Boundary Value Problems in Heat Conduction
3.13 The linear heat conductor, bounded on one side: In the preceding sections, we have already dealt with the compensation process in the case of an unbounded, on both sides infinite, linear conductor and represented the solution by
We mention here as an extension that with the substitution
it assumes the Laplace form
A comparison with D'Alembert's version (9.2) is interesting: One has in the latter case two arbitrary functions F1, F2, corresponding to the hyperbolic type of the vibration equation, in (1) and (2) one arbitrary function f, corresponding to the parabolic type of heat conduction.
In the case of the heat conductor with one boundary one is at first faced by the boundary condition to be specified at x = 0:
a) Given the temperature u(0, t), especially the isothermal condition
b) Given the heat flux (the notation is the same as in (7.9) to (7.12)), especially the adiabatic condition
c) A linear combination of these two takes into account so-called external heat conduction in the conventional form
Here, n is the outwards normal, which in our case points along the negative x-axis. Using the term external heat conduction, one combines the actions of convection, radiation into the surrounding medium and heat transfer by conduction in this medium, which in most cases is neglected. In particular, we will convince ourselves that (3c) is an approximation from the strict physical form of the Stefan Boltzmann radiation law, which is known to state: Heat radiation per time- and area-unit of a body of absolute temperature T is proportional to T4. If we denote the proportionality factor by a (the traditional constant of the law) and the bar's end* is in an environment of temperature T0, which radiates to the bar's end the heat quantity aT04 per time- and area-unit, the energy, emitted eventually in the direction of the normal by an area element ds of the bar, is
Since, as a rule, both the temperatures T and T0 lie far away from the absolute zero, one has
* We speak here of bar and bar-end, although the linear heat conductor need not have the form of a thin bar, but can be arbitrarily extended, provided that its state depends only on a single co-ordinate; naturally, in an actual bar, one has to impose the additional, adiabatic condition for such a mantle surface; cf. 3.16.
This heat quantity must be supplied by the heat flux from inside the bar, which is given by the Fourier expression (7.12), whence we write
Comparing (4) and (4a), we find
which agrees with (3c) and shows that h is a positive constant.
First of all, we shall deal with (3a) and (3b). They are fulfilled if we expand the function f, which is only given for , in a pure sin- or cos-integral or, which is by (4.11a, b) the same, if we continue f to the negative side of the x-axis as an odd or even function. By adding, like in (12.18), the time dependence and executing the integration with respect to w, we obtain instead of (1)
The second of these integrals, which initially was extended from , has already been converted by inversion of the sign of the integration variable into one from , whereby the principal solution U(x) becomes
i.e., the representation of a heat unit pole at x = -x, t = 0. Equation (6), derived following Fourier, thus becomes automatically
This Green function now satisfies all the conditions formulated at the end of 2.12. Indeed, it has in the region only a single heat pole, since the additional heat pole at x = -x lies outside this region; it also does not contradict the condition that G in the variables x, t is to satisfy the adjoined equation, because in our case t = 0 and therefore the change in sign of t is of no consequence.
Obviously, it would have been clearer to start with the single heat pole x = x and to reflect it in the border x = 0, with the negative or positive sign of U depending on (3a) or (3b). In this way, we would have constructed at first the Green function G and produced with its aid by continuous superposition of heat poles of strength f(x)dx the prescribed temperature f(x). In the sequel, we will use mostly this clear approach, i.e., restrict ourselves to the construction of the Green function, from which we can then, following the procedure leading to (8), write down the solution for arbitrary initial temperatures f(x). We will now try out this procedure in the case of the somewhat more complicated boundary condition (3c); in Exercise 3.1, as a comparison, we will deal with the same boundary condition by Fourier's method.
First of all, we will make sure that we cannot here succeed with an isolated reflection point x = -x,, but require in addition a continuous sequence of heat poles, which we introduce at all points h<-x. Let A and a(h)dh be effects of the isolated and continuous heat sources, respectively (Fig. 15). Thus, we complement (8) for G to
In addition, we obtain from (9) the derivative where we immediately replace under the integral . We then obtain for the same location x = 0
whence after an integration by parts
If we substitute (10) and (11) into (3c), where we have to interchange , then this condition must be fulfilled identically for all t > 0. If we set the individual terms of this condition, which depend differently on t, equal to zero, we obtain
as well as for a(h) the differential equation
with the solution
the last being due to (13). Thus, we have determined the constant A and the additional function a(h). It follows from 3.17 that this result is unique, i.e., that there cannot exist other solutions of this problem.
