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Third Boundary-Value Problem of the Heat Conduction Equation for a System with Plane-Parallel Boundaries

We obtained a new representation of a solution of the heat conduction equation with boundary condition of the third kind for a layer. The result is presented as a superposition of fundamental solutions for an unbounded system with variable coefficients, the explicit form of which is given. We consider the well-known problem of the evolution of the temperature field initially uniformly distributed in a layer. The distribution of the temperature field is represented in terms of the obtained functions.


I. INTRODUCTION
Theoretical description of the distribution of heat in bounded solids is based on the solution of the heat conduction equation with the corresponding initial and boundary conditions. In the case where the heat transfer between a heated body occupying the region 0 ≤ z ≤ L, −∞ < x, y < ∞ and the external medium with the zero temperature proceeds by the Newton law, the problem is reduced to the solution the heat conduction equation [1][2][3] for t > 0 with the boundary conditions of the third kind ∂T ( r, t) ∂z + λ 2 T ( r, t) = 0 z=L (3) and the initial condition Here, △ = ∂ 2 /∂x 2 +∂ 2 /∂y 2 +∂ 2 /∂z 2 is the Laplace operator, r = (x, y, z), κ is the thermal diffusivity, λ 1 and λ 2 are the relative heat transfer coefficients between the body and the external medium at z = 0 and z = L, respectively, and f ( r, t) is the density of thermal sources normalized to cρ, where c and ρ are, respectively, the specific heat capacity and the density of the substance.
The methods for solving this boundary-value problem are well-known and widely described in the literature both for the theory of heat conduction [1][2][3] and for equations of mathematical physics [4][5][6][7]. It is worth noting that a solution of this problem is presented as a series in which the summation is carried out over an infinite set of eigenvalues α , where α are positive roots of the transcendental equation which, in the general case, is solved numerically. In particular cases where the temperature of the boundary surfaces is equal to zero ( λ = ∞ , the first boundary-value problem) or the heat flow through the boundaries is absent ( λ = 0, the second boundary-value problem), the roots of Eq. (5)  For arbitrary values of λ, Eq. (5) was numerically investigated in detail and the results are presented in the form of tables and plots (see, e.g., [3]).
As far as we know, for any λ > 0 , in the literature, there is no representation of a solution of problem (1)-(3) in terms of fundamental solutions for an unbounded system similar to those for the first and second boundary-value problems for which it is not necessary to find eigenvalues α of the transcendental equation (5). The possibility of this representation for arbitrary α is shown in the second section of the paper for Green's function. Unlike the known solutions of boundary-value problems of the first and second kinds, in this case, the fundamental solutions are in the sum with the corresponding weights that are functions of coordinates and time.
As an example, in the third section, we use the obtained Green function for studying the evolution of an initially uniform temperature field in a layer.

II. GREEN'S FUNCTION OF THE THIRD BOUNDARY-VALUE PROBLEM
We consider the homogeneous heat conduction equation (1) with boundary conditions (2)-(3) for λ 1 = λ 2 ≡ λ and the initial condition which corresponds to a point instantaneous heat source situated at the point r ′ = (x ′ , y ′ , z ′ ), where 0 ≤ z ′ ≤ L, and find a solution continuous in the region 0 ≤ z ≤ L, −∞ < x, y < ∞ for t > 0. Note that in the general case of different λ 1 and λ 2 , the expression for Green's function for the third boundary-value problem (1)-(3), (6) expressed in terms of fundamental solutions for an unbounded system was obtained in [8], where the Brownian motion of particles in a layer was investigated for various values of adsorption coefficients of particles by the boundary surfaces. We also noted there that the specific case λ 1 = λ 2 should be considered in its own right because in going from the general expressions presented in [8] to the case λ 1 = λ 2 , we have to sum expressions each term of which is divergent.
Then we use the integral representation change the order of integration, and take integrals over k z and ξ. We present the final result in the following form: where By the direct substitution of solution (9) for the function F k (x). To obtain relation (14), we use the known recurrence relationships for multiply probability integrals [9] and representation (12) for the function F k (x). As the subscript k of the function F k (x) increases, the function F k (x) decreases and, for all values of the argument x, remains either positive (for even k and k = 0 ) or negative (for odd k ). The behavior of the first six functions F k (x) is shown in Fig. 1 [ k = 0, 1, 2, 3, 4, 5, the number near a curve corresponds to the subscript k of the function F k (x) ].
and Eq. (9) is reduced to the well-known solution of the first boundary-value problem for Green's function in a layer [1,10] As λ → 0, [ λ(κt)/|Z ± | ≪ 1 ], in view of Eqs. (10) and (11), we have and whence, for λ = 0 P (±) n (Z ± (n), 0) = 1 , it follows the well-known solution of the second boundary-value problem [1,10] In the limiting case of an infinitely thick layer (L → ∞) , we have and expression (9) is reduced to the known solution of the third boundary-value problem

