Coupled Equations how to solve?
Coupled Equations how to solve?
(OP)
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!!
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!!





RE: Coupled Equations how to solve?
RE: Coupled Equations how to solve?
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!!!!
RE: Coupled Equations how to solve?
If coordinate, how does it relate to x?
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RE: Coupled Equations how to solve?
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RE: Coupled Equations how to solve?
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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RE: Coupled Equations how to solve?
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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RE: Coupled Equations how to solve?
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
RE: Coupled Equations how to solve?
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.
RE: Coupled Equations how to solve?
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?
RE: Coupled Equations how to solve?
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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RE: Coupled Equations how to solve?
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?
RE: Coupled Equations how to solve?
I think Electrippete is saying that r is a function of x and t. He is not saying that r=t in his notation.