The result is
If one wants to evaluate this result numerically, the last integral can be reduced to the tables of the error integral (for example, those of Jahnke and Emde, 3. edition, Teubner 1938).
We will explain this by an example which simultaneously serves to demonstrate the transfer of our heat conduction problems into the language of diffusion
Let a cylindrical vessel be filled in its lower part 0 < x < H with a concentrated solution (for example, CuSO4) and have above it a pure layered solvent (water) up to an arbitrary height . Denote the concentration of the solution by u and set it initially arbitrarily equal to 1. Let there apply on the bottom of the cylinder for all later times the condition because the dissolved salt molecules cannot penetrate the bottom.
You meet these conditions simply by imagining the diffusion vessel extended downwards and prescribing the reflected equal initial condition as in the upper part. (In the case of a finite height of the layered water column, one must employ the somewhat more complicated reflection method of 3.16. The initial distribution for u is then
Equation (1), or somewhat more conveniently (2), yields then
The meaning of the limits z1 and z2 is obtained by setting in (1a) :
In the standard notation for the error integral
Equation (16) becomes surprisingly simple
3.14 The problem of Earth's temperature: We treat Earth's surface as a plane and assume a mean, purely periodic temperature history f(t) (its annual or daily mean). For the determination of the temperature inside Earth (neglecting, of course, the physical problem of geo-thermal depth stages as the temperature increases internally due to radio-active and nuclear physical processes), we could employ the general method of Fig. 13 by setting there x0 = 0 (Earth's surface) and (large depth) and the boundary value u0 at x = 0 equal to the empirically known function f(t). However, in this case it is more convenient to expand this function in a Fourier series, best in the complex form
and to do the same for the temperature inside Earth
Each term of this series must satisfy on its own the basic law of heat conduction, whence one obtains for un the ordinary differential equation
Moreover, in order that (2) is to become (1) at x = 0, one must have
Depending on whether n is positive or negative, we set
The general integral of (3) then becomes
Here, one must have An = 0, because otherwise the temperature would become infinite as . Moreover, one must have Bn = 1 due to (3a). Substitution of the thus specialized expression (5) into (2) yields
In order to make this real, we write for n > 0
By (1.13), was must use for n > 0 the same value of Cn, but with negative phase g. Thus (6) becomes
We see: The amplitude |Cn| of the n-th partial wave is exponentially damped on penetration, g the stronger, the larger are n and x. Simultaneously, the phase gn of the partial wave is retarded, also increasingly with n and x.
The numerical values are of interest. For a medium type of soil, the temperature conductivity is about
Hence, for the annual period T = 365?24?60?60 = 3.15?107 sec and x = 1 m = 100 cm,
At the depth of 4 m, the lag q1 of the phase is already p, the damping of the amplitude becomes 2-4 = 1/16. Already in the case of the first and most important partial wave of the temperature oscillation, rules the winter at the depth of 4 m, while it is summer at the surface; the amplitude is only a small fraction of the amplitude at the surface. In the cases of the higher order partial waves, the lag and damping of the amplitude is correspondingly high due to the factor of qn.. One can say: Soil acts as a harmonic analyzer, in that it extracts from all partial waves the fundamental one, although much weakened.
We shall consider as a special example the annual curve of an extreme continental climate, namely a uniform summer and an equal but negative winter temperature, which we will set equal to ?1. This annual curve is represented graphically by the meander line of Fig. 1 and graphically by
In order to obtain the associated series u(x, t), one would have to specialize the coefficients C in (7) according to (2.1a) so that
However, it is somewhat simpler to apply after (9) directly the algorithm (2). One finds then immediately
whence with the by q3, q5, ??? complemented numerical values (8) for x = 100 cm
and for 400 cm
A comparison of (9) with (9b,c) displays clearly the effect of depth, amplitude and phase on the temperature profile.
This result shows the usefulness of a deep cellar: It has not only temperature variations which are smaller than at Earth's surface, but is also warmer in winter than in summer ( or better, that this would be so if one could eliminate any influx of air.)
