EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 751-762
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Hybrid Cubic B-spline Method for Solving A Class of
Singular Boundary Value Problems

Ahmed Salem Heilat1,∗, Belal Batiha1, Tariq Qawasmeh1, Raed Hatamleh1

1 Department of Mathematics, Jadara University, P.O. Box(733), 21111 Irbid, Jordan

Abstract. In this paper, a class of singular two-point boundary value problems are solved using
hybrid cubic B-spline method. In this algorithm, a free parameter γ plays an important role
to give accurate converge results for the solution. This parameter are chosen via optimization.
Numerical examples are displayed to prove that our suggested method is flexible and effective, and
the numerical results are compared with other numerical methods from the literature.

2020 Mathematics Subject Classifications: 34B05, 65L10

Key Words and Phrases: Singular two-point boundary value problems, Cubic B-spline, Trigono-
metric cubic B-spline, Hybrid cubic B-spline

1. Introduction

Consider a class of singular two-point boundary value problems:

x−α(xαy′)′ = f(x, y), 0 < x ≤ 1

y′(0) = 0, y(1) = β. (1)

where β is a finite constant and α ≥ 1. It is clear that (1) has a unique solution, if
f(x, y) is continuous, ∂f

∂y exists and is continuous and ∂f
∂y ≥ 0 [1]. Second-order singular

boundary value problems arise frequently in several real life applications in chemical reac-
tion, gas dynamics, thermal explosions, electrohydrodynamics, nuclear physics and atomic
calculations can be modeled by linear and nonlinear singular differential equations. One
important class of these equations is linear singular two-point boundary value problems
which appeared clearly in many applications and has been examined in [2–11, 18, 19].

A lot of research has been carried out with the linear and non-linear singular of second-
order boundary value problems. The finite difference method had been used to solve a
class of singular two-point boundary value problems. Kumar suggested a difference method

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4725

Email addresses: ahmed heilat@yahoo.com (A. S. Heilat),
b.bateha@jadara.edu.jo (B. Batiha), ta.qawasmeh@jadara.edu.jo (T. Qawasmeh),
raed@jadara.edu.jo (R. Hatamleh)

https://www.ejpam.com 751 © 2023 EJPAM All rights reserved.



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 752

based on uniform mesh for a class of singular two-point boundary value problems [2]. a
class of singular two-point boundary value problems using fourth-order finite difference
method had been studied by Kanth and Reddy [3]. On the other hand, a class of singular
two-point boundary value problems had been widely treated by using the spline method.
Kanth and Reddy considered cubic spline method for solving a class of singular two-point
boundary value problems [4] then he treated non-linear singular two-point boundary value
problems by using Cubic spline polynomial [5]. Kadalbajoo and Aggarwal presented B-
spline method for numerically solving singular two-point boundary value problems also
they used Chebyshev polynomial to remove the singularity [6]. Singular two-point bound-
ary value problems by using B-spline had been studied by N. Caglar and H. Caglar [7]. J.
Rashidinia et al studied parametric spline method for a class of singular two-point bound-
ary value problems [8]. B-spline of non-linear singular boundary value problems arising in
physiology had been proposed by N. Caglar and H. Caglar [9]. Khuri and Sayfy suggested
a new approach implementing a modified decomposition method in combination with the
cubic B-spline collocation technique is introduced for the numerical solution of a class
of singular boundary value problems arising in physiology [10]. Extended cubic uniform
B-spline for a class of singular two-point boundary value problems had been suggested by
Goh et al [11].
In this paper, we study a hybrid cubic B-spline method (HCBSM) for solving second-
order singular boundary value problems. The approach used is basically that of [17]. The
present method is examined by taking several problems to prove its accuracy.

2. Hybrid Cubic B-spline (HCBS)

In this paper, hybrid cubic B-spline function are utilized to solve singular two-point
boundary value problems. Consider a partition π of [a, b] is equally-spaced knots xi into
n segments [xi, xi+1], i = 0, 1, ..., n, where a = x0 < x1 < ... < xn = b, such that
h = b−a

n , x0 = a, xi = x0 + ih. Then, the hybrid cubic B-spline functions can be defined
as the following relation [17]:

H4,i(x) = γB4,i(x) + (1− γ)T4,i(x) (2)

where γ ∈ R, B4,i(x) is cubic B-spline basis function [7].

