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vpde_lecture32 [2020/04/16 12:15] trinh |
vpde_lecture32 [2020/04/16 12:34] (current) trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] |
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so the energy is always decreasing. But note that $E'(0) = 0$ since $w(x, 0) = 0$. Finally, note that $E(t)$ is always $\geq 0$ by its form (the integral of a squared quantity). So the energy is always decreasing, begins from zero, and can never be negative. We have the three statements: | so the energy is always decreasing. But note that $E'(0) = 0$ since $w(x, 0) = 0$. Finally, note that $E(t)$ is always $\geq 0$ by its form (the integral of a squared quantity). So the energy is always decreasing, begins from zero, and can never be negative. We have the three statements: | ||
- | - $E'(t) \leq 0$ for all time | + | - $E'(t) \leq 0$ for all time, |
- | - $E(0) = 0$ | + | - $E(0) = 0$, |
- | - $E(t) \geq 0$ for all time | + | - $E(t) \geq 0$ for all time, |
- | Therefore | + | and you would conclude that it has to remain at its initial value, and therefore |
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E(t) \equiv 0 | E(t) \equiv 0 | ||
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- | for all time. Looking at the form of the integrand, you would conclude that | + | for all time. Looking at the form of the integrand, you would conclude that the only way this occurs is if the integrand is itself zero, or |
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w^2(x, t) = 0 | w^2(x, t) = 0 | ||
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- | for all $x\in[0, L]$ and for all $t \geq 0$. So $u(x, t) \equiv v(x, t)$ and the solutions must be the same. | + | for all $x\in[0, L]$ and for all $t \geq 0$. So $w = u - v \equiv 0$ and thus u(x, t) \equiv v(x, t)$ and the solutions must be the same. |
===== Section 19.2: Uniqueness for other BCs of the heat equation | ===== Section 19.2: Uniqueness for other BCs of the heat equation |