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

[rexasaurus]

1. Homework Statement :

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. Homework 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).

Please help!

 

[LCKurtz]

1. Homework Statement :

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. Homework Equations :

<snip>

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).

Please help!

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.

 

[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.

 

[LCKurtz]

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.

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.