B4,i(x) =
1

24h4


(x− xi)

3, x ∈ [xi, xi+1],

h3 + 3h2(x− xi+1) + 3h(x− xi+1)
2 − 3(x− xi+1)

3, x ∈ [xi+1, xi+2],

h3 + 3h2(xi+3 − x) + 3h(xi+3 − x)2 − 3(xi+3 − x)3, x ∈ [xi+2, xi+3],

(xi+4 − x)3, x ∈ [xi+3, xi+4],

(3)

and T4,i(x) is cubic trigonometric B-spline basis function [13] given as



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 753

T4,i(x) =
1

κ


p3(xi), x ∈ [xi, xi+1],

p(xi)[p(xi)q(xi+2) + p(xi+1)q(xi+3)] + p2(xi+1)q(xi+4), x ∈ [xi+1, xi+2],

q(xi+4)[p(xi+1)q(xi+3) + p(xi+2)q(xi+4)] + p(xi)q
2(xi+3), x ∈ [xi+2, xi+3],

q3(xi+4), x ∈ [xi+3, xi+4],

(4)

where p(xi) = sin(x−xi
2 ), q(xi) = sin(xi−x

2 ), κ = sin(h2 ) sin(h) sin(
3h
2 ). The values

of γ have essential role and influential in the hybrid cubic basis function. If γ = 0, the
basis function is equal to cubic trigonometric B-spline basis function and if γ = 1, the
basis function is equal to B-spline basis function.

3. Numerical method for singular boundary value problems

In linear case, the equation (1), can be taken as [14]

y′′(x) +
k

x
y′(x) + b(x)y(x) = c(x), 0 < x ≤ 1 (5)

with boundary conditions
y′(0) = 0, y(1) = β. (6)

where k ≥ 1.
By using L’Hôpital rule, we can modify Eq.(1) at the singular point x = 0, and then the
boundary value problem is transform into [3, 7]

 (k + 1)y′′(x) + b(0)y(x) = c(0), for x = 0,

y′′(x) +
k

x
y′(x) + b(x)y(x) = c(x), for x ̸= 0,

(7)

In this section, hybrid cubic B-spline is used to solve a class of singular two-boundary
value problems, the approach used is basically that of [15]. The approximate solution
S(x), to the exact solution, y(x), is considered as

S(x) =
n+1∑
i=−1

CiH4,i(x) (8)

where Ci are unknown real coefficients and H4,i(x) is hybrid cubic B-spline basis functions.
There are only three nonzero basis functions; as follows,H4,i−1(xi),H4,i(xi), andH4,i+1(xi)
on sub interval [xi, xi+1] , as a result of local support properties of B-spline basis function.
So, the approximate solution and its derivatives with respect to x are



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 754

S(xi) = A1Ci−1 +A2Ci +A1Ci+1, (9)

S
′
(xi) = A3Ci−1 −A3Ci+1, (10)

S
′′
(xi) = A4Ci−1 +A5Ci +A4Ci+1, (11)

where

Ai = γσi + (1− γ)ηi, for i = 1, 2, ..., 5, (12)

with

σ1 =
1

6
, σ2 =

4

6
, σ3 =

−1

2h
, σ4 =

1

h2
, σ5 =

−2

h2
,

η1 =
κ21

κ2κ3
, η2 =

2κ1
κ3

, η3 =
−3

4κ3
, η4 =

3(κ1 − 2κ31 + κ3)

8κ1κ2κ3
, η5 =

−3(κ4 + 2κ21κ2)

4κ1κ2κ3
(13)

where

κ1 = sin(
h

2
), κ2 = sin(h), κ3 = sin(

3h

2
), κ4 = sin(2h), (14)

Since H4,i(xi) = 0,

SH(xi) = Ci−1H4,i−1(xi) + CiH4,i(xi) + Ci+1H4,i+1(xi)

= Ci−1[γσ1 + (1− γ)η1] + Ci[γσ2 + (1− γ)η2] + Ci+1[γσ1 + (1− γ)η1] ≈ y(xi), (15)

By finding the first and second derivative of (8) and calculating at xi, to get (16) and
(17).

