Chebyshev and Fourier Spectral Methods 2nd Edition by John P Boyd ISBN 0486411834 9780486411835 by John P. Boyd 9780486411835, 0486411834 instant download after payment.
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Product details:
ISBN 10: 0486411834
ISBN 13: 9780486411835
Author: John P Boyd
Chebyshev and Fourier Spectral Methods 2nd Table of contents:
Chapter 1: Introduction to Spectral Methods
- 1.1 What Are Spectral Methods?
- 1.2 Why Use Spectral Methods? Advantages and Disadvantages
- 1.3 A Brief History of Spectral Methods
- 1.4 Overview of Chebyshev and Fourier Basis Functions
- 1.5 Structure of the Book
Chapter 2: Fourier Spectral Methods: Foundations
- 2.1 The Continuous Fourier Series and Transform
- 2.2 The Discrete Fourier Transform (DFT)
- 2.3 The Fast Fourier Transform (FFT) Algorithm
- 2.4 Properties of the DFT and FFT
- 2.5 Fourier Interpolation and Aliasing
- 2.6 Numerical Differentiation using Fourier Series
- 2.7 Applications to Periodic Problems
Chapter 3: Chebyshev Polynomials and Their Properties
- 3.1 Definition and Recurrence Relations of Chebyshev Polynomials (Tn(x))
- 3.2 Orthogonality Properties and Inner Products
- 3.3 Roots (Chebyshev Nodes) and Extrema
- 3.4 Chebyshev Series Expansion of Functions
- 3.5 Properties of Chebyshev Series Coefficients
- 3.6 Relationship to Fourier Cosine Series
Chapter 4: Chebyshev Interpolation and Differentiation
- 4.1 Chebyshev-Lagrange Interpolation
- 4.2 The Discrete Chebyshev Transform (DCT)
- 4.3 Fast Chebyshev Transform (FCT) Algorithms
- 4.4 Chebyshev Differentiation Matrices
- 4.5 Properties of Spectral Derivatives
- 4.6 Numerical Integration using Chebyshev Methods
Chapter 5: Pseudospectral Methods
- 5.1 Collocation vs. Galerkin Methods
- 5.2 The Pseudospectral (Collocation) Approach
- 5.3 Implementation of Pseudospectral Differentiation
- 5.4 Treatment of Nonlinear Terms and Products
- 5.5 Aliasing in Pseudospectral Methods and Dealiasing Techniques
Chapter 6: Solving Ordinary Differential Equations (ODEs)
- 6.1 Spectral Methods for Initial Value Problems
- 6.2 Spectral Methods for Boundary Value Problems
- 6.3 Linear ODEs with Constant Coefficients
- 6.4 Linear ODEs with Variable Coefficients
- 6.5 Nonlinear ODEs
- 6.6 Eigenvalue Problems
Chapter 7: Solving Partial Differential Equations (PDEs): Elliptic Problems
- 7.1 Introduction to Elliptic PDEs
- 7.2 The Poisson Equation in 1D, 2D, and 3D
- 7.3 Boundary Conditions: Homogeneous and Non-homogeneous
- 7.4 Spectral Methods for Dirichlet, Neumann, and Robin Boundary Conditions
- 7.5 Iterative Solvers for Spectral Systems
Chapter 8: Solving Partial Differential Equations (PDEs): Parabolic Problems
- 8.1 Introduction to Parabolic PDEs (e.g., Heat Equation)
- 8.2 Time-Stepping Schemes: Explicit and Implicit Methods
- 8.3 Spectral Discretization in Space and Finite Differences in Time
- 8.4 Operator Splitting Techniques
- 8.5 Stability and Accuracy Considerations
Chapter 9: Solving Partial Differential Equations (PDEs): Hyperbolic Problems
- 9.1 Introduction to Hyperbolic PDEs (e.g., Wave Equation)
- 9.2 Spectral Discretization for Advection-Dominated Flows
- 9.3 Treatment of Shocks and Discontinuities (Shock-Capturing Schemes)
- 9.4 Stability Conditions for Hyperbolic Problems
Chapter 10: Advanced Topics and Extensions
- 10.1 Multi-Domain Methods and Domain Decomposition
- 10.2 Spectral Element Methods
- 10.3 Orthogonal Polynomials Beyond Chebyshev (e.g., Legendre, Hermite, Laguerre)
- 10.4 Spectral Methods for Unbounded Domains
- 10.5 High-Dimensional Problems
- 10.6 Adaptive Spectral Methods
Chapter 11: Implementation Aspects and Software Considerations
- 11.1 Choosing the Right Method for the Problem
- 11.2 Error Analysis and Convergence
- 11.3 Computational Cost and Efficiency
- 11.4 Practical Tips for Implementation (e.g., aliasing control, preconditioning)
- 11.5 Overview of Available Spectral Software Libraries (if applicable)
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