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- Product code: 15831
- ISBN: 0817639675,
ISBN13: 9780817639679,
380 pages, hardback
Published by Birkhauser Verlag AG on 1997
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Description of Introduction to Partial Differential Equations with MATLAB |
At the heart of partial differential equations is an unchanging core of material, but the subject is constantly expanding and evolving. This advanced textbook reference is an introduction to partial differential equations covering the traditional subjects of the heat equation, wave equation and Laplace equation in the traditional manner using the method of separation of variables.
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Contents of Introduction to Partial Differential Equations with MATLAB |
Preface
1. Preliminaries
1.1 Elements of analysis
1.1.1 Sets and their boundaries
1.1.2 Integration and differentiation
1.1.3 Sequences and series of functions
1.1.4 Functions of several variables
1.2 Vector spaces and linear operators
1.3 Review of facts about ordinary differential equations
2. First-Order Equations
2.1 Generalities
2.2 First-order linear PDE's
2.2.1 Constant coefficients
2.2.2 Spatially dependent velocity of propagation
2.3 Nonlinear conservation laws
2.4 Linearization
2.5 Weak solutions
2.5.1 The notion of a weak solution
2.5.2 Weak solutions of u(1) + F(u)(x) =
2.5.3 The Riemann problem
2.5.4 Formation of shock waves
2.5.5 Nonuniqueness and stability of weak solutions
2.6 Numerical methods
2.6.1 Difference quotients
2.6.2 A finite difference scheme
2.6.3 An upwind scheme and the CFL condition
2.6.4 A scheme for the nonlinear conservation law
2.7 A conservation law for cell dynamics
2.7.1 A nonreproducing model
2.7.2 The mitosis boundary condition
2.8 Projects
3. Diffusion
3.1 The diffusion equation
3.2 The maximum principle
3.3 The heat equation without boundaries
3.3.1 The fundamental solution
3.3.2 Solution of the initial-value problem
3.3.3 Sources and the principle of Duhamel
3.4 Boundary value problems on the half-line
3.5 Diffusion and nonlinear wave motion
3.6 Numerical methods for the heat equation
3.7 Projects
4. Boundary Value Problems for the Heat Equation
4.1 Separation of variables
4.2 Convergence of the eigenfunction expansions
4.3 Symmetric boundary conditions
4.4 Inhomogeneous problems and asymptotic behavior
4.5 Projects
5. Waves Again
5.1 Acoustics
5.1.1 The equations of gas dynamics
5.1.2 The linearized equations
5.2 The vibrating string
5.2.1 The nonlinear model
5.2.2 The linearized equation
5.3 The wave equation without boundaries
5.3.1 The initial-value problem and d'Alembert's formula
5.3.2 Domains of influence and dependence
5.3.3 Conservation of energy on the line
5.3.4 An inhomogeneous problem
5.4 Boundary value problems on the half-line
5.4.1 d'Alembert's formula extended
5.4.2 A transmission problem
5.4.3 Inhomogeneous problems
5.5 Boundary value problems on a finite interval
5.5.1 A geometric construction
5.5.2 Modes of vibration
5.5.3 Conservation of energy for the finite interval
5.5.4 Other boundary conditions
5.5.5 Inhomogeneous equations
5.5.6 Boundary forcing and resonance
5.6 Numerical methods
5.7 A nonlinear wave equation
5.8 Projects
6. Fourier Series and Fourier Transform
6.1 Fourier series
6.2 Convergence of Fourier series
6.3 The Fourier transform
6.4 The heat equation again
6.5 The discrete Fourier transform
6.5.1 The DFT and Fourier series
6.5.2 The DFT and the Fourier transform
6.6 The fast Fourier transform (FFT)
6.7 Projects
7. Dispersive Waves and the Schrodinger Equation
7.1 Oscillatory integrals and the method of stationary phase
7.2 Dispersive equation
7.2.1 The wave equation
7.2.2 Dispersion relations
7.2.3 Group velocity and phase velocity
7.3 Quantum mechanics and the uncertainty principle
7.4 The Schrodinger equation
7.4.1 The dispersion relation of the Schrodinger equation
7.4.2 The correspondence principle
7.4.3 The initial-value problem for the free Schrodinger equation
7.5 The spectrum of the Schrodinger operator
7.5.1 Continuous spectrum
7.5.2 Bound states of the square well potential
7.6 Projects
8. The Heat and Wave Equations in Higher Dimensions
8.1 Diffusion in higher dimensions
8.1.1 Derivation of the heat equation
8.1.2 The fundamental solution of the heat equation
8.2 Boundary value problems for the heat equation
8.3 Eigenfunctions for the rectangle
8.4 Eigenfunctions for the disk
8.5 Asymptotics and steady-state solutions
8.5.1 Approach to the steady state
8.5.2 Compatibility of source and boundary flux
8.6 The wave equation
8.6.1 The initial-value problem
8.6.2 The method of descent
8.7 Energy
8.8 Sources
8.9 Boundary value problems for the wave equation
8.9.1 Eigenfunction expansions
8.9.2 Nodal curves
8.9.3 Conservation of energy
8.9.4 Inhomogeneous problems
8.10 The Maxwell equations
8.10.1 The electric and magnetic fields
8.10.2 The initial-value problem
8.10.3 Plane waves
8.10.4 Electrostatics
8.10.5 Conservation of energy
8.11 Projects
9. Equilibrium
9.1 Harmonic functions
9.1.1 Examples
9.1.2 The mean value property
9.1.3 The maximum principle
9.2 The Dirichlet problem
9.2.1 Fourier series solution in the disk
9.2.2 Liouville's theorem
9.3 The Dirichlet problem in a rectangle
9.4 The Poisson equation
9.4.1 The Poisson equation without boundaries
9.4.2 The Green's function
9.5 Variational methods and weak solutions
9.5.1 Problems in variational form
9.5.2 The Rayleigh-Ritz procedure
9.6 Projects
10. Numerical Methods for Higher Dimensions
10.1 Finite differences
10.2 Finite elements
10.3 Galerkin methods
10.4 A reaction-diffusion equation
11. Epilogue: Classification
Appendices:
A: Recipes and Formulas
A.1: Separation of variables in space-time problems
A.2: Separation of variables in steady-state problems
A.3: Fundamental solutions
A.4: The Laplace operator in polar and spherical coordinates
B: Elements of MATLAB
B.1: Forming vectors and matrices
B.2: Operations on matrices
B.3: Array operations
B.4: Solution of linear systems
B.5: MATLAB functions and mfiles
B.6: Script mfiles and programs
B.7: Vectorizing computations
B.8: Function functions
B.9: Plotting 2-D graphs
B.10: Plotting 3-D graphs
B.11: Movies
C: References
D: Solutions to Selected Problems
E: List of Computer Programs
Index
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