Coupled Equations how to solve? 1

[Promovendus]

Hi everybody,
Well actually i am not sure of getting an answer from you, as i read a former post you are not about helping MATLAB beginners....any way, maybe someone finds interesting enough my problem.

I have 2 coupled PDEs and i have no idea how to find a solution. I have checked a lot of tutorials and they just explain how to solve one PDE or, in case, a system of PDEs, but, what about 2 equations which are coupled?
I would appreciate if at least you could provide me a tutorial, article of whatever, where this case is considered.

Thank you in advanced guys!!

 

[VisiGoth]

MATLAB help talks about how thier control systems solver works. That would be a way to start. Can you give us the industrial application this is, what the physical parameters are in your equation. Some experts in that domain can then comment and possibly give you advice on how to accomplish your goals without your particular approach. You gave it so little information and did not define your application that this almost sounds like a HW problem.

 

[Promovendus]

Thank you VisiGoth for your reply. Indeed i did not want give details about the problem i need to solve as i was not sure of the importance of the post for all you guys.
In my case, i want to model the temperature profile of a combustion wave in combustion of solid fuels. For this reason, i have to work with the heat equation where the heat source contains a parameter related to the degree of conversion (from the starting fuel to the combustion products which shows an Arrhenius dependance). The equation are more or less as follows:

dCp(dT/dt) = k(d2T/dx2) + Qd(dr/dt) (1)

dr/dt = Koexp(-Ea/RT)(1-r) (2)

where d,Cp,k,Q,d,Ko,Ea and R are constants. IC and BC are not important at this moment. Eq (1) is a non linear parabolic PDE, eq (2) is a, let's say ODE. I know how to solve then separately but not coupled!!! The matlab help is so tough to find what you need!!
Does any of you know the sentence i have to use? in that case i could use the matlab help to go further....


Thank you for your replies!!!!

 

[electricpete]

x must be an independent variable (coordinate). So what is r? coordinate or constant?

If coordinate, how does it relate to x?

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[electricpete]

I'm sorry. It must be r is a dependent varaible upon time. Presumably r is not dependent upon x?

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[electricpete]

Here is one way to do it:
If you took the Laplace transform of all your equations, you could convert the variable t and derivatives with respect to t into variable s. Treat s as a constant and you have a set of coupled ODE's. Solve the ODE analytically in terms of s. Take inverse Laplac transform to put t back in.

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[electricpete]

I'm thinking r has to be a function of x otherwise equation 2 makes no sense. Guess I answered my own quesiton.

I tried the Laplace method and didn't work too well on this example. Having a dependent variable (T) as argument to another function (exp) makes this method not work here.

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[Promovendus]

Thank you electricpete for your ideas!!
r is the reaction rate and in principle depends on the temperature and time although it would as well depend on the position of the front wave (x). But lets make the problem easier and forget the dependence on x of r...The thing is that to solve eq (1) i need first solving (dr/dt) to consider this as a value for each iteration in eq(1). So far i can only solve Eq(1) considering fixed r's, but this is not a correct approximation as r depends on t and for each t you get a different value of (dr/dt) which increases or decreases the heat generation as the chemical reaction (combustion) is proceeding....i am just stuck!!
More ideas? pleaseeeee

 

[VisiGoth]

I am having problems understanding your notation. Could you make it explicit? i.e. is dCp the first derivative wrt time or is it d * Cp? For your audience to spend less time it might be good to be able to see the equation and not have to sift through your list of variables to figure this out. Also, as electricpete suggested, show the dependents. Since you said dr/dt can we assume that r is a function of t and only t? Since you said (dT/dt) I would think that T(t), but you also said (d2T/dx2) so maybe it is T( t, x ).
By substituting r_dot in the second eq. into the ifrst do you get

partial of T wrt t minus second partial of T wrt x is equal to:
( Q / Cp ) ) * k * exp( - Ea / ( R * T( t, x ) ) ) - 1 - r( t ) )

You do not show the first partial of T wrt to x, that might help if you knew it.

 

[Promovendus]

It is exactly as VisiGoth has written. I am so sorry about the misunderstanding in notation but i usually use an equation editor and not a plain text window for posting....i am really sorry!!
Eq (1) is the general form of the heat equation thus there is no term concerning the first partial of T wrt x....
Someone else did suggest me to integrate first Eq (2) at a fixed T and then to substitute it in Eq (1). Then solving Eq (1) in small steps...to be honest i do not know whether that can be done with matlab.

Is there no way of obtain solutions of Eq(1) for each solution of Eq(2), i mean, by using a handle function and a loop, like it could be in C++, fortran or turbo pascal?

 

[electricpete]

So could equation (2) could be rewritten:

dr(t)/dt = Ko*exp(-Ea/[R*T(x=r,t)])(1-r(t))

In another words, the T(x,t) in the denominator of the exponential is evaluated at x=r, right?

If that is the case, we can look at equation 2 as a nonlinear ordinary differential equation with independent variable t only.

The solution of equation 2 according to my Maple program (with T and r functions of t only and not dependent on x) was:

r(t) -1/2*r(t)^2 -Ko*integral{exp(-Ea/R/T(t)) dt} = C1

Unfortunately, it is an implicit rather than explicit solution of of r(t) in terms of T(t).

I don't think that helps much.

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[Promovendus]

Hey Electricpete, thanks a lot for your effort.
Unfortunately r has nothing to do with a spatial coordinate thus x can never be equal to r.
r is a degree of conversion (a way to measure how reactants turn into products for chemistry) and dr/dt is a kind of velocity for the conversion, that is to say a conversion rate. r is adimensional while dr/dt = [s^-1].

But really i was just thinking about the possibility of using a loop to increase r (from 0 to 1) in small steps and for a fixed r calculating the result equation of sustituing Eq(2) into Eq(1), so as the exppresion Visigoth showed in his last post. Then for each iteration r would take a fixed value and not more dependent of t....but, i really do not know whether it could work properly or whether matlab could successfully do it.
Is there any mathematician between us?

Another question, so far i know, matlab can only solve non linear parabolic PDEs in one dimension. Does anyone know which program can solve PDEs in 2 or 3D?

 

[VisiGoth]

Promoovendus,
I think Electrippete is saying that r is a function of x and t. He is not saying that r=t in his notation.