Trinh @ Bath

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vpde_lecture32 [2020/04/16 10:23]
trinh
vpde_lecture32 [2020/04/16 12:34] (current)
trinh [Section 19.1: Uniqueness for zero Dirichlet heat equation]
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 ====== Lecture 33: Uniqueness of solutions I ====== ====== Lecture 33: Uniqueness of solutions I ======
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 In this lecture, we'll prove uniqueness for two variants of the heat equation. The different variants you should consider as as follows (all posed on $x\in[0, L]$):  In this lecture, we'll prove uniqueness for two variants of the heat equation. The different variants you should consider as as follows (all posed on $x\in[0, L]$): 
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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
 $$ $$
 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  =====