Our conclusions become most detailed as we proceed from the annual to the daily mean. The qn grow then by a factor Hence there already appears a damping and phase delay, which in the year-curve belong to the depth x, in the day-curve to the depth of x/19. Equation (9c) shows that the decrease of the amplitude of 1/16 in the leading term and the inversion of the day's time (midnight instead of noon) appear already in the day curve at the depth of x = 400/19 = 21 cm. Thus, the daily temperature variation only penetrates the soil with appreciable intensity by a few centimetres. The entire process is confined to a thin surface layer.
Clearly, we are dealing here with the analogue to the important skin effect of the electrician. The fact that this is in practice mainly encountered with cylindrical wires, is not essential; we have seen in Volume III that it occurs quantitatively in almost the same manner with a conductor which like Earth is bounded by a plane. Our day curve corresponds in the electrical case to a high frequency alternating current, our year curve to one which is slower by a factor of 1/365. Indeed. we know from 2.7 that the differential equation in both cases - the thermal and electric one - are the same, where only our factor k must be interpreted (apart from a factor) as the specific resistance of the conductor.
3.15 The problem of the ring: We proceed now from the heat conductor, infinite on both sides treated in (2.12), and that bounded on one side (3.13) to one of finite length 1. However, at its ends x = ??, we will not impose the boundary conditions a), b) or c), but the much simpler
periodicity condition. We understand thereby that not only u, but also all its derivatives at the two ends are to be the same. We ensure this by making the ends of our finite bar touché¬ i.e., bending it into a ring. The shape of this ring is not important, since we assume that its mantle, as always for the linear conductor, is adiabatically shielded. Fig. 16 shows the ring as a circle.
As initial temperature distribution, we select an f(x) which is arbitrary, but is symmetric with respect to x = 0 . Its Fourier expansion is then a pure cos series which automatically satisfies the periodicity condition at the end. According to (4.1) and (4.2), where one must set a = ?,
In order to obtain the corresponding solution u(x, t) of the heat conduction equation, one need only multiply the n-th term by
i.e., write instead of (1)
We will now specialize f(x) into a d function by demanding that
Then, as we see from (1),
and we find from (2), if we still convert u into the historically sanctified ,
The symbol points at the q-function, introduced by C.G.J.Jacobi (1786 - 1856) into the theory of elliptic functions, which is for numerical computations of enormous importance.* The fact that it satisfies the heat conduction equation is employed there as an, in a certain sense, accidental property, while this property has served us just as the definition of .
*The reason for its special convergence has already been pointed out: Since the -series together with all its derivatives is periodic and therefore does not have jumps at x = +? and -?, its terms decrease with increasing n more strongly than any power of n.
We must now also adapt our notation t to the theory of -functions by setting
Of course, this quantity t has nothing in common with the t - t in the principal solution U; it does not have the dimension of time and is in our case positive imaginary. (In the theory of the elliptic functions, in general, t is complex with a positive imaginary component, in fact, it is equal to the ratio of the two periods of these functions.) Written in t, we obtain a formula, which is identical with (3):
This formula converges very well for large |t| or, what according to (4) is the same thing, for large kt. Hence it represents excellently the later phases of the fading away of the unit source on our ring; however, it deserts us for the start of the process. It is for this reason that we compliment Fourier's method, used so far, by one which, similar to the reflection process, rests on the periodic continuation of the initial state (Fig. 16, right half).
We have here unwound the cut ring onto the x-axis in sequence to the right and left hand side. Out of the heat source U0(x, t), given in the ring, there arise thereby at the locations x = n identical heat sources
Hence, we have by the series
a second representation of the damping process which converges excellently for small values of kt. In fact, for those, only U0 and its closest neighbours must be considered, while the further away Un, due to the factor exp(-n?/4kt), are inactive, Whence (7) is the required complement to (5). Fig. 16 demonstrates the type of the two representations: By the flat curve, the course for large kt following (5), by the steep curve, the course for small kt following (7).
It is strange that also (7) can be brought into a form similarly to the -function in (5). One only needs to extract the factor exp(-n?/4kt) in front of the sum and combine the terms ?n. Then (7) yields to start with
It we now introduce t instead of t, following (4), the sum becomes
This expression differs from (5) only by the fact that the argument of cos becomes x/t instead of x and that of the exponent -1/t instead of t. Hence, the bracket in (7) is nothing else but
If one also introduces in the factors ahead of the brackets t for t and takes into consideration that (5) and (7) are solutions of the same heat conduction problem, one arrives at
or conversely at
This is a famous transformation formula of the -function. It serves in the theory of elliptic functions to convert the series (x|t), which converges badly for small t, into the excellently converging series (x/t |-1/t). For us, this is the transition from the Fourier method of heat conduction to the method of heat poles. In quantum theory, the transformation formula (8) has a role in the rotation energy of bi-atomic molecules and the calculation of their specific heat at low temperatures (cf. Volume V).
