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Suppose $F:C^1(\Omega, [0,T]) \to C^1(\Omega, [0,T])$ with $$F(u) = u_t - f(x, t, u, u_x).$$

How do I calculate the Frechet derivative of $F$ at the point $w = f(x,t, 0, 0)t$?

It should be $$F'(w, v) = v_t - \frac{\partial f}{\partial z}\bigg|_{w}v - \frac{\partial f}{\partial p}\bigg|_{w}v_x$$ apparently.

Maybe another day I can do this but forming the difference and then considering another difference to get out the partial derivative is confusing me!

Thanks

Court
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  • do you want to find the expression for the derivative or show that the Frechet derivative exists? –  Jun 15 '12 at 18:38
  • @Thomas I want to find the expression. – Court Jun 15 '12 at 18:39
  • You can just differentiate formally. $u\mapsto u_t$ is linear, hence $D_v(u\mapsto u_t)= v_t$. For the $f$ -term you use the chain rule, only the third ($z$) argument depends (formally) on $u$. –  Jun 15 '12 at 18:42
  • sorry but I got the expression for what the derivative should be wrong. I edited it. – Court Jun 15 '12 at 18:47
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    that proves my comment to be wrong ;-), but your result is correct, you have to look at $(F(u+sv)-F(u))/s$, which involves $x$-derivatives. Nonetheless, if you just want the result, you only need to take directional derivatives, that is, calculate $\frac{d}{d s}F(w+sv)|_{s=0}$ –  Jun 15 '12 at 18:52
  • Thanks @Thomas. I was a bit confused about how to do it formally (as in satisfying the definition on the wikipedia article) but I'll have another go. – Court Jun 15 '12 at 19:06

1 Answers1

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If you just want the result and already do know that the functional is Frechet differentiable (or just don't care, which is not a good attitude ;-) you only have to calculate the directional derivative $$\frac{d}{ds}|_{s=0}F(w+sv) = \frac{d}{ds}|_{s=0}\left\{(w+sv)_t - f(x,t,(w+sv), (w+sv)_x)\right\} = v_t - f_zv -f_p v_x$$ (assuming $f=f(x,t,z,p)$). This reduces the question to a one dimensional differentiation task.

  • Thanks. (for some reason I kept thinking you said small $f$ in the comment above.)

    I see you've assumed $F$ is Frechet differentiable and hence Gateaux differentiable and calculated the directional derivative. But if you didn't know if it was Frechet differentiable how would you do it? (I do care!)

     – Court Jun 15 '12 at 21:07