Basics of Partial Differential Equations Summary

PDE: Separation of Variables

1) Let u(x,y)=X(x)Y(y).
2) Note that u_x=X'Y, u_y=XY', u_{xx}=X''Y, u_{yy}=XY'', u_{xy}=u_{yx}=X'Y'.
3) Rearrange the equation such that LHS is a function of x only, RHS is a function of y only.
4) Thus, LHS=RHS=some constant k.
5) Solve the two separate ODEs.

Wave Equation
\displaystyle c^2y_{xx}=y_{tt}, where y(t,0)=y(t,\pi)=0, y(0,x)=f(x), y_t(0,x)=0.

Solution of Wave Equation (with Fourier sine coefficients)
\displaystyle y(t,x)=\sum_{n=1}^\infty b_n\sin(nx)\cos(nct) where \displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx.

d’Alembert’s solution of Wave Equation
\displaystyle y(t,x)=\frac{1}{2}[f(x+ct)+f(x-ct)].

Heat Equation
\displaystyle u_t=c^2u_{xx},
u(0,t)=u(L,t)=0, u(x,0)=f(x).

Solution of Heat Equation
\displaystyle u(x,t)=\sum_{n=1}^\infty b_n\sin\left(\frac{n\pi x}{L}\right)\exp\left(-\frac{\pi^2n^2c^2}{L^2}t\right), where \displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx are Fourier sine coefficients of f(x).