Partial Differential Equations in Action From Modelling to Theory

ISBN-10: 8847007518

ISBN-13: 9788847007512

Edition: 2009

Authors: Sandro Salsa

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This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering. The main purpose is on the one hand to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences; on the other hand to give them a solid theoretical background for numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first one has a rather elementary character with the goal of developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. Ideas and connections with concrete aspects are emphasized whenever possible, in order to provide intuition and feeling for the subject. For this part, a knowledge of advanced calculus and ordinary differential equations is required. Also, the repeated use of the method of separation of variables assumes some basic results from the theory of Fourier series, which are summarized in an appendix. The main topic of the second part is the development of Hilbert space methods for the variational formulation and analysis of linear boundary and initial-boundary value problems\emph{. }% Given the abstract nature of these chapters, an effort has been made to provide intuition and motivation for the various concepts and results. The understanding of these topics requires some basic knowledge of Lebesgue measure and integration, summarized in another appendix. At the end of each chapter, a number of exercises at different levelof complexity is included. The most demanding problems are supplied with answers or hints. The exposition if flexible enough to allow substantial changes without compromising the comprehension and to facilitate a selection of topics for a one or two semester course.
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Book details

List price: $69.95
Copyright year: 2009
Publisher: Springer London, Limited
Publication date: 12/18/2009
Binding: Paperback
Pages: 556
Size: 6.00" wide x 9.00" long x 1.25" tall
Weight: 2.398
Language: English

