Integration by parts with u''(y)Φ(x-y)

[squenshl]

Homework Statement

How do I integrate $$\int_a^b$$ u''(y) $$\Phi(x-y)$$ dy by parts.

Homework Equations

The Attempt at a Solution

I let $$\int_a^b$$ u''(y) $$\Phi(x-y)$$ dy = $$\int_a^x$$ u''(y) $$\Phi(x-y)$$ dy + $$\int_x^b$$ u''(y) $$\Phi(x-y)$$ dy but when I integrate by parts I get $$\int_a^b$$ u''(y) $$\Phi(x-y)$$ dy = $$\Phi(x-b)$$u'(b) - $$\Phi(x-a)$$u'(a) + $$\Phi'(x-b)$$u(b) - $$\Phi'(x-a)$$u(a) but I am missing out a u(x) term somewhere so I can write u(x) as a subject of everything else. Please help.

 

[SammyS]

What do you know about the function, $$\Phi(x)\ ?$$

 

[squenshl]

$$\Phi(x)$$ = -|x|/2

 

[squenshl]

I think I got it, the u(y)$$\Phi'(x-y)$$ in the first integral becomes -u(x)/2 & the -u(y)$$\Phi'(x-y)$$ in the second integral becomes -u(x)/2 so adding these together we get -u(x) (assuming x < y) so u(x) can be written as a subject of the rest.