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Hence, we have the following solution to the heat equation (ignoring the initial condition for now) | 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) $$ |
Thus, if we decompose the initial condition $u_0$ as | Thus, if we decompose the initial condition $u_0$ as | ||
$$ u_0(\theta) = c + \sum_{n = 1}^\infty a_n \cos(n \theta) + b_n \sin(n \theta)$$ | $$ u_0(\theta) = c + \sum_{n = 1}^\infty a_n \cos(n \theta) + b_n \sin(n \theta)$$ | ||
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To fix the coefficients $v_{n,m}$, we use the initial conditions | To fix the coefficients $v_{n,m}$, we use the initial conditions | ||
$$ v(0,x,y) = \sum_{n,m} a_{n, | $$ v(0,x,y) = \sum_{n,m} a_{n, | ||
- | so | + | so multiply both sides by $v_{n, |
$$a_{n,m} = \frac{\int_{[0, | $$a_{n,m} = \frac{\int_{[0, | ||
+ | |||
+ | Remark: if the boundary temperature is not constant $T$ (but still time-independent), | ||
+ | $$ u(t,x,y) = U(x,y) + v(t,x,y) $$ | ||
+ | where $v(t,x,y)$ now has boundary condition $0$, and initial condition $v(0,x,y) = u(0,x,y) - U(x,y)$. | ||
+ | |||
+ | ==== Schroedinger Equation (without potential) ==== | ||
+ | $$ i \d_t u = - \Delta u $$ | ||
+ | |||
+ | We may reuse the analysis for the heat equation, except replacing $t$ in heat equation to $it$. Thus, exponential decay now become oscillation. | ||
+ | |||
+ | ==== Wave equation ==== | ||
+ | $$ \d_t^2 u = \Delta u$$ | ||
+ | Suppose $u$ lives on a domain $D$ with boundary value zero, or $u$ lives on a space without boundary, e.g $S^1$ or a torus. We may then consider eigenvalue of $\Delta$, $\lambda_1 \leq \lambda_2 \leq \cdots, $ with $\lambda_n \geq 0$, (repeated with multiplicity), | ||
+ | $$ u(t,x) = \sum_{n=1}^\infty (a_n \cos(\sqrt{\lambda_n} t) + b_n \sin(\sqrt{\lambda_n} t) ) u_n(x) $$ | ||
+ | (if $\lambda_n=0$, | ||
+ | $$ \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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