# PDE math homework help

1. Jan 17, 2010

### r.a.c.

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

We are given f $$\epsilon$$ C(T) [set of continuous and 2pi periodic functions] and PS(T) [set of piecewise smooth and 2pi periodic functions]

SOlve the BVP

ut(x,t) = uxx(x,t) ; (x,t) belongs to R x (0,inf)
u(x,0) = f(x) ; x belongs to R

Find a solution of this PDE

2. Relevant equations

The only relevant equation I can think of is a Fourier Inversion which states that if is continuous and piecewise smooth then
$$f(x) = \sum f^h(n) e^i^n^x ; where f^h = 1/2\pi \int f(x)e^-^i^n^x dx$$

3. The attempt at a solution

I have tried solving the first equation ODE till I get by seperation of varaibles
S''(x) - AS(x) = 0 and T'(t) - AT(t) = 0

A is real. Three cases: A>0 in which case the solution is S(x) = C(sin(Lx)) + B(cos(Lx))
where C,B are constants and L = (-L)^(1/2)
Then I tried using power series expansion of sin and cos to be able to relate it to the fourier series of f (seeing as f(x) = S(x)) but to only get stuck.

When A = 0 S(x) = Cx + B
Then I can do this by finding the fourier coefficients($$f^h$$) and so on

When A<0 things become complicated because we get and exponential.

Anyways, first off, how do we know which A to use!? Then I can try to reach some sort of conclusion.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 17, 2010

### kfdleb

Re: Pde

first, the best thing is to solve it using separation of variable i.e. as an ODE as u did

u get X''(x)-CX(x)=0 and T'(t)-CT=0

i think ur missing something in the given such as u(0,t)=u(2\pi,t)
if so the solution would be X''(x)+L^2X(x)=0
X(x)=Acos(Lx)+Bsin(Lx)
with L=-n^2*(pi)^2/(2pi)^2

the solution would be the sum of X_n(x)*T_n(t)

3. Jan 17, 2010

### r.a.c.

Re: Pde

No I'm not missing anything in the solution. That's the problem. If the constraints on the extremities of x were there it would be a piece of cake. And if we had them then there would be no need for f to be belong to C or PS. Even in the question it tells us to solve the PDE using fourier series.

4. Jan 18, 2010

### r.a.c.

Re: Pde

Thanks but I found the answer to this. I think this is done.