S
′
H(xi) = Ci−1[−γσ3 − (1− γ)η3] + Ci+1[γσ3 + (1− γ)η3] ≈ y

′
(xi), (16)

S
′′
H(xi) = Ci−1[γσ4+(1−γ)η4]+Ci[γσ5+(1−γ)η5]+Ci+1[γσ4+(1−γ)η4] ≈ y

′′
(xi), (17)

the simplifications of SH(x) , S
′
H(x) , and S

′′
H(x) at xi are very beneficial for solving

the problems.
In order to obtain the approximations of equations (5)-(6) at the point x = xi, we put
equations (15)-(17) into equations (7) and (6); this leads to

 (k + 1)S′′(0) + b(0)S(0) = c(0), i = 0,

S′′(xi) +
k

xi
S′(xi) + b(xi)S(xi) = c(xi), i = 1, 2, ..., n

(18)



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 755

S′(xi) = 0 for i = 0, and S(xi) = β for i = n.
To solve the singular two-point boundary value problems (5), suppose that y(x) = SH(x)
and substituting (15), (16) and (17) into (18) and collecting the terms that only contain
Ci−1, Ci, Ci+1, to get the following

{Ci−1[γσ4 + (1− γ)η4] + Ci[γσ5 + (1− γ)η5] + Ci+1[γσ4 + (1− γ)η4]}

+
k

xi
{Ci−1[−γσ3 − (1− γ)η3] + Ci+1[γσ3 + (1− γ)η3]}

+b(xi){Ci−1[γσ1+(1− γ)η1] +Ci[γσ2+(1− γ)η2] +Ci+1[γσ1+(1− γ)η1]} = c(xi). (19)

A system of (n + 3) equations with (n + 3) unknowns C−1, C0, C1, ..., Cn+1 is obtained.
This system can be written in the matrix-vector

[A](n+3)×(n+3)[Q](n+3)×1 = [R]1×(n+3), (20)

Where
Q = [C−1, C0, C1, ..., Cn+1]

T ,

R = [0, c(x0), c(x1), ..., c(xn), β]
T

and A is an (n+ 3)× (n+ 3) -dimensional tri-diagonal matrix given by

A =



−γσ3 − (1− γ)η3 0 γσ3 + (1− γ)η3 0 · · · 0 0
a0(x0) b0(x0) c0(x0) 0 · · · 0 0

0 a1(x1) b1(x1) c1(x1) 0 · · · 0
...

...
...

...
...

...
...

0 · · · 0 0 an(xn) bn(xn) cn(xn)
· · · · γσ1 + (1− γ)η1 γσ2 + (1− γ)η2 γσ1 + (1− γ)η1


as well the coefficient in the matrix A as the following

a0(x0) = (k + 1)[γσ4 + (1− γ)η4] + b(xi)[γσ1 + (1− γ)η1], i = 0
b0(x0) = (k + 1)[γσ5 + (1− γ)η5] + b(xi)[γσ2 + (1− γ)η2], i = 0
c0(x0) = (k + 1)[γσ4 + (1− γ)η4] + b(xi)[γσ1 + (1− γ)η1], i = 0

ai(xi) = [γσ4 + (1− γ)η4]− k
xi
[γσ3 + (1− γ)η3] + b(xi)[γσ1 + (1− γ)η1], i = 1, 2, ..., n

bi(xi) = [γσ5 + (1− γ)η5] + b(xi)[γσ2 + (1− γ)η2], i = 1, 2, ..., n
ci(xi) = [γσ4 + (1− γ)η4] +

k
xi
[γσ3 + (1− γ)η3] + b(xi)[γσ1 + (1− γ)η1], i = 1, 2, ..., n

where Q is the unknown vector. Therefor, Q can be solved by taking

Q = A−1R. (21)

Finally, substitute the values of Ci , for i = −1, 0, ..., n + 1 in equation (8) to get the
approximated analytical solution to (18).



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 756

4. Optimization of the free parameter γ

The optimization for the values of γ has already been calculated as in [17].

5. Numerical Examples and Discussions

Hybrid cubic B-spline method was applied on problems (1), (2), (3), and (4) with
different value of n to demonstrate the effectiveness of our proposed hybrid cubic B-spline
method . The results obtained by hybrid cubic B-spline method are compared with the
analytical solution. The discrete L∞-norm and L2-norm are defined as follows

L∞ =
n−1
max
i=1

| S(xi)− u(xi)) |

L2 =

√√√√n−1∑
i=1

[S(xi)− u(xi)]2

We found that our method in comparison with the cubic spline method [4] and cubic
B-spline method [7] and reproducing kernel spaces method [16] is much better with a view
to accuracy and efficiency.

