Thank you, professor Ijaz Durrani

I notice that the boundary condition is nonlinear. As a general rule, separation of variables only work for linear equations with linear boundary conditions.

Hello Arturo Ortiz Tapia ,

Forgive my ignorance, but there are some things I don't understand about your problem statement. If your system is 1-D, then that implies that it is infinite in the y and z directions, for example, and finite or semi-infinite in the x direction. Suppose the system is finite in the x direction, i.e., it goes from x=0 to x=L

Thomas Cuff , I stand corrected by your observation. Indeed, we have a finite length, hence the (unknown and to-be-solved) function inside the BC *is indeed* only a function of time; having said that, notice that we don't know precisely that function that has to be solved (only formally, and as you rightly so pointed out, a function of time, with x=0 or x=L), and that appears both on the heat equation and as part of the BC.

Hello Arturo Ortiz Tapia ,

I forgot to mention it in my last post, but if you are interested, there are worked examples of 1-D systems, very similar to the one you mentioned in your question, on my RG web page:

1) Laser flash diffusivity measurement (1-D region bounded by x=0 and x=L with Neumann BC's at both surfaces: Method Laser Flash Diffusivity Measurement

2) Two semi-infinite regions (x=-∞ and x=+∞) with a surface heat source sandwiched between them at x=0: Method Example#2-TwoSemi InfiniteRegionsSeparatedByAPlanarHeatSource

Regards,

Tom Cuff

I have not plunged deeply into the details of your problem, but it looks similar to the problems described in the book "Boundary Value Problems of Heat Conduction" by M. Necati Ozisik, Dover Publications. I've have used this reference for time dependent boundary condition problems for both the heat and wave equations with success. The time dependency at the boundary is addressed by the integral transform technique as discussed in Section 2-3 of the this book.

This can be solved numerically. Well, that is my guess, but based on past experience with unusual boundary conditions, and given that these represent a well-known physical process, then I would very surprised indeed if it couldn't be solved numerically.

But as Mat Hunt has said, the boundary conditions are nonlinear even though the equation is linear. With almost any method of solution you will need iteration at every time step.

Can be solved

Hello Mat Hunt ,

Can you, please, provide a citation to a book or paper supporting your contention that a linear, homogenous, parabolic PDE cannot generally be solved by separation of variables if the Neumann boundary condition is nonlinear in the dependent variable? I am asking because I am having trouble finding such a theorem and its source.

Regards,

Tom Cuff

1D heat equations can be solved by semi-analytical methods. Separation of variables in problems with the BC ~ T ^ 4 will not succeed in the form in which they usually do. In solving such complex problems, I usually use the linearization of nonlinear BC and solve the system of heat equations at small time intervals. This allows you to reduce the risks that nonlinearity puts into the solution. I note that this approach will work perfectly at small temperature differences at the boundaries of heat transfer by radiation.

So the essence of the linearization method is as follows. The BC's with (Tenv ^ 4-T (x, m) ^ 4) is factorized, namely:

Tenv ^ 4-T (x, t) ^ 4 = {Tenv-T (x, t)} * {[Tenv + T (x, t)] * [Tenv ^ 2 + T (x, t) ^ 2] }.

Assuming that the factor {[Tenv + T (x, m)] * [Tenv ^ 2 + T (x, m) ^ 2]} = const has a varying sign on each time interval Dt, we obtain a linearized boundary condition. More details can be found here https://www.ams.org/journals/qam/2006-64-02/S0033-569X-06-00970-7/. I used this approach in solving many 1D tasks with a nonlinear BC's and got excellent results. I also draw your attention to the fact that we must not forget that linearization increases the calculation error, and this error must also be known. It can be substantially reduced by reducing the intervals Dt to such a value when the comparison of the results of the final solution at different time intervals Dt will not practically differ.

Hi Alexey Romanov

Does your linearisation form part of a one-step computation per time step, or one part of an iteration scheme per time step? The latter procedure wouldn't then increase the calculation error.

