On a system without critical points arising in heat conductivity theory
INNA SAMUILIK1, FELIX SADYRBAEV1,2
1 Department of Natural Sciences and Mathematics
Daugavpils University
Parades street1
LATVIA
2 Institute of Mathematics and Computer science
University of Latvia
Rainis boulevard 29
LATVIA
Abstract: A two-point boundary value problem for the second order nonlinear ordinary
differential equation, arising in the heat conductivity theory, is considered. Multiplicity
and existence results are established for this problem, where the equation contains two
parameters.
Keywords: heat conductivity, nullclines, phase portrait, Cauchy problems, bifurcation
curves
Received: August 12, 2021. Revised: April 16, 2022. Accepted: May 17, 2022. Published: July 1, 2022.
1 Introduction
Mathematical modeling of biomass thermal
conversion in reactors of regular shape can
help in solving the problem of reduction
greenhouse emissions and have a positive
effect on climate change. The respective
mathematical models of heat transfer in
the presence of nonlinear heat sources can
lead to nonlinear boundary value problems
(BVP) for ordinary differential equations.
The problem is to detect the number of
positive solutions to BVP and trace the
change of this number under the influence
of built-in parameters. For more details the
interested reader can consult the papers
[11], [5], [4]. In this paper we consider one
of these problems.
Consider the equation
T00 +aT 0+F eT= 0,0=d
dt,(1)
together with the boundary conditions
T(−1) = 0, T (1) = 0.(2)
The parameters aand Fare real numbers, a
can be of arbitrary sign, Fis positive. The
problem is to study the existence and the
number of positive solutions. Each positive
solution of the above problem corresponds
to a temperature regime in a domain of par-
allelepipedal form. A solution T(t) (Tis the
temperature, tis treated here as a spatial
variable) is supposed to be C2-smooth func-
tion. For general reading about problems,
arising in heat conductivity theory, one may
consult [1], [2], [3], [5], [16].
2 Reduction to a system
Replace Tby x,x0by yand rewrite the
equation in the form
½x0=y,
y0=−ay −F ex.(3)
We are interested in positive solutions in the
interval I= (−1,1), vanishing at the end-
points of the interval I. Therefore, we have
three equivalent boundary value problems
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(BVP), respectively, the problem (1), (2),
the problem
x00 +ax0+F ex= 0, x =x(t),(4)
x(−1) = 0, x(1) = 0, x(t)>0,∀t∈(−1,1),
(5)
and the problem (3), (5).
We will refer to one of these problems
when convenient.
2.1 Tools
In this section we consider several tech-
niques (adapted for our case), used for
investigation of autonomous ordinary
differential equations and systems in the
plane.
2.1.1 Nullclines and the phase plane
In order to analyze the phase plane of sys-
tem (3) consider first the nullclines N1=
{(x, y) : y= 0}and N2={(x, y) : −ay −
F ex= 0}.The nullclines have not cross-
points, so the system has no critical points.
-1.0
-0.5
0.5
1.5
t
-1.5
-1.0
-0.5
0.5
1.0
1.5
2.0
x
Figure 1: Solutions of (4) for x(0) = −1,2.5≤
x0(0) ≤6, a = 1, F = 1.
Figure 2: Trajectories of (3) for x(0) = −1,
2.5≤y(0) = x0(0) ≤6, a = 1.5, F = 1.Red
- nullcline N2,green - two trajectories passing
through the points (−1,2.9) and (−1,5.0) re-
spectively, correspond to solutions of the BVP
(3), (5)
Consider the Cauchy problem
x00 +ax0+F ex= 0, a, F > 0,
x(−1) = 0, x0(−1) = p > 0.(6)
Proposition 2.1. Any solution of the
Cauchy problem (6) has the first zero t1and
a unique maximum in the interval (−1, t1).