Example 1.Consider the following boundary value problem [4] y′′(x) +
2

x
y′(x)− 4y(x) = −2, 0 < x ≤ 1,

y′(0) = 0, y(1) = 5.5
(22)

The analytic solution is y(x) = 0.5 + 5sinh2x
xsinh2 . The numerical results are tabulated in

Table 1 with h = 0.05 and compared with those obtained in [4, 7] and the exact solutions.
The computational errors, L∞ norm, and L2 norm are tabulated in Table 2.

Table 1: Comparison of proposed method with cubic spline method [4] and cubic B-spline
method [7] in the solution of example 1 with h = 0.05 and γ = 1.85127150 for S(x)

x Exact solution Cubic spline[4] Cubic B-spline[7] HCBSM
0.1 3.275624 9.0800E − 04 2.9500E − 04 1.4257E − 11
0.2 3.331322 2.8900E − 04 2.9200E − 04 3.2166E − 06
0.3 3.425641 2.8100E − 04 2.8500E − 04 4.9187E − 06
0.4 3.560864 2.7200E − 04 2.7500E − 04 6.4545E − 06
0.5 3.740271 2.5500E − 04 2.5800E − 04 7.8650E − 06
0.6 3.968246 2.3300E − 04 2.3500E − 04 8.9406E − 06
0.7 4.250393 2.0000E − 04 2.0200E − 04 9.3166E − 06
0.8 4.593706 1.5500E − 04 1.5500E − 04 8.4697E − 06
0.9 5.006766 8.9000E − 05 8.9000E − 05 5.6839E − 06
1.0 5.500000 0.0000E − 00 0.0000E − 00 0.0000E − 00



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 757

0.0 0.1 0.2 0.3 0.4 0.5

3.3

3.4

3.5

3.6

3.7

x

SH
xL

Figure 1: Numerical solution S(x) and exact solution y(x) for Example 1 with h = 1
20 and

γ = 1.85127150

Table 2: Comparison of error norms with cubic spline[4], cubic B-spline[7], and HCBSM
for Example 1 with h = 0.05 and γ = 1.85127150

Cubic spline[4] Cubic B-spline[7] HCBSM
h L∞ L2 L∞ L2 L∞ L2

0.05 9.0800E − 04 1.1190E − 03 2.9500E − 04 7.2366E − 04 9.3166E − 06 2.0219E − 05

Example 2.Consider the following boundary value problem [4] y′′(x) +
1

x
y′(x) = (

8

8− x2
)2,

y′(0) = 0, y(1) = 0
(23)

The analytic solution is y(x) = 2log( 7
8−x2 ).

We can obtain absolute errors and the errors at different knots by using suggested
hybrid cubic B-spline method with h = 0.05 and γ = 1.4125 are tabulated in Table 3
and compared with existing methods[4, 7]. The computational errors, L∞ norm, and L2

norm are tabulated in Table 4. It is clear that the current hybrid cubic B-spline method
(HCBSM) is acceptable and accurate than cubic spline method [4] and cubic B-spline
method [7]. The numerical results obtained by hybrid cubic B-spline method are shown
in Figs. 1.



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 758

Table 3: Comparison of proposed method with cubic spline method [4] and cubic B-spline
method [7] in the solution of example 2 with h = 0.05 and γ = 1.4125 for S(x)

x Exact solution Cubic spline[4] Cubic B-spline[7] HCBSM
0.1 −0.264561 2.7000E − 05 2.7000E − 05 1.2355E − 07
0.2 −0.257038 2.5000E − 05 2.6000E − 05 3.6696E − 10
0.3 −0.244435 2.5000E − 05 2.5000E − 05 1.8901E − 07
0.4 −0.226657 2.4000E − 05 2.4000E − 05 4.2654E − 07
0.5 −0.203565 2.2000E − 05 2.2000E − 05 6.8326E − 07
0.6 −0.174975 1.9000E − 05 1.9000E − 05 9.1680E − 07
0.7 −0.140651 1.5000E − 05 1.5000E − 05 1.0672E − 06
0.8 −0.100300 1.1000E − 05 1.1000E − 05 1.0503E − 06
0.9 −0.053562 6.0000E − 06 6.0000E − 06 7.4911E − 07
1.0 0.000000 0.0000E − 00 0.0000E − 00 0.0000E − 00

Figure 2: Numerical solution S(x) and exact solution y(x) for Example 2 with h = 0.05
and γ = 1.4125

Table 4: Comparison of error norms with cubic spline[4], cubic B-spline[7], and HCBSM
for Example 2 with h = 0.05 and γ = 1.4125

