Comparison of Solutions of the Nonlinear Transfer Equation
The paper considers the evolution of nonlinear equation describing the process of energy transfer by radiation. The comparison theorem for the solutions of this equation is formulated and proved.
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irk-123456789-1405802018-07-11T01:23:35Z Comparison of Solutions of the Nonlinear Transfer Equation Kholkin, A.M. Sanikidze T.A. The paper considers the evolution of nonlinear equation describing the process of energy transfer by radiation. The comparison theorem for the solutions of this equation is formulated and proved. 2017 Article Comparison of Solutions of the Nonlinear Transfer Equation / A.M. Kholkin, T.A. Sanikidze // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 344-352. — Бібліогр.: 16 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag13.04.344 Mathematics Subject Classification 2000: 35K55, 35K65 http://dspace.nbuv.gov.ua/handle/123456789/140580 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The paper considers the evolution of nonlinear equation describing the process of energy transfer by radiation. The comparison theorem for the solutions of this equation is formulated and proved. |
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Article |
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Kholkin, A.M. Sanikidze T.A. |
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Kholkin, A.M. Sanikidze T.A. Comparison of Solutions of the Nonlinear Transfer Equation Журнал математической физики, анализа, геометрии |
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Kholkin, A.M. Sanikidze T.A. |
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Kholkin, A.M. |
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Comparison of Solutions of the Nonlinear Transfer Equation |
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Comparison of Solutions of the Nonlinear Transfer Equation |
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Comparison of Solutions of the Nonlinear Transfer Equation |
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Comparison of Solutions of the Nonlinear Transfer Equation |
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Comparison of Solutions of the Nonlinear Transfer Equation |
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comparison of solutions of the nonlinear transfer equation |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Comparison of Solutions of the Nonlinear Transfer Equation / A.M. Kholkin, T.A. Sanikidze // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 344-352. — Бібліогр.: 16 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT kholkinam comparisonofsolutionsofthenonlineartransferequation AT sanikidzeta comparisonofsolutionsofthenonlineartransferequation |
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2025-07-10T10:47:32Z |
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2025-07-10T10:47:32Z |
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1837256628020183040 |
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Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No. 4, pp. 344–352
doi:10.15407/mag13.04.344
Comparison of Solutions of the Nonlinear
Transfer Equation
A.M. Kholkin and T.A. Sanikidze
Pryazovskyi State Technical University
7 Universitets’ka St., Mariupol 87500, Ukraine
E-mail: a.kholkin@gmail.com, sanukidze tariel@gmail.com
Received October 27, 2016, revised November 28, 2016
The paper considers the evolution of nonlinear equation describing the
process of energy transfer by radiation. The comparison theorem for the
solutions of this equation is formulated and proved.
Key words: nonlinear transfer equation, supersolutions, subsolutions, the
comparison theorem.
Mathematical Subject Classification 2010: 35K55, 35K65.
1. Introduction
Comparison of solutions of the Cauchy problem on the initial data is an
effective means of qualitative analysis for quasi-linear differential equations of
parabolic type [6–9, 15]. In particular, the equation describing the interaction of
self-radiation with substance and radiation energy transfer process [16] is of this
type. Currently there are not much works found on the subject. The difficul-
ties in studying nonlinear equations describing these processes appear because of
the complexity of nonlinear integral-differential equations. If the radiation path
length is comparable with the size of the heated region of space, then the non-
local nature of the interaction of radiation with substance should be taken into
account. We consider a one-dimensional heat transfer process in the substance
in the quasi-stationary approximation [16]. In this case, when the presence of
heat sources and heat sinks in dimensionless variables is supposed, the process is
described by the energy transfer equation.
2. Preliminary Notes
Let us consider the energy transfer equation (see [16])
c© A.M. Kholkin and T.A. Sanikidze, 2017
Comparison of Solutions of the Nonlinear Transfer Equation
∂E
∂t
= −∂S
∂x
+ f. (2.1)
Here t ∈ R+ is a time; x ∈ R is a straight line along which the heat is
transferred; E (T ) is a specific internal energy of the substance, which is a con-
tinuous monotonically increasing function of the temperature substance, E (0) =
0; T (x, t) > 0 is a temperature of the substance; f (T ) ∈ C (R+) is a source
(sink) function; S (x, t) is a flux density of radiant energy, which is determined
for the substance in the form of nonlinear integral operator S (x, t) = S (T ) [16],
S =
∫ +∞
−∞
T 4 (ξ, t)K (T (ξ, t)) sgnP (x, ξ)W2 (|P (x, ξ)|) dξ, (2.2)
where
P (x, ξ) ≡ P (T ) =
∫ ξ
x
K (T (ζ, t)) dζ, (x, t) ∈ Ω = R× R+;
K (T ) ∈ C (R+) is the coefficient of absorbtion of radiation by the substance,
K (T ) > 0, T ≥ 0; Wi (z) =
∫ +∞
1 exp (−zµ)µ−idµ, i = 0, 1, 2, . . . is the integral
exponential function [4].
