1. Partial Differential Equations

Hi I am stuck on a PDE boundary question, I think I have shown the equation seperates and understand what the boundarys mean but I am not sure how to show the general solution and then solve the boundary value problems. Please help

Assume solution $\displaystyle u(x,t) = X(x)T(t)$ to the modified diffusion equation $\displaystyle u_t - Du_{xx} - au = 0$.

First it says show the equation seperates, I think the follwing is correct?:

substituting the solution into the equation:
$\displaystyle u_t - au = Du_{xx}$

$\displaystyle XT' - aXT = DX''T$

so $\displaystyle \frac {T'}{T} - a = D \frac {X''}{X}$

Then it wants the general solution for X(x) and T(t)

and then assuming $\displaystyle D>0, a>=0, L>0$ solve the boundary value problem:
$\displaystyle u_t - Du_{xx} - au = 0$ for all $\displaystyle 0=<x=<L , t>=0$

$\displaystyle u(0,t) = u(L,t) = 0$ for all $\displaystyle t>=0$

$\displaystyle u(x,0) = sin(\pi x/L) + sin(2\pi x/L)$

and finally characterise the differenc between solutions for $\displaystyle a>D\pi^2 / L^2$ and $\displaystyle a<D\pi^2 / L^2$

I have found a similiar problem in Boyce's elementary Diff equations book but just cant work it out!

Thanks

Hi I am stuck on a PDE boundary question, I think I have shown the equation seperates and understand what the boundarys mean but I am not sure how to show the general solution and then solve the boundary value problems. Please help

Assume solution $\displaystyle u(x,t) = X(x)T(t)$ to the modified diffusion equation $\displaystyle u_t - Du_{xx} - au = 0$.

First it says show the equation seperates, I think the follwing is correct?:

substituting the solution into the equation:
$\displaystyle u_t - au = Du_{xx}$

$\displaystyle XT' - aXT = DX''T$

so $\displaystyle \frac {T'}{T} - a = D \frac {X''}{X}$
Assume that this is correct (your job to check), than the right hand side is a function of $\displaystyle x$ only and the left hand side is a function of $\displaystyle y$ only, so as they are equal they are also equal to a constant, which leaves you with the two ordinary differential equations:

$\displaystyle {T'}= (K+a) T$

and:

$\displaystyle X''=\frac{K}{D}X$

for some constant $\displaystyle K$

RonL

3. Thanks, ok I understand that but still dont know how to find the general solution and solve the boundary problem? Sorry if I'm missing the point here.