Mathematical Modelling
Partial Differential Equations
Well Posed Problems
Basic Notations and Facts
Smooth and Lipschitz Domains
Integration by Parts Formulas
The Diffusion Equation
The conduction of heat
Well posed problems (n = 1)
A solution by separation of variables
Problems in dimension n > 1
Integral method
Maximum principles
The Fundamental Solution
Invariant transformations
Fundamental solution (n = 1)
The Dirac distribution
Fundamental solution (n > 1)
Symmetric Random Walk (n = 1)
Preliminary computations
The limit transition probability
From random walk to Brownian motion
Diffusion, Drift and Reaction
Random walk with drift
Pollution in a channel
Random walk with drift and reaction
Multidimensional Random Walk
The symmetric case
Walks with drift and reaction
An Example of Reaction-Diffusion (n = 3)
The Global Cauchy Problem (n = 1)
The homogeneous case
Existence of a solution
The non homogeneous case. Duhamel's method
Maximum principles and uniqueness
An Application to Finance
European options
An evolution model for the price S
The Black-Scholes equation
The solutions
Hedging and self-financing strategy
Some Nonlinear Aspects
Nonlinear diffusion. The porous medium equation
Nonlinear reaction. Fischer's equation
The Laplace Equation
Well Posed Problems. Uniqueness
Harmonic Functions
Discrete harmonic functions
Mean value properties
Maximum principles
The Dirichlet problem in a circle. Poisson's formula
Harnack's inequality and Liouville's theorem
A probabilistic solution of the Dirichlet problem
Recurrence and Brownian motion
Fundamental Solution and Newtonian Potential
The fundamental solution
The Newtonian potential
A divergence-curl system. Helmholtz decomposition formula
The Green Function
An integral identity
The Green function
Green's representation formula
The Neumann function
Uniqueness in Unbounded Domains
Exterior problems
Surface Potentials
The double and single layer potentials
The integral equations of potential theory
Scalar Conservation Laws and First Order Equations
Linear Transport Equation
Pollution in a channel
Distributed source
Decay and localized source
Inflow and outflow characteristics. A stability estimate
Traffic Dynamics
A macroscopic model
The method of characteristics
The green light problem
Traffic jam ahead
Integral (or Weak) Solutions
The method of characteristics revisited
Definition of integral solution
The Rankine-Hugoniot condition
The entropy condition
The Riemann problem
Vanishing viscosity method
The viscous Burger equation
The Method of Characteristics for Quasilinear Equations
The Cauchy problem
Lagrange method of first integrals
Underground flow
General First Order Equations
Characteristic strips
The Cauchy Problem
Waves and Vibrations
General Concepts
Types of waves
Group velocity and dispersion relation
Transversal Waves in a String
The model
The One-dimensional Wave Equation
Initial and boundary conditions
Separation of variables
The d'Alembert Formula
The homogeneous equation
Generalized solutions and propagation of singularities
The fundamental solution
Non homogeneous equation. Duhamel's method
Dissipation and dispersion
Second Order Linear Equations
Characteristics and canonical form
Hyperbolic Systems with Constant Coefficients
The Multi-dimensional Wave Equation (n > 1)
Special solutions
Well posed problems. Uniqueness
Two Classical Models
Small vibrations of an elastic membrane
Small amplitude sound waves
The Cauchy Problem
Fundamental solution (n = 3) and strong Huygens' principle
The Kirchhoff formula
Cauchy problem in dimension 2
Non homogeneous equation. Retarded potentials
Linear Water Waves
A model for surface waves
Dimensionless formulation and linearization
Deep water waves
Interpretation of the solution
Asymptotic behavior
The method of stationary phase
Elements of Functional Analysis
Norms and Banach Spaces
Hilbert Spaces
Projections and Bases
Linear Operators and Duality
Linear operators
Functionals and dual space
The adjoint of a bounded operator
Abstract Variational Problems
Bilinear forms and the Lax-Milgram Theorem
Minimization of quadratic functionals
Approximation and Galerkin method
Compactness and Weak Convergence
Weak convergence and compactness
Compact operators
The Fredholm Alternative
Solvability for abstract variational problems
Fredholm's Alternative
Spectral Theory for Symmetric Bilinear Forms
Spectrum of a matrix
Separation of variables revisited
Spectrum of a compact self-adjoint operator
Application to abstract variational problems
Distributions and Sobolev Spaces
Distributions. Preliminary Ideas
Test Functions and Mollifiers
The derivative in the sense of distributions
Gradient, divergence, lapacian
Multiplication, Composition, Division, Convolution
Multiplication. Leibniz rule
Fourier Transform
Tempered distributions
Fourier transform in S'
Fourier transform in L[superscript 2]
Sobolev Spaces
An abstract construction
The space H[superscript 1] ([Omega])
The space H[superscript 1 subscript 0] ([Omega])
The dual of H[superscript 1 subscript 0]([Omega])
The spaces H[superscript m] ([Omega]), m > 1
Calculus rules
Fourier Transform and Sobolev Spaces
Approximations by Smooth Functions and Extensions
Local approximations
Estensions and global approximations
Traces of functions in H[superscript 1] ([Omega])
Traces of functions in H[superscript m] ([Omega])
Trace spaces
Compactness and Embeddings
Rellich's theorem
Poincare's inequalities
Sobolev inequality in R[superscript n]
Bounded domains
Spaces Involving Time
Functions with values in Hilbert spaces
Sobolev spaces involving time
Variational Formulation of Elliptic Problems
Elliptic Equations
The Poisson Problem
Diffusion, Drift and Reaction (n = 1)
The problem
Dirichlet conditions
Neumann, Robin and mixed conditions
Variational Formulation of Poisson's Problem
Dirichlet problem
Neumann, Robin and mixed problems
Eigenvalues of the Laplace operator
An asymptotic stability result
General Equations in Divergence Form
Basic assumptions
Dirichlet problem
Neumann problem
Robin and mixed problems
Weak Maximum Principles
Equilibrium of a plate
A Monotone Iteration Scheme for Semilinear Equations
A Control Problem
Structure of the problem
Existence and uniqueness of an optimal pair
Lagrange multipliers and optimality conditions
An iterative algorithm
Weak Formulation of Evolution Problems
Parabolic Equations
Diffusion Equation
The Cauchy-Dirichlet problem
Faedo-Galerkin method (I)
Solution of the approximate problem
Energy estimates
Existence, uniqueness and stability
The Cauchy-Neuman problem
Cauchy-Robin and mixed problems
A control problem
General Equations
Weak formulation of initial value problems
Faedo-Galerkin method (II)
The Wave Equation
Hyperbolic Equations
The Cauchy-Dirichlet problem
Faedo-Galerkin method (III)
Solution of the approximate problem
Energy estimates
Existence, uniqueness and stability
Fourier Series
Fourier coefficients
Expansion in Fourier series
Measures and Integrals
Lebesgue Measure and Integral
A counting problem
Measures and measurable functions
The Lebesgue integral
Some fundamental theorems
Probability spaces, random variables and their integrals
Identities and Formulas
Gradient, Divergence, Curl, Laplacian
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