3.16 The linear heat conductor with two ends: As we have set in 3.15 the length of the ring equal to 1, which was convenient for linking to the -formulae, we introduced thereby without comment a new dimensionless co-ordinate x' = x/l and wrote for x' again x. Hence, for the bar of length l, to be considered now, we must, in order to refer to the formulae there, replace x by x/l; in the sequel, t is the same abbreviation as in Equation (15.4).
We start with a tabular survey of the tasks and solutions for the earlier stated boundary conditions of Fourier as well as the heat pole method. Regarding the last, we note in advance that it leads in contrast to 13.13 to an infinite sequence of mirror images, because one does not have to reflect only the primary heat pole, but also its mirror images at the two ends of the bar. Consider an optical example: There does not appear only once in each of the mirrors in a hall with two parallel mirrors the chandelier, which hangs from the ceiling hanging, but it is repeated infinitely often.
As far as regards the formulae for f(x), one sees immediately that they satisfy the given boundary conditions a)a) ??? b)a) on the left hand side; the same observation applies to the associated solutions u(x, t) of the boundary value problems, which are obtained with Fourier?s method from the series for f, if the last term by term are multiplied by
On the other hand, the schemes in the top row of each of these sections demonstrate the position and sign of the heat pole according to the reflection method. In the first two cases, one sees that the heat poles have the period 2l, in the last two cases the period 4l. Their summation yields in each case the Green function G = SU and is here, for the sake of brevity, expressed in terms of the -function of the preceding section, in a)a) and b)b) by two such terms, in a)b) and b)a) by four terms, as follows from the figures. Obviously, we must replace x in the formulae of the preceding section , where the period is 1 and the heat pole is assumed to be at x = 0, by (x-xi)/2l in a)a) and by (x-xi)/4l in ab) and b)a), respectively; xi is here the position of any heat pole of the sequence, summed by (which is unimportant because of the periodicity); in particular, we have taken in our -formulae for xi the heat pole of the starting regime 0 < x < l or of one of the adjacent regions. Our knowledge of the Green function then yields, as we know, the solution of the boundary value problem for any initial distribution u(x,0) = f(x), every time according to the general rule
We will now look at the boundary condition c and deal especially with the combination a) c). In order to satisfy beforehand the condition a) at x= 0, we set
Thus, we substitute into the solution at a)a) instead of the series of the integer n the series
which we will determine so that for x = l every term of the sum (2) satisfies our condition c). Thus,
and we arrive at the transcendental equation
It is identical with (1.6.2a), if we set there a = -p/hl; their solution was illustrated by Fig. 7. Here too, we have to deal with a typical case of an-harmonic Fourier analysis. Hence we cannot use the values of the coefficients Bn in each term of the sum of (2) directly from (1.6.3b). We obtain the final solution of this boundary value problem a)c) by adding in each term of the sum of (2) the time factor, conditioned by the heat conduction equation:
We have already announced in 1.6 that we will replace the formal computation of the coefficients B there by a physically sensible one. This will happen soon, and actually in the way that we can refer to this simplest case during subsequent developments in terms of eigen-functions.
We select two terms of the series (2)
they satisfy the differential equations
The left hand side is a complete differential quotient (when dealing with Green's theorem, we said a divergence). The integration of (5b) over the basic region 0 < x < l reduces on the left hand side to the boundary points (in Green's theorem, we said to a boundary integral). From then on, the value of the integral on the right hand side can be found without manipulations; indeed, it is
The right hand side vanishes here for x = 0, because, due to (5), un = um = 0; but it also vanishes for x = l, because the boundary condition c) is valid for each term of (2) separately and hence the du/dx are proportional to the u, whence one has for the orthogonality condition (1.6.3).
It is shown in Exercise 3.2 that, starting from there, one can obtain the normalization integral (1.6.3b) almost without manipulations.
Not only our means of presentation, but also our mathematical formulation here rests on the assumption that the mantle of the bar is totally protected against loss of heat, which one has a good reason to doubt. Hence, we want still to show how our formula can be used even when this condition is not met.
