DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
SUPPLEMENT 2007
Website: www.AIMSciences.org
pp. 373–381
EXISTENCE OF POSITIVE SOLUTIONS FOR SECOND ORDER
DIFFERENTIAL EQUATIONS ARISING FROM CHEMICAL
REACTOR THEORY
Wenying Feng
Computer Science and Mathematics, Trent University
Peterborough, ON Canada K9J 7B8
Guang Zhang
School of Science, Tianjin University of Commerce
Tianjin 300134, P. R. China
Yikang Chai
Department of Mathematics, Qingdao Technological University
No. 11 Fushun Road, Qingdao 266033, P. R. China
Dedicated to Prof. J.R.L. Webb on the occasion of his 60th birthday
Abstract. In this paper, we study second-order differential equations that
represent the steady state model in an adiabatic tubular chemical reactor.
Theoretical results on existence and range of positive solutions are proved by
applying a fixed point theorem. At the mean time, numerical solutions are
obtained by computer programming. Results from mathematical analysis are
compared with the numerical solutions.
1. Introduction. As mathematical models for an adiabatic tubular chemical reactor, the following boundary value problem of second order differential equations
has been extensively studied (see [2]-[7]):
x00 − λx0 + λµ (β − x) ex = 0, t ∈ (0, 1) ,
0
0
x (0) = λx (0) , x (1) = 0.
(1)
(2)
BVP (1)-(2) describes the steady state of the reactor that processes an irreversible exothermic chemical reaction (see [5], [6]). The three parameters, λ > 0,
µ > 0 and β ≥ 0, represent the Peclet number, the Damkohler number and the
dimensionless adiabatic temperature rise, respectively. A positive solution x to the
system represents the steady state temperature of the reaction.
Most of the results on problem (1)-(2) are obtained by numerical methods ([4], [5],
[7]). For example, Madbouly, McGhee and Roach applied the Adomian’s method for
Hammerstein integral equations and the shooting method [5]. In a recent paper [7],
Saadatmandi, Razzaghi and Dehghan proposed the application of the Sinc-Galerkin
2000 Mathematics Subject Classification. Primary: 34B15; Secondary 34B18.
Key words and phrases. Steady state model, positive solution, fixed point theorem, numerical
simulation.
The first author is supported by a grant from NSERC (Natural Sciences and Engineering
Research Council of Canada).
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374
WENYING FENG, GUANG ZHANG AND YIKANG CHAI
method for solving (1)-(2). Existence of multiple solutions for parameters in some
ranges are also discussed (see [2], [3]).
The following existence and uniqueness theorem is given in [5]:
Theorem 1. For any M > 0, the problem (1)-(2) has a unique solution in B (M ) =
{x ∈ C [0, 1] : ||x||∞ ≤ M } provided
¾
½
M
1
µ < min
,
.
(3)
kKk (β + M ) eM kKk (|β − 1| + M ) eM
Here, ||x||∞ = supt∈[0,1] |x(t)|. Theorem 1 is proved by the contraction mapping
principle [4]. It is noticed that
lim µ = 0.
M →∞
Therefore, Theorem 1 specifies that to have a solution in a bigger range (M is
bigger), the value of µ should be smaller. This is not clearly suitable. In this paper,
we apply the following well known fixed point theorem [1] to obtain results on the
existence of positive solutions for (1)-(2).
Lemma 1. [1] Let E be a Banach space. Assume Ω1 and Ω2 are two bounded
open¡ sets in
¢ E such that 0 ∈ Ω1 and Ω1 ⊂ Ω2 , and P is a cone in E. Let T :
P ∩ Ω2 \Ω1 → P is completely continuous and one of the two conditions is satisfied:
1. kT xk ≤ kxk for x ∈ P ∩ ∂Ω1 and kT xk ≥ kxk for x ∈ P ∩ ∂Ω2 ;
2. kT xk ≥ kxk for x ∈ P ∩ ∂Ω1 and kT xk ≤ kxk for x ∈ P ∩ ∂Ω2 .
¡
¢
Then T has at least one fixed point in P ∩ Ω2 \Ω1 .
The proofs of the theorems are based on the construction of various open sets.
Our results not only provide sufficient conditions on existence of a positive solution, but also specify the possible ranges, thus the upper and lower bounds for the
solutions. Using Matlab programming, we are able to obtain some numerical solutions with various input parameters. Comparisons among the theoretical results
and the numerical solutions provide more information on the positive solutions. For
example, it proves that the solution is not unique for some specified parameters.
