1. Apr 30, 2012

omoplata

The attached image is from "Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations" by J.W. Thomas.

In the beginning the set of test functions, $\phi$ is defined.

They arrive at equation (9.2.11) by using $\phi(x,T) = \phi(a,t) = \phi(b,t) = 0$.

Where does this condition come from?

Thanks.

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2. Apr 30, 2012

omoplata

I guess, to find out what's going on, I should find out what this set $C_0^1$ is first. It's not defined anywhere else on the book.

3. Apr 30, 2012

Staff: Mentor

The symbol C1 means functions whose first derivative is continuous. From your attached page, I infer that C01 means functions whose first derivatives are continuous, and that are 0 outside some rectangle in R2.

4. Apr 30, 2012

omoplata

Thanks. That helps.

I guess from that you could say that, if the first derivative of $\phi$ is continuous, then $\phi$ is also continuous, so $\phi = 0$ at the edges of the rectangle? Therefore, $\phi(x,T) = \phi(a,t) = \phi(b,t) = 0$.

But in that case, why is it not required that $\phi(x,0) = 0$ ?

5. Apr 30, 2012

Staff: Mentor

I don't think you can conclude that, at least based on the document you attached. It looks like there is more to the problem than what you scanned, so perhaps the answer is there.