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
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| TEXT |
Text(s)
typically used in this course. |
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