In Section 2, we present the theorems on existence of a positive solution. Solutions from computer simulation and comparisons are given in Section 3.
2. Existence of positive solutions. The existence of solutions for (1)-(2) is
equivalent to the existence of solutions to the Hammerstein integral equation [5]
Z 1
x (t) = µ
G (t, s) (β − x (s)) ex(s) ds, 0 ≤ t ≤ 1,
(4)
0
where the Green’s function G (t, s) is defined by
½ λ(t−s)
e
, 0≤t≤s≤1
G (t, s) =
1,
0≤s≤t≤1
and satisfies the condition
e−λ ≤ G (t, s) ≤ 1.
When β = 0, equation (4) has a unique zero solution, see [3]. Thus, we will
assume that β > 0. Our first theorem gives a sufficient condition for a positive
solution that has uniform upper bound. We also prove that the maximum of the
solution is greater than a constant. In applications, this implies that the highest
temperature during the reaction is higher than the constant.
EXISTENCE OF POSITIVE SOLUTIONS FOR APPLIED ODES
375
Theorem 2. Let λ, µ and β be positive numbers with µeβ ≤ 1. Boundary value
problem (1)-(2) has at least one positive solution x such that
max x(t) >
t∈[0,1]
µβ
and x(t) ≤ β for t ∈ [0, 1].
µ + eλ
Proof: Since µeβ ≤ 1, we have µ < 1 < eλ and µβe−λ < β. Let
E = {u : u (t) ∈ C [0, 1]}
with the norm ||u||∞ = maxt∈[0,1] |u (t) | for any u ∈ E. Assume that
P = {u ∈ E : u (t) ≥ 0}
and
Ω = {u ∈ E : ||u||∞ < β} .
Clearly, P is a cone of E. Define an operator T : P → E by
Z 1
(T x) (t) = µ
G (t, s) (β − x (s)) ex(s) ds, 0 ≤ t ≤ 1,
(5)
0
which is completely continuous on Ω ∩ P . Let r =
µβ
µ+eλ
and define the open set
Ω1 = {x ∈ E : ||x||∞ < r} .
For x ∈ E and x ∈ ∂Ω1 ∩ P ,
Z
(T x)(t) =
1
µ
G (t, s) (β − x(s)) ex(s) ds
0
≥
≥
=
Z
µ
eλ
1
(β − x(s)) ex(s) ds
0
¶
Z µ
µ 1
µβ
ds
β−
eλ 0
µ + eλ
µβ
= ||x||∞ .
µ + eλ
Thus, ||T x||∞ ≥ ||x||∞ .
On the other hand, let Ω2 = Ω ⊃ Ω1 . Then, for x ∈ ∂Ω2 ∩ P , we have
Z 1
(T x)(t) = µ
G (t, s) (β − x (s)) ex(s) ds
0
Z
≤
<
1
µ
(β − x(s))ex(s) ds
0
β
µβe ≤ β = ||x||∞ .
Hence,
¡ ||T x||
¢ ∞ ≤ ||x||∞ . Lemma 1 ensures that BVP (1)-(2) has a positive solution
x ∈ Ω2 \Ω1 , which implies
max x(t) >
t∈[0,1]
The proof is complete.
µβ
and x(t) ≤ β for t ∈ [0, 1].
µ + eλ
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WENYING FENG, GUANG ZHANG AND YIKANG CHAI
In the following, we denote y (t) = 1 − x(t)
β . The integral equation (4) reduces to
Z 1
y (t) = 1 − µeβ
G (t, s) y (s) e−βy(s) ds.
(6)
0
Existence of a solution for (1)-(2) is equivalent to the following BVP:
Z 1
G(t, s)y(s)e−βy(s) ds,
y(t) = 1 − µeβ
(7)
0
y 0 (0) = λ(y(0) − 1), y 0 (1) = 0.
(8)
Our next result gives a sufficient condition on existence of a positive solution
with a minimum value less than a constant. In applications, it implies that at one
point, temperature of the reaction is lower than the constant.
Theorem 3. Let λ, µ and β be positive numbers with µeβ ≤ 1. If r is a positive
1
number that satisfies r ≤ 1+µe
β . Then BVP problem (1)-(2) has at least one positive
solution x such that
min x(t) < β(1 − r) and x(t) ≤ β for t ∈ [0, 1].
t∈[0,1]
In particular, there exists a positive solution x(t) such that
β
min x(t) <
and x(t) ≤ β for t ∈ [0, 1].
