Define f(x) where it is odd and even at two points

1. Sep 24, 2011

rexasaurus

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

A function f : R → R is called “even across x∗ ” if f (x∗ − x) = f (x∗ + x) for every x and is called “odd across x∗ ” if f (x∗ − x) = −f (x∗ + x) for every x. Deﬁne f (x) for 0 ≤ x ≤ ℓ by setting f (x) = (x^2) . Extend f to all of R (i.e., deﬁne f (x) for all real x) in such a way that it is odd across x∗ = 0 and even across x∗ = ℓ.

2. Relevant equations:

1. conservation equations (transport): concentration, ﬂux
(a) ﬂow (ﬂux = vu ) — ﬂuid, traﬃc, etc.
(b) mixing: diﬀusion/dispersion (probability; ﬂux = D ∇u )
reaction/diﬀusion systems
2. mechanics (Newton’s 3rd Law): force, potential energy, momentum
(a) wave equation; ICs and BCs
(b) beam, plate equations
3. steady state (equilibrium: balance equations)
4. some other examples . . . (e.g., Cauchy-Riemann equations)

**Also studying the heat equation/etc**

3. The attempt at a solution:

I understand the difference between "even" and "odd". I have created the following:
f(ℓ-x)=f(ℓ+x) even at "ℓ"
f(0-x)=-f(0-x) OR f(-x)=-f(x) odd at "0"

I think I need to use the above IC's to setup the BC's. Once I have the BC's determined I am not sure how to combine them to find the full equation for f (x) or what to do with f (x) = (x^2).

2. Sep 24, 2011

LCKurtz

Can you draw a picture of the graph that is y = x2 on [0,ℓ] and that is odd across 0 and even across ℓ? That's your first step.

3. Sep 25, 2011

rexasaurus

Thank you for your help. I was able to plot the graph for x^2 and the odd/even BC's not sure where to go from there. Any help is appreciated.

4. Sep 25, 2011

LCKurtz

If your function comes out to be periodic (did it?) all you have to do is define it over one period and extend it periodically.