In physical meaning, S (x, t) is a continuous function of its arguments. By
(2.2), the continuity of S follows from the function T (x, t) ( T (x, t) ∈ C (Ω)).
In addition, the temperature rise T (x, t) is limited as |x | → +∞ such that
the improper integral in (2.2) is convergent (for example, T (x, t) < A = const,
(x, t) ∈ Ω).
3. Formulation of the Problem
From the physical considerations about the smoothness of the functions
T (x, t), S (x, t), it follows that the derivatives in equation (2.1) generally do not
exist in the classical sense. Therefore, for further analysis it is more convenient
to consider the relation
L (T ) ≡
∮
Γ
E (T ) dx−
∮
Γ
S (T ) dt+
∫∫
ω
f (T ) dxdt = 0 (3.1)
instead of (2.1). It should be noticed that (3.1) does not contain derivatives [10–
14]. Here Γ ⊂ Ω is the piecewise smooth contour bounding an arbitrary simply
connected domain ω ⊂ Ω.
Bypassing the contour Γ is assumed to be single. However, the region remains
on the left. Relation (3.1) can be obtained by applying Green’s formula to (2.1).
It can also be obtained independently from (2.1) as a consequence of the law of
conservation of energy.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 345
A.M. Kholkin and T.A. Sanikidze
Let us consider the problem of the evolution of the initial heat pulse:
T (x, 0) = T0 (x) ≥ 0, T0 (x) ∈ C (R) , (3.2)∫
R
E (T0 (x)) dx = Q0 < +∞.
We assume that there exists a function T (x, t) ∈ C (Ω) satisfying conditions
(3.2) and equation (3.1), and that the heat stream S (x, t) ≡ S (T ) vanishes at
|x | → +∞,
S (x, t)→ 0, |x | → +∞. (3.3)
Let Γ be a contour projecting to the axis t and infinite with respect to x.
Then from (2.1) it can be obtained that
Q (t) ≡
∫
R
E (T (x, t)) dx = Q0 +
∫ t
0
∫
R
f (T (x, t)) dxdt. (3.4)
Hence, physically natural existence of the energy integral Q (t) < +∞ is
equivalent to the existence of the integral on the right-hand side of (3.4). We
define the function T (x, t) satisfying (3.1)–(3.4) as a generalized solution to the
Cauchy problem [3].
One of the main moments of the theory developed in this work is the possibil-
ity of estimating the integrals similar within the definition of the heat flux density
(2.2). To do this, we assume that the function K (z) satisfies simultaneously the
inequalities:
K (z1) ≤ K (z2) , K (z1) z4
1 ≥ K (z2) z4
2 , (3.5)
z1 ≥ z2, zi ∈ R+, i = 1, 2.
The first inequality in (3.5) means that the radiation path length is not de-
creasing with temperature increasing of the substance, and the second inequality
means the obvious physical requirement of impossibility of reducing the radiant
exitance of the substance when its temperature increases.
4. Super- and Subsolutions
Along with the solution T (x, t), let us consider the functions
θ (x, t) ≥ 0, θ (x, t) ∈ C (Ω) , L (θ) ≤ 0
which we call a supersolution of the Cauchy problem of equation (3.1) (see [6]),
and
τ (x, t) ≥ 0, τ (x, t) ∈ C (Ω) , L (τ) ≥ 0
346 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Comparison of Solutions of the Nonlinear Transfer Equation
which we call a subsolution of equation (3.1).
In this case, the introduction of super- and subsolutions is of essential impor-
tance since the class of functions K (T ), for which the simplest invariant solutions
of equations can be constructed (2.1), is significantly poorer in functions than the
class for the equation of radiative thermal conductivity [1].
The possibility of application of super- and subsolutions for qualitative esti-
mate of the Cauchy problem (3.1)–(3.4) is determined by the lemma below.