Thus, we impose on the area element ds of the mantle of our, for example, cylindrical bar with a circular cross-section, instead of the adiabatic boundary condition (b) the condition (c), which says that during unit time the amount of heat
escapes through ds. We apply this to a cylindrical element of the bar of height dx and cross-section radius b, the mantle of which is therefore 2p bdx and the outer normal dn of which points in the direction of the extended radius. Then, the amount of heat leaving the bar in time dt is
By adding the algebraic sum of the two covers x = const and x + dx = const of the cylindrical bar element to the amount of heat lost through heat conduction
we obtain for the heat the amount of heat lost dQ1 + dQ2 . By (2.7.9), it equals the product of and the volume under consideration, i.e.,
By (7) and (8), this yields
On the other hand, by (2.7.11), is related to the temperature drop of the element of the bar. We thus find, after division by k,
We see that external heat conduction through the mantle adds to our differential equation only the additional term on the right hand side of (11). Our conclusion was based on the assumption that the linear character of the thermal state, i.e., its sole dependence on x is not disturbed noticeably by sideboards radiation, which seems to be plausible in the case of a sufficiently small cross-section.
It is now very simple to take during the integration into account that additional term in (11) . One need only set
one then obtains, after division by exp (-lt), the differential equation
i.e., again the ordinary heat conduction equation provided one sets
All the work of this chapter applies accordingly to a bar with external heat conduction after multiplication by exp (-2hkt/b).
An elegant experimental determination of the ratio external : internal heat conduction as well as of the physically important ratio of the heat conduction, i.e., electron conduction in metals, is treated in Exercises 3.3 and 3.4.
3.17 Reflection in a plane and in space: We now desert linear heat conduction and move into spatial regions, which are bounded by planes and can be treated by the simple reflection method; for the corresponding plane regions, bounded by straight lines, similar methods apply.
The simplest case is the half-space with the boundary conditions u = 0 or Since we already know the space function of the heat pole (3.12.14), we can immediately write down Green's function G of the half-space. Letting z = 0 be its boundary and x, h, z the co-ordinates of the source point, we find
it follows that
However, also in the case of the boundary condition c), the earlier solution (3.13.15) is readily transferred to the space problem:
However, not all regions bounded by planes can be solved by reflections. It requires that during continued reflection of the initial space the entire space is covered as well as simply without gaps . We show this by the example of the wedge. If it has an angle of 60? (Fig. 17.), it is reproduced 5 times during continued reflection and the process ends. Thus, its Green function is represented by the sum of 6 heat poles, of which, for the boundary condition u=0, one half, i.e., the initial pole Q and its images Q 2 and Q4 are positive, the others, i.e., Q1, Q3, Q5, negative.
It is already clear from this figure that only those poly-hedral can be solved by reflection, for which all edge angles add up to p (and not 2p). Among the wedge angles, already that of 2p/3 = 120? leads to double, the external angle of the rectangular wedge 3p/2 to threefold coverage, everyone with a p-commensurable angle to infinitely often coverage of the space. Special interest attaches to the outside space of the half-plane, so to say, a wedge with an opening angle 2p; its treatment by the reflection method demands a study of the principal solution in a Riemann double space, the branch line of which is the edge of the half-plane .
The author gave this solution already in 1894 (Math. Ann., Vol. 45) just for heat conduction and soon afterwards transferred it to light diffraction. (Frank-Mises, 2. ed.,(8. Ed. Riemann-Weber)Vieweg, 1935, Chapter 20.)
Among the poly-hedral, one has first the cube (its internal space; its external space leads to highly complicated branching!) and as its generalization the parallelepiped. The reflections of the initially prescribed source point arrange themselves into 8, inside each other, rectangular space lattices, corresponding to the 8 sign combinations of ?x, ?h, ?z. Each of these lattices constructs by itself a threefold periodic solution of the differential equation, a higher order -function, as one might say. When you subdivide the base of a quadratic parallelepiped from the centre into four isosceles triangles, then the cylindrical column, built on top of one of them, is also a polyhedron of the type under consideration. Further examples arise from the column, erected over an equilateral triangle or one obtained by halving through the triangle's height (the regular column with 6 corners leads during reflection, due to the edge angle 2p/3, not to a simple, but double coverage of space).
