Partial Differential Equations: An Introduction to Analytical and Numerical Methods

Foreword by the Translator
Preface
Acknowledgments
About the Authors
About the Translator
Contents
List of figures
1 Modeling, or where do differential equations come from
	1.1 Mathematical modeling
		1.1.1 Modeling with partial differential equations
		1.1.2 Modeling is only the first step
	1.2 Transport processes
		1.2.1 Conservation laws
		1.2.2 From a conservation law to a differential equation
		1.2.3 The linear transport equation
		1.2.4 The convection-reaction equation
		1.2.5* Burgers\' equation
	1.3 Diffusion
	1.4 The wave equation
	1.5 The Black–Scholes equation
	1.6 Let\'s get higher dimensional
		1.6.1 Transport processes
		1.6.2 Diffusion processes
		1.6.3 The wave equation
		1.6.4 Laplace\'s equation
	1.7* But there\'s more
		1.7.1 The KdV equation
		1.7.2 Geometric differential equations
		1.7.3 The plate equation
		1.7.4 The Navier–Stokes equations
		1.7.5 Maxwell\'s equations
		1.7.6 The Schrödinger equation
	1.8 Classification of partial differential equations
	1.9* Comments
	1.10 Exercises
2 Classification and characteristics
	2.1 Characteristics of initial value problems on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
		2.1.1 Homogeneous problems
		2.1.2 Inhomogeneous problems
		2.1.3* Burgers\' equation
	2.2 Equations of second order
	2.3* Nonlinear equations of second order
	2.4* Equations of higher order and systems
	2.5 Exercises
3 Elementary methods
	3.1 The one-dimensional wave equation
		3.1.1 D\'Alembert\'s formula on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R times double struck upper R) /StPNE pdfmark [/StBMC pdfmarkR Rps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
		3.1.2 The wave equation on an interval
	3.2 Fourier series
	3.3 Laplace\'s equation
		3.3.1 The Dirichlet problem on the unit square
		3.3.2 The Dirichlet problem on the disk
		3.3.3 The elliptic maximum principle
		3.3.4 Well-posedness of the Dirichlet problem for the square and the disk
	3.4 The heat equation
		3.4.1 Separation of variables
		3.4.2 The parabolic maximum principle
		3.4.3 Well-posedness of the parabolic initial-boundary value problem on the interval
		3.4.4 The heat equation in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript d) /StPNE pdfmark [/StBMC pdfmarkRdps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	3.5 The Black–Scholes equation
	3.6 Integral transforms
		3.6.1 The Fourier transform
		3.6.2* The Laplace transform
	3.7 Outlook
	3.8 Exercises
4 Hilbert spaces
	4.1 Inner product spaces
	4.2 Orthonormal bases
	4.3 Completeness
	4.4 Orthogonal projections
	4.5 Linear and bilinear forms
		4.5.1* Extensions and generalizations
	4.6 Weak convergence
	4.7 Continuous and compact operators
	4.8 The spectral theorem
	4.9* Comments on Chapter 4
	4.10 Exercises
5 Sobolev spaces and boundary value problems in dimension one
	5.1 Sobolev spaces in one variable
	5.2 Boundary value problems on the interval
		5.2.1 Dirichlet boundary conditions
		5.2.2 Neumann boundary conditions
		5.2.3 Robin boundary conditions
		5.2.4 Mixed and periodic boundary conditions
		5.2.5 Non-symmetric differential operators
		5.2.6* A variational approach to singularly perturbed problems and the transport equation
	5.3* Comments on Chapter 5
	5.4 Exercises
6 Hilbert space methods for elliptic equations
	6.1 Mollifiers
	6.2 Sobolev spaces on ΩRd
	6.3 The space H10 (Ω)
	6.4 Lattice operations on H1(Ω)
	6.5 The Poisson equation with Dirichlet boundary conditions
	6.6 Sobolev spaces and Fourier transforms
	6.7 Local regularity
	6.8 Inhomogeneous Dirichlet boundary conditions
	6.9 The Dirichlet problem
	6.10 Elliptic equations with Dirichlet boundary conditions
	6.11 H2-regularity
	6.12* Comments on Chapter 6
	6.13 Exercises
7 Neumann and Robin boundary conditions
	7.1 Gauss\'s theorem
	7.2 Proof of Gauss\'s theorem
	7.3 The extension property
	7.4 The Poisson equation with Neumann boundary conditions
	7.5 The trace theorem and Robin boundary conditions
	7.6* Comments on Chapter 7
	7.7 Exercises
8 Spectral decomposition and evolution equations
	8.1 A vector-valued initial value problem
	8.2 The heat equation: Dirichlet boundary conditions
	8.3 The heat equation: Robin boundary conditions
	8.4 The wave equation
	8.5 Inhomogeneous parabolic equations
	8.6* Space/time variational formulations
	8.7* Comments on Chapter 8
	8.8 Exercises
9 Numerical methods
	9.1 Finite differences for elliptic problems
		9.1.1 FDM: the one-dimensional case
		9.1.2 FDM: the two-dimensional case
	9.2 Finite elements for elliptic problems
		9.2.1 The Galerkin method
		9.2.2 Triangulation and approximation on triangles
		9.2.3 Affine functions on triangles
		9.2.4 Norms on triangles
		9.2.5 Transformation into a reference element
		9.2.6 Interpolation for finite elements
		9.2.7 Finite element spaces
		9.2.8 The Poisson problem on polygons
		9.2.9 The stiffness matrix and the linear system of equations
		9.2.10 Numerical experiments
	9.3* Extensions and generalizations
		9.3.1 The Petrov–Galerkin method
		9.3.2 Further extensions
	9.4 Parabolic problems
		9.4.1 Finite differences
		9.4.2 Finite elements
		9.4.3* Error estimates via space/time variational formulations
	9.5 The wave equation
		9.5.1 Finite differences
		9.5.2 Finite elements
	9.6* Comments on Chapter 9
	9.7 Exercises
10 Maple®, or why computers can sometimes help
	10.1 Maple®
		10.1.1 Elementary examples
		10.1.2 Solutions via Fourier transforms
		10.1.3 Laplace transforms
		10.1.4 It can also be done numerically
		10.1.5 Calculating function values
	10.2 Exercises
Appendix
	A.1 Banach spaces and linear operators
	A.2 The space C(K)
	A.3 Integration
	A.4 More details on the Black–Scholes equation
		A.4.1 Basics of stochastics
		A.4.2 Black–Scholes model
		A.4.3 The fair price
References
Index of names
Index of symbols
Index