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wen l fourier methods science engineering 2022

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Textbook in PDF format This innovative book discusses and applies the generalized Fourier Series to a variety of problems commonly encountered within science and engineering, equipping the readers with a clear pathway through which to use the Fourier methods as a solution technique for a wide range of differential equations and boundary value problems. Beginning with an overview of the conventional Fourier series theory, this book introduces the generalized Fourier series (GFS), emphasizing its notable rate of convergence when compared to the conventional Fourier series expansions. After systematically presenting the GFS as a powerful and unified solution method for ordinary differential equations and partial differential equations, this book expands on some representative boundary value problems, diving into their multiscale characteristics. This book will provide readers with the comprehensive foundation necessary for solving a wide spectrum of mathematical problems key to practical applications. It will also be of interest to researchers, engineers, and college students in various science, engineering, and mathematics fields. Preface Acknowledgments Authors Introduction Scales and Scale Effects Multiscale Phenomena and Multiscale Analysis Methods Fourier Series Methods in Scientific and Engineering Applications Scope of this Book References Fourier Series Expansions of Functions Periodic Functions and their Fourier Series Expansions Convergence of Fourier Series Expansionsin Fourier Series for the Derivatives of Functions References The Generalized Fourier Series with Accelerated Convergence Improving the Convergence of Fourier Series The Generalized Fourier Cosine Series Expansion with Accelerated Convergence The Generalized Fourier Sine Series with Accelerated Convergence The Generalized Fourier Series Expansion with Accelerated Convergence References The Generalized Fourier Series Solutions of the Euler-Bernoulli Beam Equation Linear Differential Equations with Constant Coefficients Characteristic Solutions of the Beam Equation Fourier Series Solutions of the Beam Problems The Generalized Fourier Series Solutions Convergence Assessment Numerical Examples References Fourier Series for the Derivatives of One-Dimensional Functions Integral Formulas Regarding the Derivatives of Functions Fourier Coefficients for the Derivatives of Functions Examples Sufficient Conditions for the Term-by-Term Differentiations of Fourier Series Fourier Series for the Partial Derivatives of Two-Dimensional Functions Integral Formulas Regarding Higher Order Partial Derivatives of Two-Dimensional Functions Fourier Coefficients for Partial Derivatives of Two-Dimensional Functions The Full-Range Fourier Series for the Partial Derivatives The Half-Range Fourier Series for the Partial Derivatives Examples Sufficient Conditions for Term-by-Term Differentiations of the Fourier Series of Two-Dimensional Functions Appendix: Additional Integral Formulas The Generalized Fourier Series of Functions Structural Decompositions of One-Dimensional Functions Generalized Fourier Series of One-Dimensional Functions The Generalized Full-Range Fourier Series of One-Dimensional Functions The Generalized Half-Range Fourier Cosine Series of One-Dimensional Functions The Generalized Half-Range Fourier Sine Series of One-Dimensional Functions The Polynomial-Based Generalized Fourier Series for One-Dimensional Functions Examples Error Indexes of the Series Approximations Convergence Characteristics Reproducing Property of Polynomials Approximation Accuracy The Generalized Fourier Series of Two-Dimensional Functions Structural Decompositions of Two-Dimensional Functions The Generalized Fourier Series Expansions for Two-Dimensional Functions The Generalized Full-Range Fourier Series The Generalized Half-Range Fourier Sine-Sine Series The Polynomial-Based Generalized Fourier Series Expansions Numerical Characteristics of the Generalized Fourier Series Error Norms of Simultaneous Series Approximations Convergence Characteristics The Accuracy of the Generalized Fourier Series Multiscale Fourier Series Methods for Linear Differential Equations The Generalized Fourier Series Solutions of One-Dimensional Boundary Value Problems The Generalized Full-Range Fourier Series Solutions The Generalized Half-Range Fourier Cosine Series Solutions The Generalized Half-Range Fourier Sine Series Solutions The Generalized Fourier Series Solutions for Two-Dimensional Boundary Value Problems The Generalized Full-Range Fourier Series Solutions The Generalized Half-Range Fourier Sine-Sine Series Solutions Limitations of the Polynomial-Based Generalized Fourier Series Methods Determination of the General Solution The General Solutions of One-Dimensional Boundary Value Problems The General Solutions of Two-Dimensional Boundary Value Problems Equivalent Transformation of the Solution Introduction of the Supplementary Solution The Multiscale Characteristic of the Solution Solution Schemes Multiscale Fourier Series Method for the Convection-Diffusion-Reaction Equation Multiscale Fourier Series Solution for One-Dimensional Convection-Diffusion-Reaction Equation Description of the Problem The General Solution The Supplementary Solution The Particular Solution The Multiscale Fourier Series Solution One-Dimensional Numerical Examples Convergence Characteristics Computational Efficiency Multiscale Characteristics Multiscale Fourier Series Solution for Two-Dimensional Convection-Diffusion-Reaction Equationthe Description of the Problem The General Solution The Supplementary Solution The Particular Solution The Multiscale Fourier Series Solution Two-Dimensional Numerical Examples Convergence Characteristics Multiscale Characteristics References Bending of Thick Plates on Elastic Foundations Description of the Problem The Multiscale Fourier Series Solutions The General Solution of the Transverse Displacement The General Solution of the Stress Function Expressions of the Multiscale Fourier Series Solutions Equivalent Transformation of the Solutions Expressions of Stress Resultants The Solution Obtained from the Energy Principle Numerical Examples Convergence Characteristics Multiscale Characteristics References Wave Propagation in Elastic Waveguides Description of the Problem The Multiscale Fourier Series Solutions The Differential Equations of Modal Functions Structural Decomposition of Modal Functions Expressions of Boundary Functions Expanded along the y-Direction Expressions of Boundary Functions Expanded along the x-Direction Expressions of Internal and Corner Functions Expressions of the Multiscale Fourier Series Solution Expressions of Stress Resultants Solving for Solutions Numerical Examples Wave Propagations in a Square Waveguide Frequency Spectra Wave Modes References Index

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