Results in Applied Mathematics 19 (2023) 100388
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Results in Applied Mathematics
journal homepage: www.elsevier.com/locate/results-in-applied-mathematics
Numerical modeling of the boundary value problem of an
ordinary differential equation with a small parameter at the
highest derivative by Chebyshev polynomials of the second
kind
Chori Begaliyevich Normurodov, Barno Abdiyevna Tursunova
∗
Termez State University, Uzbekistan
article
info
Article history:
Received 16 May 2023
Received in revised form 26 June 2023
Accepted 1 July 2023
Available online 16 July 2023
Keywords:
Small parameter
Chebyshev polynomials
Boundary conditions
Polynomial nodes
Spectral solution
a b s t r a c t
In the present work the application of the spectral method with Chebyshev polynomials
of the second kind is considered for solving a boundary value problem of an ordinary
differential equation with a small parameter at the highest derivative. Numerical and
graphical results illustrate the effectiveness of the applied method for various values of
polynomials.
© 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
The problem of constructing a uniformly convergent algorithm for the numerical solution to a boundary value problem
for an ordinary differential equation with a small parameter at the highest derivative was posed and solved in [1] using
a special difference scheme with weights on a uniform grid. In [2,3], the method proposed in [1], was extended to the
solution of more complex (parabolic, multidimensional) boundary value problems. The numerical calculation on a uniform
grid does not describe the boundary layer well, as for a sufficiently small parameter the first grid node already lies outside
the boundary layer. For practical applications, it is important to know the structure of the boundary layer, so, it is necessary
to build a non-uniform difference grid, which is condensed in the area of the boundary layer [4]. Papers [5,6] present a
unified convergence analysis for solving singularly perturbed problems using the standard Galerkin finite element method
on an unconventional Shishkin-type mesh that completely separates boundary layers from other subdomens. In recent
years, along with difference schemes, spectral [7–10] and spectral-grid methods [11–16] have been successfully used,
which provide high accuracy and efficiency of calculations for boundary layer problems.
2. Statement of the problem
In [4], a boundary value problem for a second-order ordinary differential equation with a small parameter at the highest
derivative was considered.
∗ Corresponding author.
E-mail addresses: ch.normurodov@gmail.com (Ch.B. Normurodov), barno.xusnora.1947@gmail.com (B.A. Tursunova).
https://doi.org/10.1016/j.rinam.2023.100388
2590-0374/© 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.
org/licenses/by-nc-nd/4.0/).
Ch.B. Normurodov and B.A. Tursunova
Results in Applied Mathematics 19 (2023) 100388
In the present paper, to solve the problem posed in [4], the spectral method with Chebyshev polynomials of the second
kind is used. In the spectral method, the boundary layer zones are not distinguished and an approximate solution to the
problem is sought in the form of a finite series in Chebyshev polynomials of the second kind. The use of Chebyshev
polynomials of the second kind for solving ordinary differential equations was described in [12–14].
Solve approximately the following differential equation using Chebyshev polynomials of the second kind
ε
d2 u
+
dy2
1 du
2 dy
=
1
8
(y + 1) , y ∈ (−1, +1) ,
(1)
with the boundary conditions
u(−1) = u(+1) = 0,
(2)
where ε is a small parameter.
The exact solution for problem (1)–(2) has the following form [4]:
)
ε − 0.5 (
y+1
(y + 1)2
−(y+1)/2ε
−ε
1−e
+
.
−1/ε
1−e
2
8
u(y) =
3. Solution method
An approximate solution to the differential problem (1)–(2) is sought in the form of a series
ua (y) =
N
∑
an Un (y),
(3)
n=o
where Un (y) are Chebyshev polynomials of the second kind:
U0 (y) = 1, U1 (y) = 2y, Un+1 (y) = 2yUn (y) − Un−1 (y), n = 1, 2, . . . ,
an (n = 0, 1, 2, . . . , N) are unknown coefficients.
The derivatives of function ua (y) from (3) are determined by the following formula
d q ua
=
q
dy
N
∑
a(nq) Un (y), where a(n0) = an .
n=o
For Chebyshev polynomials of the second kind, the following recursive formula is valid:
2Un (y) =
1
n+1
1
Un′ +1 (y) −
n+1
Un′ −1 (y) for n ≥ 0.
q
Taking into account this relation, we write the series for ddyuqa in the following form:
N
d ∑
dy
N
−1)
a(q
Un (y) =
n
n=0
d 1∑
dy 2
an(q)
n=0
[
1
n+1
Un+1 (y) −
1
n+1
]
Un−1 (y) .
