Fourier Series and PDEs Flashcards
Define a periodic function
The function f: R → R is a periodic function if there exists p > 0 such that
f(x + p) = f(x) for all x ∈ R.
Define the prime period of a function
The smallest p where f(x + p) = f(x) for all x ∈ R.
Define a periodic extension
Given any real x and a periodic function f with period α, there exists a unique integer m(x) such that x - m(x)p is between α and α + p. Then, the periodic extension F is F(x) = f(x - m(x)p)
Define an even function
f is even if f(x) = f(-x) for all x in the reals
Define an odd function
f is odd if f(x) = -f(-x) for all x in the reals
Give the orthogonality relations
Integral of cos(mπx/L)cos(nπx/L)dx from -L to L = Lδ_(mn)
Integral of cos(mπx/L)sin(nπx/L)dx from -L to L = 0
Integral of sin(mπx/L)sin(nπx/L)dx from -L to L = Lδ_(mn)
Give Fourier’s Convergence Theorem
f(x) ~ (a_0)/2 +SUM[a_ncos(nx) + b_nsin(nx)], where:
a_0 = 1/Lintegral from -L to L of f(x)dx
a_n = 1/Lintegral from -L to L of f(x)cos(nπx/L)dx
b_n = 1/L*integral from -L to L of f(x)sin(nπx/L)dx
Give the assumptions used when deriving the heat equation.
The lateral surfaces are insulated so no heat escapes
T is a function that depends only on distance along the rod and time
Fourier’s Law
Define the specific heat of a material
The energy required to heat one unit of mass by one unit of temperature.
Define heat flux
The rate which thermal energy is transported through a cross-section of the rod per unit cross-sectional area per unit time
Give Fourier’s Law
q = -k(∂T/∂x), where q is the flow of heat per unit area per unit time, T is temperature and k is the conductivity.
Give the assumptions used when deriving the wave equation.
The string is homogeneous, extensible and elastic
The effects of gravity and air resistance are negligeable
The transverse displacement is small, so |∂y/∂x| «_space;1
