Introduction to the subject of partial differential equations: First order equations (linear and nonlinear) and second order equations (heat equation, wave equation and Laplace equation). Method of characteristics for hyperbolic problems. Solution of initial boundary value problems using separation of variables and eigenfunction expansions. Qualitative properties of solutions.
1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH246, MATH341).
Level of Rigor
Introduction to Partial Differential Equations, by Walter Strauss
First Course in Partial Differential Equations, by Weinberger.
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Students interested in grad school in AMSC should strongly consider this course
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First order equations (linear and nonlinear): Method of characteristics.
Second order equations: Diffusion (heat) equation, Wave equation and Laplace equation
Homogeneous and inhomogeneous problems
Qualitative properties of solutions (Energy, mean value property and maximum principle for harmonic functions)
Initial boundary value problems on the whole line and on the half line (Dirichlet and Neumann boundary conditions).
Initial boundary value problems on a finite interval: Method of separation of variables
Laplace equation in the rectangle and disk
Heat and wave equations in higher dimensions (eigenfunction expansions)
Nonlinear conservation laws, derivations, shock waves
Numerical methods, CFL condition, Crank-Nicolson scheme
Finite element method
Derivation from equations of gas dynamics or from equations of the vibrating string
Derivation from Fourier's Law of cooling or Fick's law of diffusion
Traveling wave solutions to a nonlinear heat equation
Series solution and Poisson kernel representation of solution of the Dirichlet problem in the disk
Harnack inequality and Liouville's theorem
Green's function for the Poisson equation in R2 and R3