PDE: Separation of Variables
1) Let .
2) Note that , , , , .
3) Rearrange the equation such that LHS is a function of only, RHS is a function of only.
4) Thus, LHS=RHS=some constant .
5) Solve the two separate ODEs.
where , , .
Solution of Wave Equation (with Fourier sine coefficients)
d’Alembert’s solution of Wave Equation
Solution of Heat Equation
where are Fourier sine coefficients of .