BConsider the vector field {y, −ay −
F ex},directed as shown in Fig. 2 for
any positive aand F. The vector field is
directed vertically downward only on the
x0=y= 0 axis. Since there are no critical
points, any trajectory, starting at x= 0
with y > 0,can be continued to x= 0 with
y < 0.C
Figures (1), (2) and (3) provide the first
impressions on the behavior of solutions of
the Cauchy problems (6). These figures
shows that t1(p) (the first zero) monoton-
ically increases together with p, and, after
reaching some critical value p∗, monotoni-
cally decreases. In Fig. 1 the value of t1(p∗)
is less than 1. This is indication that there
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-1.0
-0.5
0.5
1.5
t
-0.5
0.5
1.0
1.5
2.0
x
Figure 3: Trajectories of (3) for x(0) = −1,
2.5≤y(0) = x0(0) ≤6, a = 1.5, F = 1.Two
solutions of the BVP correspond to x0(0) = 1.9,
x0(0) = 5.0.
are no positive solutions of the BVP (4,)
(5) for a= 1, F = 1.On the other hand,
Fig. 3 shows that t1(p∗) is greater than 1.
Therefore t1(p) passes the value t= 1 twice,
giving rise to two solutions of the BVP (4,)
(5) for a= 1.5, F = 1.The two trajectories,
corresponding to solutions of the BVP, are
depicted in Fig. 2 (in green).
2.1.2 Energy dissipation
Conservative (Newtonian) systems are
known to save energy along trajectories.
This is not the case for system (3).
Proposition 2.2. Any solution of the
Cauchy problem (6) has the first zero t1and
the initial value pis greater than |x0(t1(p))|
by 2aRt1
−1x02(s)ds.
BOne has for equation (4)
x00 +ax0+F ex= 0,
2x0x00 + 2ax02+ 2F exx0= 0,
d(x02+ 2F ex) = −2ax02dt,
x02(t)+2F ex(t)=x02(−1) + 2F ex(−1)
−2aRt
−1x02(s)ds,
x02(t1)+2F ex(t1)=x02(−1) + 2F ex(−1)
−2aRt1
−1x02(s)ds,
x02(t1)+2F=x02(−1) + 2F
−2aRt1
−1x02(s)ds,
x02(t1)−x02(−1) = −2aRt1
−1x02(s)ds < 0,
x02(t1)−p2=−2aRt1
−1x02(s)ds < 0.
C
2.1.3 Continuity of t1
Consider the Cauchy problem (6). Then the
first zero t1(p) exists. The value t1depends
on three parameters, respectively a, F and
p. So we may write t1(a, F, p.) This function
is continuous on an open set of parameters,
say, for a > 0, F > 0, p > 0.One has that
t1(1,1,3) <1 (by calculation). On the other
hand, t1(0.1,0.1,4) >1 (by calculation).
By continuity of t1,there exists at least one
point (a∗, F∗, p∗) on any continues 3D-curve,
lying in {(a, F, p) : a > 0, F > 0, p > 0}
and connecting the points (0.1,0.1,4) and
(1,1,3).
Generally, solutions x(t;a, F, p) of the
Cauchy problem (6) continuously depend
on the parameters (a, F, p).Consider inter-
section of a ball Bε={a2+F2+p2=ε2}
with R3
+:= {(a, F, p) : a > 0, F > 0, p > 0}.
For ε2tending to zero, solutions
x(t;a, F, p) tend to a solution of
T00 = 0, T (−1) = 0, T 0(−1) = 0,that is,
to the trivial solution. Similarly, solutions
x(t;an, Fn,1) tend to a solution of the
problem T00 = 0, T (−1) = 0, T 0(−1) = 1,
as anand Fntend to zero.
The following therefore is true.
Proposition 2.3. Solutions x(t;a, F, 1) of
the Cauchy problems
T00 +aT 0+F eT= 0,
T(−1) = 0, T 0(−1) = 1,(7)
where a2+F2=ε2,have not zeros in the
interval (−1,1] for ε2sufficiently small.
2.1.4 Considering partial cases:F= 1
Numerical experiments show that: a) let
abe fixed and less than approximately
a0= 1.3 (this constant can be made more
precise). Let t1(p) be the first zero of the
Cauchy problem (6). The values t1(p) are
monotonically increasing together with p
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until p=p0(a) (can be calculated). The
values t1(p) are monotonically decreasing
for p∈(p0,+∞).The value p0(a) reaches
t= 1 for a0.The following assertion is
-1.0
-0.5
0.5
1.0
1.5
t
-0.5
0.5
1.0
x
Figure 4: Solutions of (6) for x(0) = −1,
0.25 ≤x0(0) ≤3.5, a = 1.28, F = 1.A unique
solution of the BVP exists.
confirmed by calculations using Wolfram
Mathematica.