Hi D. Andrew S. Rees

Detailed answer in the attached file

Hi Alexey Romanov

Many thanks. Is your document a solution of the original question? I can't see the linl - perhaps it is the Friday effect!

The manner in which I would "linearise" Arturo Ortiz Tapia 's problem is a little different. If I were summarise his BC as q+h(Tenv-T)+sigma*M(Tenv^4-T^4)=0, and assume that the solution at tau=tau_n is known, then I could linearise in the following way. Let

T_{n+1} = T^{i} + delta

where T^{i} is the current iterate for T_{n+1} and delta is the exact error. So the first iterate could simply be T_{n}.

Then the BC becomes

q + h(Tenv-T^{i}-delta) + sigma*M(Tenv^4-T^{i}^4 - 4T^{i}^3 delta -.... = 0

where the dots are higher powers of delta which we ignore. Hence the correct to T^{i} is

delta = [ q + h(Tenv-T^{i}) +sigma*M(Tenv^4-T^{i}^4)] / [ h+4sigma*M*T^{i}^3].

A small number of iterations would be enough to find the BC satisfied perfectly.

There may be slight differences of implementation depending on whether the interior nodes are updated using forward Euler or Crank-Nicolson or BDF2.

Hi D. Andrew S. Rees

You immediately kill “two birds with one stone” in your approach and solve the task only once, and I have a set of solutions “one by one”. But intuition tells me that if you use any simple numerical methods in the solution, then the number of possible options for obtaining the solutions themselves will increase. Therefore, in my approach, I didn't resort to any numerical method, but settled on a semi-analytical approach. I like it here that solutions easily stitch at the boundaries of time intervals. Nevertheless, I believe that Arturo Ortiz Tapia has the right to choose any approach to solving the problem, using what was offered to him, and will get excellent results.

I agree.

Separation of variables works only with linear systems.

Hello D. Andrew S. Rees and Alexey Romanov ,

Let me accept your contention that even if the differential equation is linear, but the boundary condition(s) is(are) nonlinear, that the overall system (DE + BC's) is nonlinear. It is my understanding that nonlinear systems do not satisfy the uniqueness theorem. How then are you both so sure that your respective numerical solution or semi-analytical solution is the only solution or even the relevant solution for the physical system being modelled?

Given the well-known difficulties with numerical solutions of nonlinear systems, how can you possibly be sure that your numerical solution is even correct, i.e., there is usually no explicit solution with which to compare it for correctness. Here, it seems to me, in my crude way of thinking, that you are in an infinite regress of trying different numerical techniques in the hope that the apparent convergence you might see is real, i.e., that it converges to a true solution and not to some incorrect solution.

With respect to the semi-analytical approach, how do you know that the nonlinearity is not an essential nonlinearity, which cannot be linearized - in any way - and still truly represent the original physical problem? Again, it seems to me that it is an infinite regress.

I believe that without mentioning all of these caveats, the question posed by Arturo Ortiz Tapia is not really being answered or that it is being answered in a very facile way with all the real issues being glossed over in some fashion or other. And I accept that I have been part of the problem by not being as rigorous as I should have been. I apologize.

Regards,

Tom Cuff

I guess that a pure mathematician might be able to provide the definitive answer to this question.

I'll make a few comments. In the original question there are some uncertainties and errors.

First, the location of the boundary conditions have not been specified, but one could, in the first instance, assume that the solid, whose temperature field is being sought, remains of the same length for all time.

Second, it is stated that it is a Robin boundary condition. This is not the case because if it were, then an x-derivative of T should also be included. My suspicion is that this has been accidentally omitted.

Third, as the boundary conditions are presently stated, the boundary value for T may be found in terms of the four values, q0, hconv, sigma and em. If those four values are constants in time, then the BC becomes T=something constant, a Dirichlet condition, and the problem may now be solved using separation of variables for it is now no longer nonlinear.

Fourth, there is no initial condition stated.

