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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 ===== |

Last modified: 2020/04/16 13:34