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--- math121b:04-01 [2020/04/01 00:10]
pzhou
+++ math121b:04-01 [2020/04/01 09:05] (current)
pzhou
@@ Line 14: / Line 14: @@
 Hence, we have the following solution to the heat equation (ignoring the initial condition for now)
-$1, \z{ and } e^{-n^2 t} \sin(n \theta), e^{-n^2 t} \cos(n \theta) $$
+$$1, \z{ and } e^{-n^2 t} \sin(n \theta), e^{-n^2 t} \cos(n \theta) $$
 Thus, if we decompose the initial condition $u_0$ as
 $$ u_0(\theta) = c + \sum_{n = 1}^\infty a_n \cos(n \theta) + b_n \sin(n \theta)$$
@@ Line 50: / Line 50: @@
 ==== Wave equation ====
 $$ \d_t^2 u = \Delta u$$
-Suppose $u$ lives on a domain with boundary value zero, or $u$ lives on a space without boundary, e.g $S^1$ or a torus. We may then consider eigenvalue of $\Delta$, $\lambda_1, \lambda_2, \cdots, $ with $\lambda_i > 0$, (repeated with multiplicity), with eigenfunction $u_1(x), u_2(x), \cdots,$ then we may write the general solution
+Suppose $u$ lives on a domain $D$ with boundary value zero, or $u$ lives on a space without boundary, e.g $S^1$ or a torus. We may then consider eigenvalue of $\Delta$, $\lambda_1 \leq \lambda_2 \leq \cdots, $ with $\lambda_n \geq 0$, (repeated with multiplicity), with eigenfunction $u_1(x), u_2(x), \cdots,$ then we may write the general solution
-$$ u(t,x) = \sum_{n=1}^\infty a_n e^{i \sqrt{\lambda_n} t} u_n(x) + b_n e^{-i \sqrt{\lambda_n} t} u_n(x) $$
+$$ u(t,x) = \sum_{n=1}^\infty (a_n \cos(\sqrt{\lambda_n} t)   + b_n \sin(\sqrt{\lambda_n} t) ) u_n(x) $$
-(if $\lambda_n=0$,  then $e^{i \sqrt{\lambda_n} t} = e^{-i \sqrt{\lambda_n} t}=1$, hence we may $b_n=0$.) To fix the coefficient, we use initial condition $u(0, x)$ and $\dot u(0, x)$.
+(if $\lambda_n=0$,  then we may set $b_n=0$.) To fix the coefficient, we use initial condition $u(0, x)$ and $\dot u(0, x)$:
+$$ \int_D u(0,x) u_n(x) dx = a_n \int u_n^2 dx $$
+$$ \int_D \dot u(0,x) u_n(x) dx = \sqrt{\lambda_n} b_n \int u_n^2 dx $$
 Example: 1-dim string vibration on an interval.

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