Fourier Analysis and Nonlinear Partial Differential Equations 1st Edition by Hajer Bahouri, Jean Yves Chemin, Raphael Danchin ISBN 9783642168307

Fourier Analysis and Nonlinear Partial Differential Equations 1st Edition by Hajer Bahouri, Jean Yves Chemin, Raphael Danchin – Ebook PDF Instant Download/Delivery. 9783642168307
Full download Fourier Analysis and Nonlinear Partial Differential Equations 1st Edition after payment

Product details:

ISBN 10: 
ISBN 13: 9783642168307
Author: Hajer Bahouri, Jean Yves Chemin, Raphael Danchin

In recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations.  It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity.

It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.

Fourier Analysis and Nonlinear Partial Differential Equations 1st Table of contents:

1. Basic Concepts of Nonlinear Partial Differential Equations

   • Introduction to partial differential equations.
   • Linear vs nonlinear PDEs: Definitions and distinctions.
   • First-order and higher-order nonlinear PDEs.
   • Applications in physics and engineering.

2. Fourier Series and Fourier Transforms

   • Fourier series for periodic functions.
   • Fourier transform for non-periodic functions.
   • Convergence and properties of Fourier transforms.
   • The Fourier inversion theorem.

3. Nonlinear Wave Equations and Fourier Analysis

   • The role of Fourier analysis in nonlinear wave equations.
   • Wave phenomena in nonlinear PDEs.
   • Existence and uniqueness of solutions to nonlinear wave equations.
   • Solving nonlinear wave equations using Fourier transforms.

4. Nonlinear Parabolic and Hyperbolic Equations

   • Fourier analysis applied to parabolic equations (e.g., heat equation).
   • Fourier transform methods for hyperbolic equations (e.g., Burgers’ equation).
   • Nonlinear diffusion equations and their solutions.
   • Techniques for analyzing long-time behavior of solutions.

5. The Role of Nonlinearity in Fourier Analysis

   • How nonlinearity affects the Fourier analysis.
   • Stability and instability of nonlinear PDEs.
   • The impact of nonlinearity on Fourier series and transforms.
   • Nonlinear effects in wave propagation and other phenomena.

6. Asymptotic Methods in Nonlinear PDEs

   • Asymptotic expansions and methods in nonlinear PDEs.
   • Use of Fourier analysis in asymptotic approximations.
   • Solving nonlinear PDEs in the long-time limit.
   • The method of multiple scales and its application in nonlinear PDEs.

7. Nonlinear Fourier Transforms and Their Applications

   • Introduction to nonlinear Fourier transforms.
   • Applications to nonlinear Schrödinger equations.
   • Fourier analysis of solitons and shock waves.
   • Nonlinear evolution equations in Fourier space.

8. Numerical Methods for Nonlinear PDEs Using Fourier Analysis

   • Discretization methods for nonlinear PDEs.
   • Fourier-based numerical techniques: spectral methods, FFT.
   • Solving nonlinear PDEs numerically with Fourier transforms.
   • Stability and error analysis in numerical methods.

9. Applications to Physical Phenomena

• Nonlinear PDEs in fluid dynamics (e.g., Navier-Stokes equations).
• Nonlinear PDEs in material science (e.g., elasticity and plasticity).
• Fourier methods in signal processing and wave propagation.
• Modeling and simulation of real-world problems using Fourier analysis.

11. Advanced Topics in Fourier Analysis and Nonlinear PDEs

• Nonlinear Fourier analysis in higher dimensions.
• Multiscale analysis and nonlinear PDEs.
• Solitary waves and integrable systems.
• The connection between chaos theory and nonlinear PDEs.

People also search for Fourier Analysis and Nonlinear Partial Differential Equations 1st:

fourier analysis and nonlinear partial
    
fourier analysis and nonlinear partial differential equations pdf
    
what is fourier analysis
    
fourier analysis explained
    
types of fourier analysis