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Partial derivatives of a strong solution are also solutions?

  1. Feb 23, 2013 #1

    cjc

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    1. The problem statement, all variables and given/known data

    For the heat equation [itex]u_{t}=\alpha^{2}u_{xx}[/itex] for [itex]x\in\mathbb{R}[/itex] and [itex]t>0[/itex], show that if [itex]u(x,t)[/itex] is a strong solution to the heat equation, then [itex]u_{t}[/itex] and [itex]u_{x}[/itex] are also solutions.

    2. Relevant equations

    [itex]u_{t}=\alpha^{2}u_{xx}[/itex]

    3. The attempt at a solution

    I've considered the following theorem which shows that the integral defining [itex]u[/itex] and the integrals of partial derivatives are all uniformly convergent. It goes as follows:

    Let [itex]g(r,s)[/itex] be defined and continuous on [itex]\mathbb{R}\times I[/itex], where [itex]I[/itex] is an interval, and suppose [itex]\partial g/ \partial s[/itex] exists and is continuous. If the improper integral for [itex]G(s)=\int_{\mathbb{R}} g(r,s) dr[/itex] and the improper integral [itex]\int_{\mathbb{R}}\frac{\partial g}{\partial s}(r,s)dr[/itex] are uniformly convergent on [itex]I[/itex], then [itex]G[/itex] is differentiable and [itex]G'(s)=\int_{\mathbb{R}}\frac{\partial g}{\partial s}(r,s)dr[/itex].

    My question would be how do I begin to apply this to the given problem?

    Thank you so much! =]
     
  2. jcsd
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