Partial Differential Equations (odd&even functions)

[icouv1]

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 where to start with f (x) = (x^2).

Please help!

 

[mighty2000]

Firstly, great job on understanding the concept of even and odd functions! You are on the right track with using the initial conditions to set up the boundary conditions. To find the full equation for f(x), you can use the fact that it needs to be even across x* = ℓ and odd across x* = 0.

For the even case, you can set up the boundary condition as f(ℓ-x) = f(ℓ+x) and use the function f(x) = (x^2) as a starting point. From there, you can manipulate the equation to satisfy the boundary condition. For example, you can try setting f(ℓ-x) = (ℓ-x)^2 and f(ℓ+x) = (ℓ+x)^2 and see if you can find a way to combine them to get back to f(x) = (x^2).

For the odd case, you can set up the boundary condition as f(0-x) = -f(0-x) or f(-x) = -f(x). Again, you can use the function f(x) = (x^2) as a starting point and manipulate the equation to satisfy the boundary condition. For example, you can try setting f(-x) = -(x^2) and see if you can find a way to combine it with f(x) = (x^2) to get back to the original function.

Remember to always check if your final equation satisfies both the even and odd conditions at their respective points. I hope this helps and good luck with your homework!