Neta B. Partial Differential Equations

Lecture Notes. — Monterey, California: Department of Mathematics Naval Postgraduate School, 2002. — 355 p.Introduction and Applications.
Basic Concepts and Definitions.
Applications.
Conduction of Heat in a Rod.
Boundary Conditions.
A Vibrating String.
Boundary Conditions.
Diffusion in Three Dimensions.
Classification and Characteristics.
Physical Classification.
Classification of Linear Second Order PDEs.
Canonical Forms.
Equations with Constant Coefficients.
Linear Systems.
General Solution.
Method of Characteristics.
3.1 Advection Equation (first order wave equation).
3.2 Quasilinear Equations.
Second Order Wave Equation.
Separation of Variables-Homogeneous Equations.
Parabolic equation in one dimension.
Other Homogeneous Boundary Conditions.
Eigenvalues and Eigenfunctions.
Fourier Series.Orthogonality.
Computation of Coefficients.
Relationship to Least Squares.
Convergence.
Fourier Cosine and Sine Series.
Term by Term Differentiation.
Term by Term Integration.
Full solution of Several Problems.
Sturm-Liouville Eigenvalue Problem.Boundary Conditions of the Third Kind.
Proof of Theorem and Generalizations.
Linearized Shallow Water Equations.
Eigenvalues of Perturbed Problems.
PDEs in Higher Dimensions.Heat Flow in a Rectangular Domain.
Vibrations of a rectangular Membrane.
Helmholtz Equation.
Vibrating Circular Membrane.
Laplace’s Equation in a Circular Cylinder.
Laplace’s equation in a sphere.
Separation of Variables-Nonhomogeneous Problems.
Inhomogeneous Boundary Conditions.
Method of Eigenfunction Expansions.
Forced Vibrations.
Poisson’s Equation.
Fourier Transform Solutions of PDEs.
Motivation.
Fourier Transform pair.
Heat Equation.
Fourier Transform of Derivatives.
Fourier Sine and Cosine Transforms.
Fourier Transform in 2 Dimensions.
Green’s Functions.One Dimensional Heat Equation.
Green’s Function for Sturm-Liouville Problems.
Dirac Delta Function.
Nonhomogeneous Boundary Conditions.
Fredholm Alternative And Modified Green’s Functions.
Green’s Function For Poisson’s Equation.
Wave Equation on Infinite Domains.
Heat Equation on Infinite Domains.
Green’s Function for the Wave Equation on a Cube.
Laplace Transform.Solution of Wave Equation.
Finite Differences.
Taylor Series.
Finite Differences.
Irregular Mesh.
Thomas Algorithm.
Methods for Approximating PDEs.
Eigenpairs of a Certain Tridiagonal Matrix.
Finite Differences.Difference Representations of PDEs.
Heat Equation in One Dimension.
Two Dimensional Heat Equation.
Laplace’s Equation.
Vector andMatrix Norms.
Matrix Method for Stability.
Derivative Boundary Conditions.
Hyperbolic Equations.
Inviscid Burgers’ Equation.
Viscous Burgers’ Equation.
Numerical Solution of Nonlinear Equations 330.Bracketing Methods.
Fixed Point Methods.
Example.
Appendix.