What we have said here about the subdivision of the parallelepiped applies, of course, also to the cube. However, beyond this, the cube yields in the case of suitable subdivisions into certain tetra-hedral, which are accessible to reflection, the tetra-hedral 1/6 and 1/24 of Gabriel Llama (1795 - 1870), the first of which just fills the cube after 6-fold, the second after 24-fold reflection, as well as a tetrahedron, discovered by Sch?flies during his general examination of crystal structures.
G.Llama, Lemons usr la theory de la choler, Paris 1861; however, he does not employ the reflection method, but Fourier's method by continuing suitably the arbitrary initial distribution of the starting region. Regarding Sch?flies' tetrahedron, cf. Math. Ann. Vol. 34.
All these topics have in common that for them not only the problem of heat conduction, but also the problems of arbitrary other physical processes with isotropic symmetry, for example, acoustic, optical, electrical problems can be solved by the reflection method. In particular, the term reflection points already to optical problems.
The manifold nature of this topic grows substantially when we do not pose, as hitherto, boundary value problems, but, as in the case of the ring in 3.15, the much simpler problems of periodicity. We can then deal instead of with the parallelepiped with its throughout right edge angles with an arbitrary parallelepiped; for this purpose, we need only repeat periodically the pole of the Green function, prescribed in the initial region, in all regions, arising through the translation group. The place of the elliptic -function of the ring is then taken by higher -functions (Abel's hyper- elliptic functions); however, we will not deal with this here, because it has no immediate physical application.
What has been said here about spatial problems, is without difficulty transferred to plane regions, i.e., to the cases when the state only depends on the two co-ordinates x and y. Then, the place of the half-space is taken by the half-plane, the place of the parallelepiped by the rectangle, the place of a column erected on an isosceles triangle the triangle itself. In Equation (1) for the Green function of the half-space, one need only replace the 3/2-Th power on the left hand side by the first power and the three-dimensional distance square of the right hand side by the corresponding distance r? of the pole and initial point in the plane.
Unfortunately, it is not possible to transfer the reflection method for heat conduction problems to spherically bounded (in two-dimensional terms, circular) regions(cf.(4.23)).
3.18 Uniqueness of the solution in the case of an arbitrarily formed heat conductor: This work might appear to the physicist to be superfluous; nevertheless, we shall present it briefly due to its methodical importance and mathematical elegance.
For this work, Green's theorem of potential theory is sufficient, as we have written it out in the formulation of Exercise 2.2 as its second form. The parabolic character of the heat conduction equation does then not have a special role; it would only start to enter when we, as in Fig. 13, have given boundary conditions, which vary in time, while we will here in essence limit ourselves to the boundary conditions a), b), c).
Our heat conductor has now an arbitrarily formed outer boundary in which we include also the surfaces of, say, internal spaces. Let there be prescribed at this total boundary s an arbitrary distribution of the boundary conditions
(non-homogeneous boundary conditions in contrast to the earlier homogeneous ones with the right hand side 0; f1, f2, f3 are arbitrary functions of the location on s). Moreover, let the initial temperature u be given as an arbitrary function of the location f(x, y, z).
Now, let u1 and u2 be two different solutions of the heat conduction equation for these boundary and initial conditions. Their difference u1 - u2 = w then satisfies just as do u1 , u2 the differential equation
and the homogeneous boundary conditions, distributed over s,
as well as the initial condition
We now set in Green's theorem of Exercise 2.2 u as well as v equal to our w and obtain
Due to 1) and (2), this becomes
where DWG is the so-called first differential parameter
The last term of (5), as is indicated by the subscript c of Ds , is only extended over that part of the surface s where the boundary condition c) was to apply.
Equation (5) displays already an internal contradiction: The right hand side is negative when h > 0, as was especially stated in connection with (13.5). (We did not have to assume that h is a constant; it can vary on the surface sc. depending on its local state.) In contrast, the left hand side of (5) is certainly positive for small t, because, by (3), u? is zero for t = 0 and can from there on only increase with growing t. In order to emphasise the contradiction more and to extend it to arbitrary values of t, one can integrate (5) with respect to t:
The only possibility for removal of this contradiction is to conclude that
This uniqueness result can also be expressed as follows: In the field of heat conduction, there do not exist Eigen-functions for any shape of the heat conductor (cf. Chapter V). In this respect, heat conduction and the different, analogous to it, equilibrating processes differ characteristically from vibration processes.