III. EVOLUTION OF A UNIFORM TEMPERATURE DISTRIBUTION
The obtained expression (9) for Green's function can be used for the solution of various boundary-value problems for a layer. As an example, we consider the simplest wellknown problem on the evolution of the temperature in an initially uniformly heated layer [T 0 ( r) = T 0 ] with boundary conditions (2) and (3) for λ 1 = λ 2 ≡ λ. The required distribution T ( r, t) is defined by the integral By virtue of Eqs. (9)-(13), relation (22) can be represented in the following form: where T (z, t, λ = ∞) is the known solution of the first boundary-value problem [1,3] T and the additional term δT (z, t, λ) caused by nonzero value of the quantity λ −1 has the Here, l = L/(4κt) 1/2 , ξ = z/(4κt) 1/2 , and R k,n = 1 It is worth to note that expressions (23)-(25) can be written at once by using Eq. (72) in [8] for the density of Brownian particles in a layer with absorbing boundaries and carrying out changes in the corresponding notation in it.

for various
Biot numbers Bi = Lλ/2. With regard for the symmetry of the system at hand about the z = L/2 plane, the distribution of temperature is shown only for the 0 ≤ z ≤ L/2. It is quite natural that the layer cools down with maximum rate in the case corresponding to the first-boundary problem ( Bi = ∞ ) when the boundaries of the layer are kept at zero temperature for the entire time (Fig. 2a). Note that a noninstanteneous drop in the temperature shown in Fig. 2a from the initial value T 0 to zero at the initial time ( t = 0 ) as the observation point approaches the boundary of the layer is not the result of calculations but appears due to the used uniformity of the space and time grid for the temperature field.
For finite great values of the Biot number (Fig. 2b), the distribution of temperature inside the layer, in fact, does not change, whereas it takes place the smooth but fast cooling of the boundary of the layer. For the time of order of 0.1τ, where τ = L 2 /4κ, the surface temperature is as small as 6% of its initial value. As the Biot number decreases ( Bi = 10, Since new representation (9) enables one to carry out analytic investigations not using graphic and table data, it can be useful for various physical problems for which, in parallel with numerical calculations, analytic estimations of solutions must be performed. In particular, we can find the first additional terms caused by the account of the heat transfer of the layer with the environment to the known results determined under the assumption of a given temperature of the surface or the absence of the thermal flow through it. In the paper, we illustrate the possibility of finding correction terms in powers of λ −1 , where λ > 1, for a special problem of the evolution of the initially uniform temperature field in the layer.
In addition to analytic investigations, Green's function (9) can be also useful for the numerical analysis of various boundary-value problems of the third order. Using recurrence relations (14), we significantly simplify this analysis by reducing it to the calculation of probability integrals and the majorant estimation of the remainder terms of the series given in the Appendix.

APPENDIX:
We prove that solution (9) satisfies the homogeneous heat conduction equation (1) with boundary conditions (2) and (3). For this purpose, first, we prove the uniform convergence of the series in solution (9) at 0 ≤ z ≤ L and t > 0. In view of the following representation of the quantity β k (|Z ± (n)|, λ) : which is valid for |n| > (κt) 1/2 /L, we obtain that for finite values of λ, for all |n| ≥ N, where N ≫ λκt/L, the coefficients P (±) n (Z ± (n), λ) ≡ P (±) n (λ) can be represented as follows: As |n| → ∞, the following estimate for the coefficients holds: Thus, starting from |n| ≥ N, the series ∞ |n|=N P (± n ) exp −Z 2 ± (n)/4κt in solution (9) can be majorized by the uniformly convergent series ∞ |n|=N exp −Z 2 ± (n)/4κt . If follows from here that series in solution (9) are also uniformly convergent. Therefore, we can integrate solution (9) term by term and use the majorant estimate for the remainder of the series which determines Green's function (9).