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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.