Author | Cooper, Jeffery. author |
---|---|
Title | Introduction to Partial Differential Equations with MATLAB [electronic resource] / by Jeffery Cooper |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1998 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1754-1 |
Descript | XV, 541 p. online resource |
1 Preliminaries -- 1.1 Elements of analysis -- 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 PDEโs -- 2.3 Nonlinear conservation laws -- 2.4 Linearization -- 2.5 Weak solutions -- 2.6 Numerical methods -- 2.7 A conservation law for cell dynamics -- 2.8 Projects -- 3 Diffusion -- 3.1 The diffusion equation -- 3.2 The maximum principle -- 3.3 The heat equation without boundaries -- 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 -- 5.2 The vibrating string -- 5.3 The wave equation without boundaries -- 5.4 Boundary value problems on the half-line -- 5.5 Boundary value problems on a finite interval -- 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 -- 6.6 The fast Fourier transform (FFT) -- 6.7 Projects -- 7 Dispersive Waves and the Schrรถdinger Equation -- 7.1 Oscillatory integrals and the method of stationary phase -- 7.2 Dispersive equations -- 7.3 Quantum mechanics and the uncertainty principle -- 7.4 The Schrรถdinger equation -- 7.5 The spectrum of the Schrรถdinger operator -- 7.6 Projects -- 8 The Heat and Wave Equations in Higher Dimensions -- 8.1 Diffusion in higher dimensions -- 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 -- 8.6 The wave equation -- 8.7 Energy -- 8.8 Sources -- 8.9 Boundary value problems for the wave equation -- 8.10 The Maxwell equations -- 8.11 Projects -- 9 Equilibrium -- 9.1 Harmonic functions -- 9.2 The Dirichlet problem -- 9.3 The Dirichlet problem in a rectangle -- 9.4 The Poisson equation -- 9.5 Variational methods and weak solutions -- 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