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vpde_lecture32 [2020/04/16 12:33] trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] |
vpde_lecture32 [2020/04/16 12:34] (current) trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] |
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| E(t) \equiv 0 | E(t) \equiv 0 | ||
| $$ | $$ | ||
| - | 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 |
| $$ | $$ | ||
| w^2(x, t) = 0 | w^2(x, t) = 0 | ||
| $$ | $$ | ||
| - | for all x∈[0,L] and for all t≥0. So u(x,t)≡v(x,t) and the solutions must be the same. | + | for all x∈[0,L] and for all t≥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 | ||