P.H. Trinh

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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]
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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 $x\in[0, L]$ 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 ​ =====