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Full Description
Distributions in the Physical and Engineering Sciences is a comprehensive exposition on analytic methods for solving science and engineering problems which is written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important to practioners and researchers. The goal of the books is to give the reader, specialist and non-specialist usable and modern mathematical tools in their research and analysis.
Volume 1 provides detailed coverage of asymptotic methods, including the stationary phase and steepest descent methods, for Fourier and other integral transforms from an application perspective. Other topics covered include fractional calculus, the uncertainty principle, wavelets, and multiresolution analysis.
Volume 2 contains an analysis of the three basic types of linear PDEs - elliptic, parabolic, and hyperbolic - as well as chapters on first-order nonlinearPDEs and conservation laws. Nonlinear waves, Burger's equations, KdV equations, and the equations of gas dynamics and porous media are also covered.
Volume 3 extends the scope to the use of distributional tools in the theory of generalized stochastic processes and fields, and in anomalous fractional random dynamics. Chapters cover topics such as probability distributions; generalized stochastic processes, Brownian motion, and the white noise; stochastic differential equations and generalized random fields; Burgers turbulence and passive tracer transport in Burgers flows; and linear, nonlinear, and multiscale anomalous fractional dynamics in continuous media.
The careful explanations, accessible writing style, many illustrations/examples and solutions also make it suitable for use as a self-study reference by anyone seeking greater understanding and proficiency in the problem solving methods presented. The book isideal for a general scientific and engineering audience, yet it is mathematically precise.
Contents
Volume 1: Distributional and Fractal Calculus, Integral Transforms, and Wavelets.- Part I: Distributions and Their Basic Applications.- Basic Definitions and Operations.- Basic Applications: Rigorous and Pragmatic.- Part II: Integral Transforms and Divergent Series.- Fourier Transform.- Asymptotics of Fourier Transforms.- Stationary Phase and Related Method.- Singular Integrals and Fractal Calculus.- Uncertainty Principle and Wavelet Transforms.- Summation of Divergent Series and Integrals.- Answers and Solutions.- References.- Index.- Volume 2: Linear and Nonlinear Dynamics in Continuous Media.- Part III: Potentials, Diffusions, and Waves.- Potential Theory and Fundamental Solutions of Elliptic Equations.- Diffusions and Parabolic Evolution Equations.- Waves and Hyperbolic Equations.- Part IV: Nonlinear Partial Differential Equations.- First-Order Nonlinear PDEs and Conservation Laws.- Generalized Solutions of First-Order Nonlinear PDEs.- Nonlinear Waves and GrowingInterfaces: 1-D Burgers-KPZ Models.- Other Standard Nonlinear Models of Higher Order.- Answers and Solutions.- References.- Index.- Volume 3: Random and Anomalous Fractional Dynamics in Continuous Media.- Part V: Random Dynamics.- Basic Distributional Tools for Probability Theory.- Random Distributions: Generalized Stochastic Processes.- Dynamical and Statistical Characteristics of Random Fields and Waves.- Forced Burgers Turbulence and Passive Tracer Transport in Burgers Flows.- Probability Distributions of Passive Tracers in Randomly Moving Media.- Part VI: Anomalous Fractional Dynamics.- Levy Processes and their Generalized Derivatives.- Linear Anomalous Fractional Dynamics in Continuous Media.- Nonlinear and Multiscale Anomalous Fractional Dynamics in Continuous Media.- Appendix: Basic Facts about Distributions.- References.- Index.
