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 vpde_lecture32 [2020/04/16 13:33]trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] vpde_lecture32 [2020/04/16 13:34]trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] Both sides previous revision Previous revision 2020/04/16 13:34 trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] 2020/04/16 13:33 trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] 2020/04/16 13:33 trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] 2020/04/16 13:15 trinh 2020/04/16 11:23 trinh 2020/04/16 09:45 trinh created Next revision Previous revision 2020/04/16 13:34 trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] 2020/04/16 13:33 trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] 2020/04/16 13:33 trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation] 2020/04/16 13:15 trinh 2020/04/16 11:23 trinh 2020/04/16 09:45 trinh created Line 82: Line 82: 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 $x\in[0, L]$ 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 ​ =====