Partial Differential Equations
Lecture Notes
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\[u_t=\kappa \,\Delta u,\qquad \Phi (x,t)=\frac {1}{(4\pi \kappa t)^{n/2}}\,e^{-|x|^2/4\kappa t}\]
Contents
1 First-Order Equations and Characteristics
1.1 Partial differential equations
1.2 The transport equation
1.3 Linear equations in two variables
1.4 The Cauchy problem
1.5 Variable speed and conservation laws
1.6 An application: age-structured populations
1.7 Characteristics in software
1.8 Practice Problems
2 Non-linear Conservation Laws and Shocks
2.1 Quasilinear equations and breaking
2.2 Weak solutions and the Rankine–Hugoniot condition
2.3 The Riemann problem and the entropy condition
2.4 Traffic flow
2.5 Conservation laws in software
2.6 Practice Problems
3 Classification of Second-order Equations
3.1 The discriminant and its invariance
3.2 Characteristics and canonical forms
3.3 Well-posed problems
3.4 Classification in software
3.5 Practice Problems
4 The Wave Equation in One Dimension
4.1 D’Alembert’s formula
4.2 Causality: domains of dependence and influence
4.3 Energy
4.4 Sources: Duhamel’s principle
4.5 Boundaries and reflection
4.6 The wave equation in software
4.7 Practice Problems
5 Waves in Higher Dimensions
5.1 Spherical means and Kirchhoff’s formula
5.2 Poisson’s formula by descent
5.3 Huygens’ principle
5.4 Energy and sources
5.5 Waves in software
5.6 Practice Problems
6 The Heat Equation
6.1 The fundamental solution
6.2 The maximum principle
6.3 Periodic heating: Fourier’s cellar
6.4 The heat equation in software
6.5 Practice Problems
7 Separation of Variables and Fourier Series
7.1 Separation of variables
7.2 Fourier series
7.3 The Gibbs phenomenon
7.4 Justifying the series solution
7.5 Fourier series in software
7.6 Practice Problems
8 Laplace’s Equation
8.1 Harmonic functions and the fundamental solution
8.2 The mean value property
8.3 The disc and Poisson’s formula
8.4 Laplace’s equation in software
8.5 Practice Problems
9 Green’s Functions and Images
9.1 Green’s representation formula
9.2 The Green’s function
9.3 The method of images
9.4 Green’s functions in software
9.5 Practice Problems
10 Fourier and Laplace Transform Methods
10.1 The Fourier transform
10.2 Solving equations on the line
10.3 The Laplace transform
10.4 Transforms in software
10.5 Practice Problems
11 Separation in Polar, Cylindrical and Spherical Coordinates
11.1 The Laplacian in curvilinear coordinates
11.2 The vibrating drum
11.3 Heat in a cylinder
11.4 Spherical harmonics and Legendre polynomials
11.5 The age of the Earth
11.6 Special functions in software
11.7 Practice Problems
12 Energy Methods, Weak Solutions and the Dirichlet Principle
12.1 The Dirichlet principle
12.2 The Poincaré inequality
12.3 Weak solutions
12.4 Galerkin’s method and finite elements
12.5 Rayleigh quotients and energy decay
12.6 Variational methods in software
12.7 Practice Problems
13 Finite Difference Methods
13.1 Difference quotients and consistency
13.2 Stability: the von Neumann method
13.3 Implicit schemes
13.4 Hyperbolic equations and the CFL condition
13.5 Finite differences in software
13.6 Practice Problems
14 Reaction–Diffusion, Travelling Waves and Review
14.1 Reaction–diffusion equations
14.2 Travelling waves
14.3 Turing instability
14.4 Reaction–diffusion in software
14.5 Review of the course
14.6 Practice Problems
References
References
1 First-Order Equations and Characteristics
1.1 Partial differential equations
1.2 The transport equation
1.3 Linear equations in two variables
1.4 The Cauchy problem
1.5 Variable speed and conservation laws
1.6 An application: age-structured populations
1.7 Characteristics in software
1.8 Practice Problems
2 Non-linear Conservation Laws and Shocks
2.1 Quasilinear equations and breaking
2.2 Weak solutions and the Rankine–Hugoniot condition
2.3 The Riemann problem and the entropy condition
2.4 Traffic flow
2.5 Conservation laws in software
2.6 Practice Problems
3 Classification of Second-order Equations
3.1 The discriminant and its invariance
3.2 Characteristics and canonical forms
3.3 Well-posed problems
3.4 Classification in software
3.5 Practice Problems
4 The Wave Equation in One Dimension
4.1 D’Alembert’s formula
4.2 Causality: domains of dependence and influence
4.3 Energy
4.4 Sources: Duhamel’s principle
4.5 Boundaries and reflection
4.6 The wave equation in software
4.7 Practice Problems
5 Waves in Higher Dimensions
5.1 Spherical means and Kirchhoff’s formula
5.2 Poisson’s formula by descent
5.3 Huygens’ principle
5.4 Energy and sources
5.5 Waves in software
5.6 Practice Problems
6 The Heat Equation
6.1 The fundamental solution
6.2 The maximum principle
6.3 Periodic heating: Fourier’s cellar
6.4 The heat equation in software
6.5 Practice Problems
7 Separation of Variables and Fourier Series
7.1 Separation of variables
7.2 Fourier series
7.3 The Gibbs phenomenon
7.4 Justifying the series solution
7.5 Fourier series in software
7.6 Practice Problems
8 Laplace’s Equation
8.1 Harmonic functions and the fundamental solution
8.2 The mean value property
8.3 The disc and Poisson’s formula
8.4 Laplace’s equation in software
8.5 Practice Problems
9 Green’s Functions and Images
9.1 Green’s representation formula
9.2 The Green’s function
9.3 The method of images
9.4 Green’s functions in software
9.5 Practice Problems
10 Fourier and Laplace Transform Methods
10.1 The Fourier transform
10.2 Solving equations on the line
10.3 The Laplace transform
10.4 Transforms in software
10.5 Practice Problems
11 Separation in Polar, Cylindrical and Spherical Coordinates
11.1 The Laplacian in curvilinear coordinates
11.2 The vibrating drum
11.3 Heat in a cylinder
11.4 Spherical harmonics and Legendre polynomials
11.5 The age of the Earth
11.6 Special functions in software
11.7 Practice Problems
12 Energy Methods, Weak Solutions and the Dirichlet Principle
12.1 The Dirichlet principle
12.2 The Poincaré inequality
12.3 Weak solutions
12.4 Galerkin’s method and finite elements
12.5 Rayleigh quotients and energy decay
12.6 Variational methods in software
12.7 Practice Problems
13 Finite Difference Methods
13.1 Difference quotients and consistency
13.2 Stability: the von Neumann method
13.3 Implicit schemes
13.4 Hyperbolic equations and the CFL condition
13.5 Finite differences in software
13.6 Practice Problems
14 Reaction–Diffusion, Travelling Waves and Review
14.1 Reaction–diffusion equations
14.2 Travelling waves
14.3 Turing instability
14.4 Reaction–diffusion in software
14.5 Review of the course
14.6 Practice Problems
References
References