math121b:04-01
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math121b:04-01 [2020/04/01 00:15] pzhou [Wave equation] |
math121b:04-01 [2020/04/01 09:05] (current) pzhou |
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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)$$ | ||
math121b/04-01.1585725305.txt.gz · Last modified: 2020/04/01 00:15 by pzhou
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