Consider the BVP (1), (2). Let F=
1.Consider the Cauchy problem (6). Let
t1(a, p) be the first zero function.
Proposition 2.4. The function t1(a, p)for
fixed a<a0is monotonically increasing for
p∈(0, p0)and monotonically decreasing for
p∈(p0,+∞)with a maximum at p0.For
a < a0one has t1(a, p0)<1,for a>a0
t1(a, p0)>1.
Therefore no positive solutions of BVP
(1), (2) for a < a0(recall F= 1).
There exists exactly one positive solution
for a=a0.
There exist exactly two positive solutions
for a > a0.
2.1.5 Time intervals evaluation
Consider the equation
x00 +ax0+F ex= 0 (8)
together with the initial conditions
x(−1) = 0, x0(−1) = p. (9)
A solution of this problem has the first zero
t1(p).It has a unique point of maximum
tmax(p).A solution is monotonically increas-
ing in the interval [−1, tmax] and monotoni-
cally decreasing in [tmax, t1].In the intervals
of monotonicity the function x(t) has an in-
verse function t(x).Using the standard for-
mulas for inverse functions
t0
x(x) = 1/xt(t), t00
x2=−x00
t2/(x0
t)3(10)
the equation (8) can be rewritten as
−t00 +at02+F t03ex= 0, t =t(x),0=d
dx.
(11)
Replacing t0=u,
−u0+au2+Fu3ex=0, u =u(x).(12)
This is the first order (non-autonomous)
equation and it is easier to treat numeri-
cally.
Let us make comparison of two solutions,
x1(t) and x2(t),of the equation (8), satisfy-
ing the initial conditions
x1(−1) = 0, x0
1(−1) = p1(13)
x2(−1) = 0, x0
2(−1) = p2.(14)
Let p1= 1, p2= 3.Both solutions are de-
picted in Fig. 5.
Figure 5: Solutions of (13) and (14), a= 1.8,
F= 1.
Let t1max and t2max be points of maximum
for x1(t) and x2(t) respectively. For our
choice of p1and p2we have t1max < t2max.
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Let us consider the inverse functions t1(x)
and t2(x),defined respectively in the inter-
vals [0, x(t1max) and [0, x(t2max).The func-
tions t0
1(x) and t0
2(x) are solutions of the
Cauchy problems
u0=au2+F u3ex,
u=u(x),0=d
dx,u(0) = 1/p1= 1
(15)
and
v0=av2+F v3ex,
v=v(x),0=d
dx,v(0) = 1/p2= 1/3.
(16)
By comparison theorem for the first order
equations u(x) = t0
1(x)> v(x) = t0
2(x) on
the interval (0, x1max) and, due to t0
1(0) >
t0
2(0), t1(x)> t2(x) for x∈(0, x1max).The
graphs of both functions t1(x) and t2(x) are
depicted in Fig. 6.
Figure 6: Solutions t1(x)(blue) and t2(x) (red)
of (11), a= 1.8, F = 1, t1(0) = t2(0) = −1,
t0
1(0) = 1, t0
2(0) = 1/3, t0
1= +∞at x≈0.222,
t0
2= +∞at x≈0.904.
Similar treatment is possible for the equa-
tion (8) on the intervals [tmax, t1],where
both tmax and t1are dependent on x0(−1) =
p.
2.1.6 Polar coordinates
Sometimes passage to polar coordinates can
be useful. Introduce polar coordinates, us-
ing the formulas
x=ρsin ϕ, y =ρcos ϕ. (17)
The system
½x0=y,
y0=−ay −F ex(18)
takes the form
ρ0=ρsin ϕcos ϕ−aρ cos2ϕ−Fcos ϕeρsin ϕ,
ϕ0= cos2ϕ+asin ϕcos ϕ+1
ρsin ϕF eρsin ϕ.
(19)
Figure 7: Solutions ρ(t) (green) and ϕ(t) (red),
a= 1.8, F = 1.
Notice that polar function ρ(t) need not
to be monotone. That means that the value
x2(t)+ x02(t) can have maxima and minima.
3 Phase portrait analysis
The typical phase portrait for system (3) is
depicted in Fig. 8 below.
