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MATH 462 (Partial Differential Equations for Scientists and Engineers)

DESCRIPTION Introduction to the subject of partial differential equations: first order equations (linear and nonlinear), heat equation, wave equation, and Laplace equation.  Examples of nonlinear equations of each type. Qualitative properties of solutions.  Method of characteristics for hyperbolic problems.  Solution of initial boundary value problems using separation of variables and eigenfunction expansions. Some numerical methods.
PREREQUISITES Calculus I, II, III, one semester of ordinary differential equations (MATH 241, MATH 246)
TOPICS First order equations
   First order linear equations (with method of characteristics)
   Weak solutions
   Nonlinear conservation laws, derivations, shock waves
   Linearized equations
   Numerical methods, CFL condition.
Diffusion (heat equation) in one space variable
   Derivation from Fourier's Law of cooling or Fick's law of diffusion
   Maximum principle, Weierstrass kernel, qualitative properties of solutions
   Traveling wave solutions to a nonlinear heat equation, Bergers' equation or reaction diffusion equations
   Initial boundary value problems on the half line
   Initial boundary value problems on a finite interval, method of separation of variables, linear operators and expansions of solutions in terms of orthogonal eigenfunctions
   Inhomogeneous problems
   Numerical methods, Crank-Nicolson scheme
The wave equation on the line
   Derivation from equations of gas dynamics or from equations of the vibrating string
   Characteristics, d'Alembert's formula, domains of influence and dependence
   Half line problems, reflections of waves by Dirichlet and Neumann boundary conditions
   Initial-boundary value problems using separation of variables
   Numerical methods
Heat and wave equations in higher dimensions
   Solutions of initial value problem on R2 and R3, Weierstrass kernel for heat equation and Kirchoff's formula for the qave equation
   Boundary value problems in the rectangle and disk, eigenfunction expansions, Bessel functions
Laplace equation
   Mean value property and maximum principle for harmonic fuctions
   Series solution and Poisson kernel representation of solution of the Dirichlet problem in the disk
   Harnack inequality and Liouville's theorem (used to prove the uniqueness of solutions of Poisson's equation)
   Green's function for the Poisson equation in R2 and R3
   Green's function for the disk, half plane, sphere
   Numerical methods
Epilogue: classification of second order linear equations

TEXT Text(s) typically used in this course.