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.
Wave Equation
where
,
,
.
Solution of Wave Equation (with Fourier sine coefficients)
where
d’Alembert’s solution of Wave Equation
Heat Equation
,
.
Solution of Heat Equation
where
are Fourier sine coefficients of
.