## 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 $f(x)$.
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