2
t∈[0,1]
Proof. We prove that there exists a fixed point for the operator T1 : E → E defined
by
Z 1
(T1 y)(t) = 1 − µeβ
G(t, s)y(s)e−βy(s) ds.
(9)
0
Let Ω3 = {y ∈ E : ||y||∞ < 1}. Note that βy < eβy and e−λ ≤ G(t, s) ≤ 1 for
t, s ∈ [0, 1], thus µeβ ≤ 1 ensures that
Z 1
0 ≤ 1 − µeβ
G(t, s)y(s)e−βy(s) ds ≤ 1 for y ≥ 0 and ||y||∞ ≤ 1.
0
So, T1 : P ∩ Ω3 → P and ||T1 y||∞ ≤ ||y||∞ for y ∈ P ∩ ∂Ω3 . Let Ω4 be the open set
defined by
Ω4 = {y ∈ E : ||y||∞ < r}.
Then for y ∈ P ∩ ∂Ω4 , we have
Z 1
Z 1
µeβ
G(t, s)y(s)e−βy(s) ds ≤ µeβ
y(s)e−βy(s) ds ≤ µeβ r.
Since r ≤
0
1
,
1+µeβ
0
(T1 y)(t) ≥ 1 − µeβ r ≥ r = ||y||∞ .
By Lemma 1, T1 has a fixed point y1 ∈ Ω3 \Ω4 , which is a positive solution of (7)-(8).
Let
x1 (t) = β(1 − y1 (t)), t ∈ [0, 1];
then x1 is a positive solution of (1)-(2). Obviously, x1 (t) ≤ β for t ∈ [0, 1]. Also,
maxt∈[0,1] y1 (t) > r implies that minx∈[0,1] x1 (t) < β(1 − r) .
1
In particular, let r = 12 ≤ 1+µe
β . We obtain that there exists a positive solution
that satisfies
β
and x(t) ≤ β for t ∈ [0, 1].
min x(t) <
2
t∈[0,1]
EXISTENCE OF POSITIVE SOLUTIONS FOR APPLIED ODES
377
Theorem 4. Assume that µ < βe1−β . For any λ > 0, the BVP (1)-(2) has at least
one positive solution x(t) such that
min x(t) < µeβ−1 and x(t) ≤ β for t ∈ [0, 1].
t∈[0,1]
Proof. Similar to the proof of Theorem 3, we prove that there exists a fixed point
for the integral operator T1 . Define the open set Ω5 as the following:
¾
½
µeβ−1
.
Ω5 = y ∈ E : ||y||∞ < 1 −
β
Then Ω5 ⊂ Ω3 . Consider the function
f (x) = xe−βx ,
we have
f 0 (x) = e−βx (1 − βx).
The unique solution of f 0 (x) = 0, x ∈ (0, +∞) is x = 1/β. Therefore,
1
sup xe−βx =
.
βe
x∈R
For any y ∈ P ∩ ∂Ω5 , we have
(T1 y)(t)
=
Z
1 − µeβ
1
G(t, s)y(s)e−βy(s) ds
0
β−1
Z 1
µe
G(t, s)ds
β
0
µeβ−1
≥ 1−
= ||y||∞ .
β
Thus, ||T1 y||∞ ≥ ||y||∞ for y ∈ P ∩ ∂Ω5 . Again, by Lemma 1, T1 has a fixed point
in Ω3 \Ω5 , which is a positive solution for problem (7)-(8). The rest of the proof
follows the proof of Theorem 3.
≥
1−
The following corollary can be easily verified by Theorem 4.
Corollary 1. (a) Assume that µ < β ≤ 1. Then for any λ > 0, the BVP (1)-(2)
has a positive solution x such that
min x(t) < µeβ−1 and x(t) ≤ β for t ∈ [0, 1].
t∈[0,1]
(b) In particular, let β = 1 and µ < 1. Then for any λ > 0, there exists a positive
solution of (1)-(2) that satisfies
min x(t) < µ and max x(t) ≤ 1 for t ∈ [0, 1].
t∈[0,1]
t∈[0,1]
Remark 1. In Theorem 1, let M = β, condition (3) is equivalent to the following:

1

,
if 0 < β ≤ 32 ,

2kKk
β
µe <
1
1


<
, if β > 32 .
