On Nonlinear Conductive Heat Transfer Equation

[CSAPL]

I am looking for an analytical solution to the 1D nonlinear heat transfer equation

d^2T/d^2X = c(T) dT/dX

as you can see nonlinearity appears as c is function of temperature; i.e c(T). I want analytical solution . Can you guid me to a sourec where I find the analtical solution of this 1D difuusion equation.
I already posted this in Heat Transfer Forum.

 

[CSAPL]

let me tell what I am doing regading this issue so that we can share knowledge. As you know that the nonlinear equation is d^2T/d^2x=C*dT/dt
and the nasty C(T) =(dS/dT) * T + S + A(dS/dT). Now if the relation between S and T is emperical (fitting curve),why not find S=f(T) for which
(dS/dT) * T + S + A(dS/dT)= constant (C) So I am just solving this equation so that I will get families of S =f(t)for which the nonlinear equation above becomes linear
afterwards I will optimize the constants of this simple differentail equations to find the best curve that fits my emperical data S and T. Makes sense to you.Your comment is appreciated
P.S zekeman: you can find info about Integral Methods for NLHT problem in any HT book

Thanks

 

[zekeman]

What you are saying, is that you will enter a universe of constants, each equalling C, to get a universe of f(x)s and pick the one that comes closest to the empirical f(x). If you are extraordiarily lucky, you may have an infinitesimal shot, although coming close is no guarantee of a proper solution.
Why are you going through all this labor? What is wrong with a numerical solution?
If you can share this with us, what is the application? Maybe one of us could shed some light on it.
And by the way, I am aware of integral methods, but they will not solve general nonlinear PDEs as I mentioned. Want to win a Nobel Prize in mathematics, just find a general solution to your problem. Oh, they don't give the prize in mathematics, so make it a problem in economics and then you may.
Good luck.

 

[CSAPL]

zekeman,
Well the idea of trying to get T given in an analytical formula is because I am going to use it in another close formula. Plus you talk about developing numerical solution as if it were piece of cake and requires few minutes to finish without any troubles.
P.S. I have no intention now to develop Zekeman's (or soory Goodman's) Integral Method in heat transfer But if i decide and win Nobel price I will give you some bucks

 

[chicopee]

To CSAPL- This reply is a bit late,however, I became interested in your problem and determined that you may already have the answer if you had DE in college. Esbach,3rd Edition has a series of solutions for second order DE on pages 326 thru 327.
You stated that C(T)=(dS/dT)*T + S +A(dS/dT). I am quiet certain that dS and dX are the same, therefore, plugging c(T) in your equation d^2T/dX^2 would yield
d^2T/dX^2 = T + A + X(T)dT/dX
which could be solved adding the results of Esbach's case 1 and case 3 the latter being shown by Zekman in your reply.

 

[chicopee]

Further digging into your question. Your DE and C(T) equations that you presented simplifies to a non-linear DE:

T" = x(T)T'+T+A; A is a constant per your definition

let T+A=I(P*dX); I is the symbol for integral

differentiate T+A becoming dT=PdX then P'= T"

make substitution to above DE: P'=x(T)P+I(pdX)
then: P'/P=x(T)+(I(PdX))/P
=x(T)+I(dX)
then: dP/P=x(T)dX+(I(dX))dX
I(dP/P)=I(x(T)dX)+I(I(dX))dX
since x(T)=F(T) then dx=f(T)dT
I(x(T)dX)=I(F(T)dT)
Final differential equation:
I(dP/P)=I(F(T)dT)+I(I(dX))dX
Make sure imits of integrations are appropriately changed.

 

[Cansand]

chicopee
I do not think CSAPL meant in his equation that dS and dX are the same" Why you said "I am quiet certain dS and dX are not the same. According to CSAPL explaination S is a function of T while X is the coordinate

CSAPL can correct me if i am wrong
d^2T/d^2X = c(T) dT/dt
C(T)=(dS/dT)*T + S +A(dS/dT)

 

[chicopee]

helppoorpeople- what is the significance of S?

 

[chicopee]

helppoorpeople- Yep, somehow I missed CSALP equation for S.