Lemma 1. Let the source function f (z) satisfy at least one of the conditions:
f (z 1) ≥ f (z2) , z1 ≤ z2, zi ∈ R+, i = 1, 2, (4.1)
i.e., it is nonincreasing with respect to the temperature, f (z) ∈ C (R+), f (0) =
0, or there exists such a > 0 that
Φ (z) ≤ 0, z ∈ [0; a] , (4.2)
where Φ (z) = f (z) + δ (z) for the supersolution, and Φ (z) = f (z)− γ (z) for the
subsolution, and δ (z) and γ (z) , are defined by the inequalities
L (θ) ≤
∫∫
ω
δ (θ) dxdt; δ (θ) < 0, θ > 0; δ (0) = 0; δ (θ) ∈ C (R+) , (4.3)
L (τ) ≥
∫∫
ω
γ (τ) dxdt; γ (τ) > 0, τ > 0; γ (0) = 0; γ (τ) ∈ C (R+) . (4.4)
Thus, if inequalities (3.5) are fulfilled and
τ (x, 0) ≤ T0 (x) ≤ θ (x, 0) , x ∈ R, (4.5)
then
τ (x, t) ≤ T (x, t) ≤ θ (x, t) , (x, t) ∈ Ω. (4.6)
Proof. We prove the lemma for the supersolution θ (x, t). For the subsolution
the proof is similar.
Following [2, 10–12] the lemma will be proved by contradiction. Suppose that
in the domain Ω there is Ω1 ⊂ Ω such that
T (x, t) > θ (x, t) , (x, t) ∈ Ω1\∂Ω1 and (x, t) = θ (x, t) , (x, t) ∈ ∂Ω1 = l.
As the area ω from (3.1), we choose the area ω1 bounded by the contour Γ1 com-
posed of the arc of the curve l and the segment of the straight line m projecting
to the time axis t (see Fig. 1).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 347
A.M. Kholkin and T.A. Sanikidze
Fig. 1: Contour Γ1.
Then, using (3.1) and the definition of supersolution, we can get the residual
L (θ)− L (T ) ≡
∮
Γ1
(E (θ)− E (T )) dx−
∮
Γ1
(S (θ)− S (T )) dx
+
∫∫
ω1
(f (θ)− f (T )) dxdt ≤ 0. (4.7)
Let us estimate each integral from (4.7). It is easy to estimate the first integral∮
Γ1
(E (θ)− E (T )) dx =
∮
m
(E (θ)− E (T )) dx > 0, (4.8)
because the direction of integration coincides with the direction of the contour
Γ1 bypass. The second integral from (4.7) can be reduced to the form∮
Γ1
(S (θ)− S (T )) dt =
∫ t1
t0
∫ 1
0
((q1 − 1) (q11 + q12) + ∆q1) dµdt
−
∫ t1
t0
∫ 1
0
((q2 − 1) (q21 + q22) + ∆q2) dµdt,
where
qi (µ, t) = exp
(
− 1
µ
|pi (x1 (t) , x2 (t))|
)
,
qi1 (µ, t) =
∫ x1(t)
−∞
T 4
i (ξ, t)K (Ti (ξ, t)) exp
(
− 1
µ
|pi (x1 (t) , ξ)|
)
dξ,
qi2 (µ, t) =
∫ +∞
x2(t)
T 4
i (ξ, t)K (Ti (ξ, t)) exp
(
− 1
µ
|pi (x2 (t) , ξ)|
)
dξ,
∆qi (µ, t) =
∫ x2(t)
x1(t)
T 4
i (ξ, t)K (Ti (ξ, t))
(
exp
(
− 1
µ
|pi (x2 (t) , ξ)|
)
348 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Comparison of Solutions of the Nonlinear Transfer Equation
+ exp
(
− 1
µ
|pi (x1 (t) , ξ)|
))
dξ,
T1 (x, t) = θ (x, t) , T2 (x, t) = T (x, t) , pi = p (Ti) , i = 1, 2.
By (3.5), the integrands in (2.2), taken in absolute value, are nondecreasing with
respect to temperature. Then, using the definitions, we can obtain the following
inequalities:
0 < q1 (µ, t) ≤ q2 (µ, t) < 1, q11 (µ, t) ≥ q21 (µ, t) ,
q12 (µ, t) ≥ q22 (µ, t) , ∆q1 (µ, t) > ∆q2 (µ, t) .