(2β − 1)kKk
2kKk
1
Therefore, when (1) kKk > 12 or (2) kKk ≤ 12 and β > max{ 23 , 2kKk
+ 12 }, the
condition of Theorem 2 is weaker than that of Theorem 1. It is easy to find examples
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WENYING FENG, GUANG ZHANG AND YIKANG CHAI
that satisfies Theorem 2 but Theorem 1 can not be applied. In addition, Theorem
2 gives the lower bound of kxk for a solution of problem (1)-(2). As mentioned
earlier, this implies the highest temperature during the reaction is higher than a
constant.
Remark 2. Note that βe1−β − e−β = (βe − 1)e−β implies βe1−β > e−β for βe > 1,
and that βe1−β < e−β for 0 < βe < 1. Thus, Theorem 4 is different from Theorems
2 and 3.
3. Comparison of numerical solutions. In this section, we solve for numerical
solutions for given parameters by applying computer simulations. Let f (x) =
λµ (β − x) ex . By a simple difference calculation, we can obtain the following
system of algebraic equations:
½
(2 + λh) xk+1 − 4xk + (2 + λh) xk−1 + 2h2 f (xk ) = 0, k = 1, 2, ..., n,
(10)
x1
x0 = 1+hλ
, xn+1 = xn ,
where the step length h = 1/ (n + 1).
The simulation is done by using Matlab. The program used for the procedure is
given below (here, b stands for β):
function f=fff(x,lam,mu,b)
f=zeros(size(x));
n=length(x);
h=1/(n-1);
f(1)=x(2)-(1+h*lam)*x(1);
x1=x(1:n-2);
x2=x(2:n-1);
x3=x(3:n);
f(2:n-1)=(2+lam*h).*x1-4.*x2+(2-lam*h).*x3
+2*h*h*lam*mu.*(b-x2).*exp(x2);
f(n)=x(n)-x(n-1);
Given the values of λ, µ and β, the program enables us to obtain sample solutions for (1)-(2). Some interesting facts are observed by comparison between the
numerical solutions and the theoretical results obtained in Section 2.
First, choose λ = 10, µ = 0.55 and β = 0.57. The numerical positive solution
obtained is shown in Figure 1.
The solution satisfies x(t) > 0.289 > 0.5β for t ∈ [0, 1]. Such a solution satisfies
Theorem 2 but does not satisfies Theorem 3. Therefore, Theorem 3 actually ensures
that the equation has at least two positive solutions.
On the other hand, with the inputs λ = 0.05, µ = 0.5, β = 0.6, BVP (1)-(2) has
the numerical solution shown in Figure 2.
Such a solution satisfies both Theorems 2 and 3 In fact,
β
µβ
= 0.3 > x(t) > 0.23 >
= 0.1974, for all t ∈ [0, 1].
2
µ + e−λ
Second, let λ = 5, µ = 0.05, and β = 0.53 (which satisfy all conditions of
Theorem 2), two numerical positive solutions are obtained as shown in Figures 3
and 4 respectively.
EXISTENCE OF POSITIVE SOLUTIONS FOR APPLIED ODES
379
Figure 1. Numerical solution when λ = 10, µ = 0.55, β = 0.57
Figure 2. Numerical solution when λ = 0.05, µ = 0.5, β = 0.6
Figure 3. First solution when λ = 5, µ = 0.05, β = 0.53
It is noticed that the solution in Figure 3 does satisfy the bound conditions of
Theorem 2. The solution in Figure 4, however, is not consistent with any of the
380
WENYING FENG, GUANG ZHANG AND YIKANG CHAI
Figure 4. Second solution when λ = 5, µ = 0.05, β = 0.53
theorems of Section 2 because of ||x||∞ > β. This raises the open problem that
does BVP (1)-(2) have at least one positive solution with kxk∞ > β?
Third, when λ = 5, µ = 0.7, β = 0.8, it can be verified that all conditions of
Theorem 4 hold but the conditions of Theorem 2 and 3 are not satisfied. In this
case, the numerical solution can also be obtained. Please see Figure 5.
Figure 5. Numerical solution when λ = 5, µ = 0.7, β = 0.8
At the last, we give an example of numerical solution with the parameter λ < 0.
In fact, speaking from the mathematical side, our theorems are valid when λ < 0.
As such an example, Figure 6 shows a solution with inputs λ = −3, µ = 0.05 and
β = 0.6.
EXISTENCE OF POSITIVE SOLUTIONS FOR APPLIED ODES
381
Figure 6. Numerical solution when λ = −3, µ = 0.05, β = 0.6
Acknowledgements. The authors thank both referees for valuable comments, especially the correction of Theorem 2 and the suggestion of Remark 1.
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Received September 2006; revised March 2007.
E-mail address: wfeng@trentu.ca