Fifth, if I am right that a dT/dx term has been omitted, then we have a physical problem with boundary conditions that allow for heat loss/gain from a surrounding medium (i.e. a Newton's law of cooling), and for radiative loss of heat. Fourier's equation is also dissipative. Therefore there is no physical mechanism here which would cause a thermal runaway, although if the values of q0 on the two boundaries are unbalanced, then a linear growth of T in time will occur. My gut instinct (but lack of pure mathematical knowledge) suggests to me that the nonlinearity is not of the sort that could cause an instability, and therefore there would be no second potential solution given the same initial conditions.

I think that the above exhausts everything useful that I can say about the problem! I shall observe how this develops....

Hello D. Andrew S. Rees ,

It appears that we still cannot even agree as to the identity of the type of BC:

1) Dirichlet BC, where the temperature at the boundary is either fixed at a predetermined value or a predetermined function of time, see [1] David L. Powers; Boundary Value Problems, Second Edition; Academic Press; 1979; p. 89.

2) Neumann BC, where ∂T(x,t)/∂x|x=boundary is either constant or a predetermined function of time. Since the heat flux at the boundary is J(x=boundary,t)=-k*∂T(x,t)/∂x|x=boundary this means the Neumann BC is physically akin to specifying the heat flux at the boundary, e.g., if the boundary is insulated, then J=0 → ∂T(x,t)/∂x|x=boundary =0. [1] David L. Powers; Boundary Value Problems, Second Edition; Academic Press; 1979; p. 90. Arturo Ortiz Tapia has decided, in his problem, to specify the heat flux at one or both boundaries. The heat flux is due to a constant surface heat flux term, a convective heat flux term, and a radiative heat flux term. In other words, J(x=boundary,t)=-k*∂T(x,t)/∂x|x=boundary = A + hconvection(Tambient-T(x=boundary,t)) + B(Tambient4-T4(x=boundary,t)) → ∂T(x,t)/∂x|x=boundary = -(1/k)[A + hconvection(Tambient-T(x=boundary,t)) + B(Tambient4-T4(x=boundary,t))].

3) Robin BC, where both the temperature and the heat flux are specified at the same boundary. Arturo Ortiz Tapia claims he is using the Robin BC even though the temperatures - except the ambient temperature - in his expression for J are not specified ahead of time, i.e., they are not predetermined but depend on the temperature solution of the PDE togther with the IC and the BC's, only the functional forms of the surface heat flux contributions are predetermined.

If we cannot agree on the type of BC, how will we ever agree on the more difficult problem of does a nonlinear BC (due to the radiation term) invalidate the use the technique of separation of variables for the solution of the homogeneous, linear, parabolic PDE? I have seen such confusion before in the statement of a question on RG, but I was hoping not to see it yet again. Like, D. Andrew S. Rees I am also finished with this question.

Regards,

Tom Cuff

Hi Thomas Cuff

The heat equation simulates a whole class of physical phenomena and the “general” case may not have physical meaning. The same applies to BC's. In this “general” case, the heat equation doesn't say anything about how heat transfer occurs in a 1D-body. This explains the non-uniqueness of the solution of PDE's, including the considered heat equation. In order to avoid obtaining the non-uniqueness of the solution, in our case, the equation is given a certain physical meaning, introduced not only by specific notation of constants, variables and functions, but also by BC's. Such conditions, known for the conditions of uniqueness, allow one to exclude non-uniqueness of a solution. I will list the main ones. The conditions of unambiguity in our case include:

1. Geometric conditions. In a 1D-task, the body of the form has no, but it has boundaries. A nonlinear BC is specified on one of the boundaries.

2. Boundary conditions. If not a single BC were set, then we would be interested only in the solution of the heat equation itself. Nonlinearity here could be manifested in an additional term on the right side of the equation describing internal heat sources. But they are not in the task, because their localization and heat exposure would be known.

3. The most important thing is the physical conditions by which it can be judged that the heat equation in parabolic form must be solved. Such conditions include thermal conductivity, heat capacity and density. They can be taken as constant values ​​or as continuous, depending on the temperature.

4. Initial conditions. Nothing was said about them in the task either, but, if something could be the reason for the non-uniqueness of the solution, then that is it. I will explain in more detail.