Hence, we have the estimate∮
Γ1
(S (θ)− S (T )) dt < 0. (4.9)
The estimate of the last integral from (4.7) is trivial if the following inequality
holds: ∫∫
ω
(f (θ)− f (T )) dxdt ≥ 0, (4.10)
since f (θ) ≥ f (T ), θ < T , (x, t) ∈ ω1.
If (4.2), (4.3) are satisfied instead of (4.1), then we can define the domain
Ω2 = {(x, t) | F (θ, T ) = f (θ)− f (T )− δ (θ) ≥ 0} .
Due to the conditions (4.2), (4.3), the continuity of function F (θ, T ) ∈ C (Ω),
and inequality F (θ, T ) ≥ 0, (x, t) ∈ l, the strip Ω3 = Ω2 ∩ Ω1 closes nowhere
and as a boundary it has the line l = ∂Ω1. Assuming ω1 ⊂ Ω3, we arrive at the
inequality ∫∫
ω1
(Φ (θ)− f (T )) dxdt ≥ 0. (4.11)
The estimates of (4.8)–(4.11) show that inequality (4.7) does not hold for an
arbitrary domain, which proves the lemma.
Remark 1. The lemma remains valid if we set Φ (z) = f (z) for the super-
and subsolutions in (4.2) and consider δ = δ (x, t), γ = γ (x, t) in (4.3) and (4.4).
The resulting lemma can be applied not only directly. It can also be used as
a basis for the proofs of various comparison theorems for the Cauchy problem
(3.1)–(3.4). In particular, we can extend the class of source functions if we do
not take into account conditions (4.1), (4.2).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 349
A.M. Kholkin and T.A. Sanikidze
5. The Main Result
Let us consider two solutions of the Cauchy problem (3.1)–(3.4), Ti (x, t) ∈
C (Ω) ∩ L1 (Ω), determined by the initial conditions Ti (x, 0) = Toi (x), i = 1, 2.
The following theorem holds.
Theorem 1. Let E = T , and the source function f (z) ∈ C (R+) satisfy the
Lipschitz condition
|f (z1)− f (z2)| ≤M |z1 − z2| , zi > 0, i = 1, 2, M = const > 0. (5.1)
Suppose also that there exist positive constants β > 1, a, M1 such that
f (z) ≤M1z
β, z ∈ [0; a] . (5.2)
Thus, if T01 (x) ≥ T02 (x), x ∈ R, then
T1 (x, t) ≤ T2 (x, t) , (x, t) ∈ Ω. (5.3)
Proof. Consider the auxiliary Cauchy problem, which is obtained by substi-
tuting f (T ) in (3.1) by fci = f (Tci) − CTci, i = 1, 2, C = const > 0. From
condition (19) and the lemma, we get the inequalities:
Ti (x, t) ≥ Tci (x, t) , Tc1 (x, t) ≥ Tc2 (x, t) , (x, t) ∈ Ω, i = 1, 2. (5.4)
By taking Γ as a contour, normal over t and infinite over x and reducing the
double integrals to the repeated integral, from (3.1), one can obtain∫
R
(Ti − Tci) dx =
∫ t
0
∫
R
(f (Ti)− f (Tci)) dxdt+ C
∫ t
0
∫
R
Tci dxdt.
Applying formulas (5.1), (5.4) to the equality, we obtain the inequality∫
R
(Ti − Tci) dx ≤M
∫ t
0
∫
R
(Ti − Tci) dxdt+ CNit,
where Ni = sup
t
∫
R Tidx. Hence, by the Gronwall inequality [5] and formulas
(5.4), we obtain
0 ≤
∫
R
(Ti − Tci) dx ≤ CNit exp (Mt) . (5.5)
Passing in (5.5) to the limit at C → 0, by (5.4), we obtain the desired inequality
(5.3) for the continuous functions Ti (x, t) ∈ C (Ω), i = 1, 2. This proves the
theorem.
350 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Comparison of Solutions of the Nonlinear Transfer Equation
Under the conditions of Theorem 1, the uniqueness of the solution of the
Cauchy problem (3.1)–(3.4) in the class of continuous functions follows immedi-
ately.
Theorem 2. Let the conditions of Theorem 1 be satisfied. Then the solution
of the Cauchy problem (3.1)–(3.4) is unique.
If there are two solutions of the Cauchy problem (3.1)–(3.4), then by (5.3)
these solutions coincide.
6. Comment
All conclusions of the theory developed here remain true if in the definition
of the radiant flux (2.2) and conditions (3.5) if the function T 4 is substituted by
an arbitrary monotonically increasing function ϕ (T ) .
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