Comparing the coefficients for the same polynomials Un (y) for n ≥ 2, we have
1
1
(q−1)
(q)
a
−
a(q) = 2an−1 , n ≥ 2.
n − 1 n−2
n+1 n
(1)
Then an = 2(n + 1)
∑N
p=n+1
p+n≡1(mod2)
ap , n ≥ 0, where a ≡ b(mod2) means that a − b is divided by 2 without remainder. From
these equations, we obtain
a(n2) = 2(n + 1)
N
∑
m=n+1
m+n≡1(mod2)
×
N
∑
N
∑
a(m1) = 4(n + 1)
p=n+2
p≡n(mod2)
ap
p−1
∑
(m + 1) = (n + 1)
m=n+1
m+n≡1(mod2)
(p − n)(p + n + 2)ap
p=n+2
p≡n(mod2)
Then the derivatives of the differential equation (1) are written as:
dua
dy
=
N
∑
n=0
⎛
2(n + 1) ⎝
⎜
N
∑
⎞
ap ⎠ Un (y),
(4)
⎟
p=n+1
p+n≡1(mod2)
2
Ch.B. Normurodov and B.A. Tursunova
Results in Applied Mathematics 19 (2023) 100388
Table 1
Comparison of exact and spectral solutions.
d 2 ua
dy2
=
l
yl -nodes
u-exact solution
ua -approximate solution
|u − ua |-error
0
5
10
15
20
−1.0
−0.70711
6.12323e−17
0.0
−0.48074
−0.37
−0.13426
0.0
1.38777e−17
−0.48275
−0.37238
−0.13161
2.77555e−17
1.38777e−17
0.00201
0.00238
0.00265
2.77555e−17
0.70711
1.0
⎞
N
∑
⎟
⎜
(p − n)(p + n + 2)ap ⎠ Un (y).
(n + 1) ⎝
⎛
N
∑
(5)
p=n+2
p≡n(mod2)
n=0
Substituting series (4) and (5) into the differential equation (1) and equating the coefficients at the same powers of
Chebyshev
⎞
⎛
⎞ equation:
⎛ polynomials, we obtain the following
⎜
ε ⎝(n + 1)
N
∑
(p − n)(p + n + 2)ap ⎠ Un (y) + (n + 1) ⎝
⎜
⎟
p=n+1
p+n≡1(mod2)
p=n+2
p≡n(mod2)
=
1
16
U1 (y) +
1
8
N
∑
ap ⎠ Un (y) =
⎟
(6)
U0 (y)
for n = 0,1,. . . ,N−2.
Boundary conditions (2) with the series (3) are written as:
N
∑
(n + 1)an = 0,
n=0
N
(7)
∑
(−1)n (n + 1)an = 0.
n=0
The main Eq. (6) with boundary conditions (7) can be written as an algebraic system with respect to unknown coefficients:
AX = B,
(8)
where X T = (a0 , a1 , . . . , aN ), BT = (b0 , b1 , . . . , bN ).
4. Discussion of results
Let us present the results of numerical calculations obtained by the above spectral method with Chebyshev polynomials
of the second kind, when the value of the small parameter is ε = 10−2 and the number of polynomials are N = 19 and
49.
Table 1 shows the results of numerical calculations obtained at the nodes of Chebyshev polynomials yl = cos Nπ+l1 , l =
0, 1, 2, . . . , N + 1 by the spectral method, when the value of the small parameter is ε = 10−2 and the number of
polynomials is N = 19.
Table 1 shows that the exact solutions to the differential problem (1)–(2) are found with an accuracy of 10−3 . The
results shown in Table 1 are most clearly illustrated in Fig. 1.
Fig. 1 shows that the minimax properties of Chebyshev polynomials are stably preserved.
Table 2 shows the results of numerical calculations obtained by the spectral method, when the value of the small
parameter is ε = 10−2 and the number of Chebyshev polynomials is N = 49.
Table 2 shows that as the number of polynomials increases from 19 to 49, the approximate solution ua coincides exactly
with the exact solution.
The results given in Table 2 are most clearly illustrated in Fig. 2
Fig. 2 shows that the graphs of the exact and approximate solutions are almost indistinguishable.
Thus, the spectral method by Chebyshev polynomials of the second kind is a reliable mathematical tool for solving a
boundary value problem for an equation with a small parameter at the highest derivative.
5. Conclusion
An efficient and high-precision algorithm based on the spectral method for the numerical solution of equations with
a small parameter at the highest derivative has been developed that describes the boundary layer quite well.
3
Ch.B. Normurodov and B.A. Tursunova
Results in Applied Mathematics 19 (2023) 100388
Table 2
Comparison of the exact and spectral solutions.
l
yl -nodes
u-exact solution
ua -approximate solution
|u − ua |-error
0
5
10
15
20
25
30
35
40
45
50
−1.0
−0.95
−0.81
−0.59
−0.31
6.1e−17
0.0
−0.44754
−0.48636
−0.47082
−0.43377
−0.37
−0.28235
−0.18281
−0.08998
−0.02393
0.0
1.55499e−16
−0.44754
−0.48636
−0.47082
−0.43377
−0.37
−0.28235
−0.18281
−0.08998
−0.02393
2.05808e−16
1.55499e−16
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
2.05808e−16
0.31
0.59
0.81
0.95
1.0
Fig. 1. Dynamics of exact and spectral solutions with Chebyshev polynomials of the second kind (N = 19).
Fig. 2. Dynamics of exact and spectral solutions with Chebyshev polynomials of the second kind (N = 49).
4
Ch.B. Normurodov and B.A. Tursunova
Results in Applied Mathematics 19 (2023) 100388
CRediT authorship contribution statement
Chori Begaliyevich Normurodov: Conceptualization, Methodology, Supervision, Validation, Writing – review & editing.
Barno Abdiyevna Tursunova: Writing – original draft, Software, Computational experiment, Visualization.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Data availability
No data was used for the research described in the article.
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