Cubic spline[4] Cubic B-spline[7] HCBSM
h L∞ L2 L∞ L2 L∞ L2

0.05 2.7000E − 05 6.1498E − 05 2.7000E − 05 6.1911E − 05 1.0672E − 06 2.0841E − 06

Example 3.Consider the following boundary value problem [7] − y′′(x)− 2

x
y′(x) + (1− x2)y(x) = x4 − 2x2 + 7,

y′(0) = 0, y(1) = 0
(24)

The analytic solution is y(x) = 1 − x2. The numerical results with h = 0.05 and
γ = 0.999999 are given in Table 5 and Figs. 2. The computational errors, L∞ norm, and
L2 norm are tabulated in Table 6.



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 759

Table 5: Comparison of proposed method with cubic spline method [4] and cubic B-spline
method [7] in the solution of example 3 with h = 0.05 and γ = 0.999999 for S(x)

x Exact solution Cubic spline[4] Cubic B-spline[7] HCBSM
0.1 0.990000 2.0000E − 06 1.1405E − 07 1.1405E − 07
0.2 0.960000 2.0000E − 06 4.5295E − 08 4.5295E − 08
0.3 0.910000 2.0000E − 06 2.5596E − 08 2.5596E − 08
0.4 0.840000 2.0000E − 06 1.6158E − 08 1.6158E − 08
0.5 0.750000 1.0000E − 06 1.0650E − 08 1.0650E − 08
0.6 0.640000 1.0000E − 06 7.0547E − 09 7.0547E − 09
0.7 0.510000 1.0000E − 06 4.5235E − 09 4.5235E − 09
0.8 0.360000 1.0000E − 06 2.6397E − 09 2.6397E − 09
0.9 0.190000 1.0520E − 07 1.1767E − 09 1.1767E − 09
1.0 0.000000 0.0000E − 00 0.0000E − 00 0.0000E − 00

Figure 3: Numerical solution S(x) and exact solution y(x) for Example 3 with h = 0.05
and γ = 0.999999

Table 6: Comparison of error norms with cubic spline[4], cubic B-spline[7], and HCBSM
for Example 3 with h = 0.05 and γ = 0.999999

Cubic spline[4] Cubic B-spline[7] HCBSM
h L∞ L2 L∞ L2 L∞ L2

0.05 2.0000E − 06 4.4734E − 06 1.1405E − 07 1.2715E − 07 1.1405E − 07 1.2715E − 07

Example 4.Consider the following boundary value problem [16] y′′(x) +
1

x
y′(x) + y(x) = 4− 9x+ x2 − x3,

y(0) = 0, y(1) = 0
(25)

The analytic solution is y(x) = x2 − x3.
Hybrid cubic B-spline method is used to solve this problem numerically with h = 1

26 .
Table 7 list approximate solutions. The numerical results obtained by hybrid cubic B-
spline method are shown in Figs. 3. The computational errors, L∞ norm, and L2 norm
are tabulated in Table 8. The results of this problem is compared with Minggen Cui [16]



A. S. Heilat et al. / Eur. J. Pure Appl. Math, 16 (2) (2023), 751-762 760

which is listed in Table 7. The table denotes that HCBSM with γ = 0.999999 gives better
results to this problems.

Table 7: Comparison of proposed method with reproducing kernel spaces method [16] in
the solution of example 4 with h = 1

26 and γ = 0.999999 for S(x)
x Exact Solution Reproducing kernel spaces [16] HCBSM

0.001 9.99E − 07 2.0000E − 11 2.6190E − 09
0.08 0.005888 2.3000E − 06 2.1606E − 08
0.16 0.021504 1.1600E − 05 1.7831E − 08
0.32 0.069632 5.5800E − 05 4.6929E − 08
0.48 0.119808 1.1900E − 04 5.4917E − 08
0.64 0.147456 1.6800E − 04 4.9914E − 08
0.80 0.128000 1.5700E − 04 3.2371E − 08
0.96 0.036864 4.5000E − 05 6.9600E − 09
1.00 0.000000 0.0000E − 00 0.0000E − 00

Figure 4: Numerical solution S(x) and exact solution y(x) for Example 4 with h = 1
26 and

γ = 0.999999

Table 8: Comparison of error norms with reproducing kernel spaces method [16], and
HCBSM for Example 4 with h = 1

