Methods of Applied Mathematics with a MATLAB Overview

"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 topics...it 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

• 1 Introduction.- 1.1 An Overview.- 1.2 Topics by Chapter.- 1.3 Applying Mathematics.- References.- 2 Fourier Series.- 2.1 Introduction.- 2.2 Inner Products and Fourier Expansions.- 2.3 Convergence of Fourier Series.- 2.4 Pointwise and Uniform Convergence of Fourier Series.- 2.5 Gibb’s Phenomenon and Summation Methods.- 2.6 Summation Methods.- 2.7 Fourier Series Properties.- 2.8 Periodic Solutions of Differential Equations.- 2.9 Impedance Methods and Periodic Solutions.- 2.10 Power Spectrum and Parseval’s Theorem.- References.- 3 Elementary Boundary Value Problems.- 3.1 Introduction.- 3.2 The One-Dimensional Diffusion Equation.- 3.3 The Wave Equation.- 3.4 The Potential Equation.- 3.5 Discrete Models of Boundary Value Problems.- 3.6 Separation of Variables.- 3.7 Half-Range Expansions and Symmetries.- 3.8 Some Matters of Detail.- References.- 4 Sturm-Liouville Theory and Boundary Value Problems.- 4.1 Further Boundary Value Problems.- 4.2 Selfadjoint Eigenvalue Problems.- 4.3 Sturm-Liouville Problems.- 4.4 Power Series and Singular Sturm-Liouville Problems.- 4.5 Cylindrical Problems and Bessel’s Equation.- 4.6 Multidimensional Problems and Forced Systems.- 4.7 Finite Differences and Numerical Methods.- 4.8 Variational Models and Finite Element Methods.- 4.9 Computational Finite Element Methods.- References.- 5 Functions of a Complex Variable.- 5.1 Complex Variables and Analytic Functions.- 5.2 Domains of Definition of Complex Functions.- 5.3 Integrals and Cauchy’s Theorem.- 5.4 Cauchy’s Integral Formula, Taylor Series, and Residues.- 5.5 Complex Variables and Fluid Flows.- 5.6 Conformal Mappings and the Principle of the Argument.- References.- 6 Laplace Transforms.- 6.1 Introduction.- 6.2 Definitions of the Laplace Transform.- 6.3 Mechanical Properties of LaplaceTransforms.- 6.4 Elementary Transforms and Fourier Series Calculations.- 6.5 Elementary Applications to Differential Equations.- 6.6 Convolutions, Impulse Responses, and Weighting Patterns.- 6.7 Vector Differential Equations.- 6.8 Impedance Methods.- References.- 7. Fourier Transforms.- 7.1 Introduction.- 7.2 Basic Fourier Transforms.- 7.3 Formal Properties of Fourier Transforms.- 7.4 Convolutions and Parseval’s Theorem.- 7.5 Comments on the Inversion Theorem.- 7.6 Fourier Inversion by Contour Integration.- 7.7 The Laplace Transform Inversion Integral.- 7.8 An Introduction to Generalized Functions.- 7.9 Fourier Transforms, Differential Equations and Circuits.- 7.10 Transform Solutions of Boundary Value Problems.- 7.11 Band-limited Functions and Communications.- References.- 8 Discrete Variable Transforms.- 8.1 Some Discrete Variable Models.- 8.2 Z-Transforms.- 8.3 Z-Transform Properties.- 8.4 z-Transform Inversion Integral.- 8.5 Discrete Fourier Transforms.- 8.6 Discrete Fourier Transform Properties.- 8.7 Some Applications of Discrete Transform Methods.- 8.8 Finite and Fast Fourier Transforms.- 8.9 Finite Fourier Properties.- 8.10 Fast Finite Transform Algorithm.- 8.11 Computing The 1-1.1.- References.- 9 Additional Topics.- 9.1 Local Waveform Analysis.- 9.2 Uncertainty Principle.- 9.3 Short-Time Fourier Transforms.- 9.4 Function Shifts and Scalings.- 9.5 Orthonormal Shifts.- 9.6 Multi-Resolution Analysis and Wavelets.- 9.7 On Wavelet Applications.- 9.8 Two-Sided Transforms.- 9.9 Walsh Functions.- 9.10 Geometrically Based Transforms.- References.- A Linear Algebra Overview.- A.1 Vector spaces.- A.2 Linear Mappings.- A.3 Inner Products.- A.4 Linear Functionals and Dual Spaces.- A.5 Canonical Forms.- References.- B Software Resources.- B.1 Computational andVisualization Software.- B.2 MATLAB Data Structures.- B.3 MATLAB Operators and Syntax.- B.4 MATLAB Programming Structures.- B.5 MATLAB Programs and Scripts.- B.6 Common Idioms.- B.7 Graphics.- B.8 Toolboxes and Enhancemants.- References.- C Transform Tables.- C.1 Laplace Transforms.- C.2 Fourier Transforms.- C.3 Z Transforms.- C.4 Discrete Fourier Transforms.