Modelling with PDEs - Boundary Condition

I am working on this question for my Partial Differential Equation seminar and would like some help in understanding how to create boundary conditions.

An endothermic chemical reaction takes place within the region x [0,L], consuming

heat energy at a rate R = x(L − x). The ends of the region are perfectly insulated

and the region is initially heated to A degrees.

(a) Write down a partial differential equation, together with boundary and initial

conditions governing the temperature u(x, t) in the region.

[Hints: The rate of heat leaving the region at x = 0,L is assumed to be propor-

tional to du/dx (partial, curly d) . Your equation should contain an inhomogeneity corresponding to the heat sink, R, which is independent of u.]

So I worked out that the initial condition will be u(x,0)=A and from there on I am a little stuck. I believe that because of the hint that the boundary conditions will be Neumann conditions (ie u'(0,t)=? and u'(L,t)=? ) but I have looked at all kinds of examples and cannot work out how to find what they are equal to. I think either they would equal 0 but my instinct tells me they should maybe be a function of t since surely the temperature would depend on what time it is?

For the equation i have u_t = Du_xx -x(L-x) but i am not sure whether i need the D at the front or not.

Any help would be greatly appreciated, i would really like to understand what I am doing

Thanks

Re: Please help with modelling with PDEs - Boundary Condition

Do **you** understand the **u'** makes no sense here? You are not thinking in terms of more than one variable. u is a function of two variables and you **must** state whether you are differentiating with respect to x or t. Saying that the ends are insulated means there is no heat flow over them. "Heat flow", in one direction or another, is the change of temperature with respect to x. Your condition should be $\displaystyle u_x= 0$ at the ends of the interval.

Quote:

For the equation i have u_t = Du_xx -x(L-x) but i am not sure whether i need the D at the front or not.

What does "D" represent? I suspect it is the "derivative operator" but, again, that makes no sense in a problem with two variables. You already have the derivatives with respect to x and t.

Re: Please help with modelling with PDEs - Boundary Condition

Perfectly insulated means

$\displaystyle u_x(0,t) = 0, u_x(L,t) = 0.$

Re: Please help with modelling with PDEs - Boundary Condition

oh woops little notation error, i knew it was wrt x, thanks for reminding me. The D is meant to be the proportionality constant, often set to one to give the heat equation in 1D only in this situation there is also a sink and i can't tell from the question whether the D is meant to be one or whether i should just leave it general as D. Thanks for your help.

so u_x(0,t)=0 and u_x(L,t)=0 because the ends of the region are insulated?

the thing i can't get my head round is that if the boundary conditions describe the temperature at any time at x=0,L why is there no t in the conditions, because surely the temperature would vary depending on what t is? or is this why we talk about u_x and not u. sorry if i'm talking nonsense i'm just struggling to get my head round some of these bits.

Re: Please help with modelling with PDEs - Boundary Condition

We usually set $\displaystyle D = 1$ because we can scale t. BTW - The solution will depend on t! That's what you are required to find - the solution of the PDE!

Re: Please help with modelling with PDEs - Boundary Condition

so now i proceed with separation of variables but because of the source term this can't be done directly right? i do the u(x,t)=v(x)+w(x,t) ? or is there a way of doing it with normal SOV?

Re: Please help with modelling with PDEs - Boundary Condition

Without the source term, a separation of variables would lead to

$\displaystyle u = \sum_{n=0}^\infty a_n e^{-k^2t} \cos \frac{ n \pi}{L} x$ where $\displaystyle k = \frac{D n \pi}{L}$.

So, with the source term we try a solution of the form

$\displaystyle u = \sum_{n=0}^\infty T_n(t) \cos \frac{n \pi}{L} x$.

If we expand the source term itself, i.e.

$\displaystyle Q = \sum_{n=0}^\infty q_n \cos \frac{n \pi}{L} x$

for appropriate $\displaystyle q_n$, then substitute and compare like terms of $\displaystyle \cos \frac{n \pi}{L} x$. This gives us a series of ODEs for $\displaystyle T_n$.

Re: Please help with modelling with PDEs - Boundary Condition

ok i feel totally lost now

Using u=T(t)X(x) on the equation without the source term i get X(x)=cos(npix/L) but i really have no idea what i should do after this. I can get to u=(cos(npix/L))e^((-n^2pi^2/L^2)*t) but i've no idea where to go from there. how do i involve my source term again as well as use my initial condition?

Re: Please help with modelling with PDEs - Boundary Condition

ok i actually think i may be onto something now.

i put u(x,t)=sum(1,inf) of u_n(t).cos(n*pi*x/L)

then subbed this into my original equation, then will take fourier series of source and then compare like cos terms

have i got it now? sorry its taken so long if i have!

Re: Please help with modelling with PDEs - Boundary Condition

right so this is what i have to show:

By seeking separable solutions, u(x, t) = T(t)X(x), show that the temperature

is given by

u(x, t) = u0(t)/2+sum(1,inf) u_n(t)cos((n*pi*x)/L) (1)

so far i have got to:

Using u=T(t)X(x) on the equation without the source term i get X(x)=cos(npix/L) but i don't know how to show it goes to (1). Is it to do with the linearity of solutions? andi can see it looks like a fourier series and cos is an even function which would explain u_0 being there and why there is a cos term but i feel that this isnt really relevent to what i'm trying to show.

Any Ideas?

Re: Please help with modelling with PDEs - Boundary Condition

I got all the way through to the very end and my u_0 and u_n for my fourier series are wrong due to one sign that is incorrect! is there any way i can scan in images of my work so someone can look for the incorrect sign?

Re: Please help with modelling with PDEs - Boundary Condition

I just wanted to let everyone know i solved this problem and thankyou for all your help