software Resources 693
B.1 Computational and Visualization Software 693
B.2 MATLAB Data Structures 694
B.3 MATLAB Operators and Syntax 696
B.4 MATLAB Programming Structures 698
B.5 MATLAB Programs and Scripts 699
B.6 Common Idioms 700
B.7 Graphics 702
B.8 Toolboxes and Enhancemants 709
References 710
C Transform Tables 711
C.1 Laplace Transforms 711
C.2 Fourier Transforms 713
C.3 Z Transforms 714
C.4 Discrete Fourier Transforms 715
Index 716
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