26 and γ = 0.999999
reproducing kernel spaces method [16] HCBSM

h L∞ L2 L∞ L2
1
26

1.6800E − 04 2.6891E − 04 5.4917E − 08 9.7968E − 08

6. Conclusions

In this paper, a class of singular two point boundary value problems has been suc-
cessfully solved using HCBSM. The hybrid cubic B-spline method for finding a numerical
solution is easy and acceptable and flexible to apply. The numerical results presented
indicate the proposed method is an accurate and reliable approach for the solution of a
class of singular two point boundary value problems. Further, minimizing the one-norm
term is sufficient to obtain the optimized values of γ. [17]



REFERENCES 761

References

[1] Russell, R. D. and Shampine, L. F. (1975). Numerical methods for singular boundary
value problems. SIAM Journal on Numerical Analysis, 12(1), 13-36.

[2] Kumar, M. (2003). A difference method for singular two-point boundary value prob-
lems. Applied mathematics and computation, 146(2), 879-884.

[3] Kanth, A. R. and Reddy, Y. N. (2004). Higher order finite difference method for a
class of singular boundary value problems. Applied mathematics and computation,
155(1), 249-258.

[4] Kanth, A. R. and Reddy, Y. N. (2005). Cubic spline for a class of singular two-point
boundary value problems. Applied mathematics and coputation, 170(2), 733-740.

[5] Kanth, A. R. (2007). Cubic spline polynomial for non-linear singular two-point bound-
ary value problems. Applied mathematics and computation, 189(2), 2017-2022.

[6] Kadalbajoo, M. K. and Aggarwal, V. K. (2005). Numerical solution of singular bound-
ary value problems via Chebyshev polynomial and B-spline. Applied mathematics and
computation, 160(3), 851-863.

[7] Caglar, N. and Caglar, H. (2006). B-spline solution of singular boundary value prob-
lems. Applied mathematics and computation, 182(2), 1509-1513.

[8] Rashidinia, J., Mahmoodi, Z. and Ghasemi, M. (2007). Parametric spline method
for a class of singular two-point boundary value problems. Applied mathematics and
computation, 188(1), 58-63.

[9] Çağlar, H., Çağlar, N. and Özer, M. (2009). B-spline solution of non-linear singular
boundary value problems arising in physiology. Chaos, Solitons and Fractals, 39(3),
1232-1237.

[10] Khuri, S. A. and Sayfy, A. (2010). A novel approach for the solution of a class of
singular boundary value problems arising in physiology. Mathematical and Computer
Modelling, 52(3), 626-636.

[11] Goh, J., Majid, A. A., Ismail, A. I. M. (2011). Extended cubic uniform B-spline for
a class of singular boundary value problems. nuclear physics, 2, 4.

[12] Mat Zin, S., Abd Majid, A., Ismail, A. I. M., and Abbas, M. (2014). Application
of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear
Klien-Gordon Equation. Mathematical Problems in Engineering, 2014.

[13] Abbas, M., Majid, A. A., Ismail, A. I. M., and Rashid, A. (2014). The application
of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems.
Applied Mathematics and Computation, 239, 74-88.



REFERENCES 762

[14] Kadalbajoo, M. K. and Kumar, V. (2007). B-spline method for a class of singular
two-point boundary value problems using optimal grid. Applied mathematics and
computation, 188(2), 1856-1869.

[15] Prenter, P. M. (1989). Splines and variational methods. John Wiley and Sons.

[16] Cui, M. and Geng, F. (2007). Solving singular two-point boundary value problem
in reproducing kernel space. Journal of Computational and Applied Mathematics,
205(1), 6-15.

[17] Heilat, A. S., Zureigat, H., and Batiha, B. (2021). New Spline Method for Solving
Linear Two-Point Boundary Value Problems. European Journal of Pure and Applied
Mathematics, 14(4), 1283-1294.

[18] Batiha, B., Ghanim, F., Alayed, O., Hatamleh, R. E., Heilat, A. S., Zureigat, H.,
and Bazighifan, O. (2022). Solving Multispecies Lotka–Volterra Equations by the
Daftardar-Gejji and Jafari Method. International Journal of Mathematics and Math-
ematical Sciences.

[19] Visuvasam, J., Meena, A., and Rajendran, L. (2020). New analytical method for
solving nonlinear equation in rotating disk electrodes for second-order ECE reactions.
Journal of Electroanalytical Chemistry, 869, 106-114.