Methods of Applied Mathematics with a MATLAB Overview, Softcover reprint of the original 1st ed. 2004 Applied and Numerical Harmonic Analysis Series

Broadly organized around the applications of Fourier analysis, Methods of Applied Mathematics with a MATLAB Overview covers both classical applications in partial differential equations and boundary value problems, as well as the concepts and methods associated to the Laplace, Fourier, and discrete transforms. Transform inversion problems are also examined, along with the necessary background in complex variables. A final chapter treats wavelets, short-time Fourier analysis, and geometrically-based transforms. The computer program MATLAB is emphasized throughout, and an introduction to MATLAB is provided in an appendix. Rich in examples, illustrations, and exercises of varying difficulty, this text can be used for a one- or two-semester course and is ideal for students in pure and applied mathematics, physics, and engineering.

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.

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