Any trajectory starts at (x, y) = (0, p).It
rotates clock-wise following the vector field
(y, −ax−F ex).The end point of any trajec-
tory corresponds to (x(1), y(1)).If the end
point is on the axis x= 0,then this trajec-
tory corresponds to a solution of the BVP
(3), (5). Then equation (4) is
x00 +F ex= 0, x =x(t).(20)
This equation together with the boundary
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Figure 8: Phase plane with the nullclines N1
(black) and N2(red)
Figure 9: Segments of trajectories parameter-
ized by t∈[−1,1], a = 0.7, F = 1.0.
conditions (5) was previously studied1. The
following result is true due to findings of the
above mentioned article.
Theorem 3.1. There exists F0≈0.878458
with the properties:
1) if 0< F < F0,then the problem (20),(5)
has exactly two positive solutions;
(2) if F=F0,then the problem (20),(5)
has exactly one positive symmetrical solu-
tion xF0with xmax =x(0) = 1.18684;
1S.-Y. Huang, S.-H. Wang, Proof of a Conjec-
ture for the One-Dimensional Perturbed Gelfand
Problem from Combustion Theory, Arch. Ration.
Mech. Anal., Vol. 222, No. 2, 2016, pp. 769825.
https://link. springer.com/article/10.1007/s00205-
016-1011-1
Figure 10: Phase plane with the nullclines N1
(black) and N2(red)
Figure 11: Segments of trajectories parame-
terized by t∈[−1,1], a = 1.8, F = 1.0.
(3) if F > F0,then the problem (20),(5) has
no positive solutions.
The exact bifurcation curve Fagainst
x(0) is available for positive solutions of
(20),(5).
3.1 a < 0
The independent variable change tto −t
turns equation x00 +ax0+F ex= 0 to the
equation X00 −aX0+F eX= 0,where the
coefficient at X0is positive. The bound-
ary conditions remain unchanged. There-
fore this case is reduced to the previously
studied one, where a > 0.
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Figure 12: Phase plane with the nullclines N1
(black) and N2(red)
Figure 13: Segments of trajectories parame-
terized by t∈[−1,1], a = 2.5, F = 1.0.
4 Bifurcation curves
An alternative method can be used to study
bifurcations in the problem (20),(5).
Consider the Cauchy problem
x00 +ax0+F ex= 0,
x(−1) = 0, x0(−1) = p > 0.(21)
Assume that a > 0 is given. The first zero
function t1(F, p) known also as the time-
map function could be studied. The equa-
tion
t1(F, p) = 1 (22)
Figure 14: Phase plane: a= 1.8, F = 1.0.
Figure 15: Phase plane: a=−1.8, F = 1.0.
defines the bifurcation curve that bears in-
formation on the number of positive solu-
tions and values of x0(−1) = pwhich pro-
duce positive solutions of (20),(5).
This bifurcation curve can be constructed
numerically for particular values of the
coefficient a. Below the bifurcation curves,
x0(−1) = pagainst Fare shown for a= 2
and a=−2.
Material in this section is based on the
private communication by A. Gritsans.
The technique of bifurcation curves for
detecting of solutions of boundary value
problems was used in [6], [7], [8], [10],
[11]. The number of solutions in related
problems were studied in [9], [12], [13], 14],
[15].
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Figure 16: a=2,two solutions for F= 0.6
Figure 17: a=−2
5Conclusions
Both cases a > 0 and a < 0 are symmetri-
cal. It is enough to consider the case a > 0.
The following observations were made for
the problem (1), (2) with a > 0:
•There are at most two positive solu-
tions of the problem;
•For some values of parameters aand F
the existence of a single positive solu-
tion and no positive solutions are pos-
sible;
•For a given pair of parameters aand F
a complete numerical analysis can be
made;
•Bifurcation curves were constructed if
one of the parameters aand Fis given;
•The approximate initial values T0(−1)
for the positive solutions of the problem
can be found by numerical inspection;
•Turning points on bifurcation curves,
corresponding to transition from no so-
lutions to two solutions can be found
numerically;
•Generic properties of the time-map
function t1(p, a, F ) can be described.
The analysis (both theoretical and numeri-
cal) of similar tasks for ordinary differential
equations can help to treat problems, aris-
ing in the theory of heat transfer and fluid
mechanics with applications to clean energy
production, biomass thermal conversion and
utilization of waste.
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