MAT 341 : Applied Real Analysis

Chapter 1 : Fourier

f(x) ~ Σ f(x)·VnVn·Vn Vn = Σf(x)·cos(nx)cos(nx)·cos(nx)cos(nx)+f(x)·sin(nx)sin(nx)·sin(nx)sin(nx) = f(x)·1 + Σ f(x)·cos(nx)πcos(nx) + f(x)·sin(nx)πsin(nx)

f(x) ~ π f(x) dx + Σ π f(x) cos(nx) dxπcos(nx) + π f(x) sin(nx) dxπsin(nx) = ao + Σ ancos(nx) + bnsin(nx)
ao = 1 π f(x) dx
an = 1π π f(x) cos(nx) dx
bn = 1π π f(x) sin(nx) dx

A*Dₓ² + B*DₓDt + C*Dt² + ... = 0
B²-AC=0 (Parabolic)>0 (Hyperbolic)<0 (Elliptic)

Chapter 2 : Heat Equation

Chapter 3 : Wave Equation

Chapter 4 : Laplace Equation

d²udx² = 1kdudt

ddt Ψ = -ħ²2mdx² Ψ

d²udx² = 1d²udt²

d²udx² = 0