Math 257/316: List of Core Skills
A student successfully completing Math 257/316 (Spring 2015) should know:
• Series solutions of second-order linear ODE:
– if given a point is an ordinary point, a regular singular point, or an irregular
singular point for a particular ODE
– how to construct a power series solution (general solution, or solution of an
initial value problem) centred at an ordinary point
– how to construct a Frobenius series solution (general, or IVP) centred at a
regular singular point (including how to determine the indicial equation)
• PDEs and the method of Separation of Variables:
– the heat, wave, and Laplace equations (and variants) and what they model;
basic PDE problems: initial-boundary-value problems for heat and wave,
and boundary value problems for Laplace
– superposition, and how to use the method of separation of variables to find
the general solution of homogeneous PDE problems as infinite series
– how to handle non-zero (time-independent) BCs and/or source term by first
finding the steady-state (or particular solution, if non-zero Neumann BCs)
– d’Alembert’s formula for the solution of the wave equation on the line
– solving BVPs for Laplace’s equation on rectangular and circular domains
• Fourier Series:
– how to represent functions as Fourier full-range, sine, or cosine series, and
the connection to even and odd periodic extensions
– the use of Fourier series in solving PDE problems: satisfying initial conditions (heat and wave), or a non-zero boundary condition (Laplace)
– the statement of the Fourier Convergence Theorem
• The method of Eigenfunction Expansion:
– use of eigenfunction expansion to find the general solution of a heat/wave
problem with a time-dependent source term, or time-dependent BCs
– Sturm-Liouville eigenfunctions, their expansion property (which generalizes
Fourier sine and cosine series), and its use in satisfying the initial conditions
(heat, wave) or a non-zero boundary condition (Laplace) in PDE problems
• Finite Difference approximations to PDE problems:
– finite-difference approximations to first and second derivatives
– finite-difference numerical schemes for approximating solutions of initialboundary-value problems for heat and wave equations with various BCs
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