From a mathematical point of view, the initial condition defines the distribution in the form of a coordinate function, in the particular case a constant value T (x, t = 0) = T0. The functions that describe the solution of the heat equation at times other than t = 0 sometimes don't have physical meaning at t = 0 and may have gaps in the coordinates. Therefore, the limit transition at t--->0 is always permissible for them, guaranteeing the continuity of the temperature distribution function at the initial moment. This in no way contradicts the other conditions of uniqueness and leaves researchers hope that they will not have to recall the theory of equations of mathematical physics every time they solve the initial-boundary-value heat transfer problems by the previously proposed methods.

Hello Arturo Ortiz Tapia and Alexey Romanov and D. Andrew S. Rees ,

[1] J.C. Jaeger; Conduction of heat in a solid with a power law of heat transfer at its surface; Mathematical Proceedings of the Cambridge Philosophical Society; Vol. 46; No. 4; October 1950; pp. 634-641; Can be purchased from the URL: https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/issue/8B295840F92AEE8C55F7567FBE77A7D4

[2] W. Robert Mann, František Wolf; Heat transfer between solids and gases under nonlinear boundary conditions; Quarterly of Applied Mathematics; Vol. 9; No. 2; 1951; pp. 163-184; Can be downloaded for free from URL: https://www.ams.org/journals/qam/1951-09-02/S0033-569X-1951-42596-7/S0033-569X-1951-42596-7.pdf

Both the above citations were found in the following book:

[3] H.S. Carslaw, J.C. Jaeger; Conduction of Heat in Solids, Second Edition; Oxford at the Clarendon Press; 1959; p. 21.

These same citations might also be in the book by M. Necati Özisik mentioned by Douglas Vinson Nance , but I do not have access to this book, currently. This book can be purchased as an ebook at the URL: https://books.google.com/books?id=3bjDAgAAQBAJ . Douglas, would you, please, check if citations [1-2] are in this book for us, if you have the time and inclination?

Arturo Ortiz Tapia , thanks for an interesting question. I am sorry I wasted your time. I should have immediately consulted Carslaw and Jaeger. I was simply too indolent to do so.

Goodbye, finally,

Tom Cuff

Interesting discussion

Is time not the dimension? You can solve 1-D problem by consider if simply you have term partial u/partial t, then by taking u(t)= 1+exp(iota epsilon t) and thus can add real and imaginary parts of d.e.

Yes we can solved it. If the problem is w'll posed then you may solved such problem easily by separation of variables. Despite this you may try to exercise the Laplace method as well we can easily tackled such problems. Finally, if still you have some issue then sent me I will help you manually.

Umair Khan : The boundary conditions are nonlinear! This means that neither separation of variables nor Laplace Transforms can be used.

This is a fairly straightforward problem to solve using finite differences, particularly if forward differencing in time is employed although some iteration will be required to find the boundary temperatures.

Dear Dr. Andrew S. Rees,

Yes you are right if the boundary is nonlinear. I think the boundary is linear then my way of solving is right. If the boundary is non linear then we can solve it by the numerical technique. We can think like that as well if we suppose the solution in general and make our boundary homogeneous. I did such problems when the boundary or equation is non homogeneous and make it homogeneous according to the above letting steady + transient part and then apply separation of variables

Umair Khan : I agree with you. I will add, though, that the use of separation of variables when one of the boundary conditions is a Robin condition becomes more difficult than for the standard Dirichlet or Neumann cases because of the need to find the set of orthogonal functions.

Hi, Ijaz Durrani

I have read what is stated in the PDF source. Unfortunately, you didn't understand the essence of the topic problem and decided that we are talking simply about solving the 1D-equation of heat conduction. In fact, the key problem is the analytical solution of the 1D heat equation with a nonlinear BC. In your source, the solution of standard 1D equations is shown and doesn't add anything new to the solution of the problem raised in the topic.

Ijaz Durrani

Thank you very much for the interesting selection of the solution to the heat equation