EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 7128
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Modified Optimal Homotopy Asymptotic Method for
Singular Two-Point Boundary Value Problems

Nidal Anakira1,∗, Sameer Bawaneh2, Areen Al-khateeb3, Ala Amourah1,
Tala Sasa4

1 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar
311, Oman
2 Department of Computer Science, Faculty of Information Technology, Jadara University,
Irbid, 21110, Jordan
3 Department of Mathematics, Faculty of Science and Technology, Jadara University, Irbid
21110, Jordan
4 Department of Mathematics, Faculty of Science, Private Applied Science University,
Amman, Jordan

Abstract. This work applies the Modified Optimal Homotopy Asymptotic Method (MOHAM) to
two benchmark singular two–point boundary value problems that were previously analyzed using
the standard OHAM. The modification introduces a refined homotopy framework in which non-
linear and singular terms are systematically distributed across successive embedding orders, while
the auxiliary function is optimally tuned through residual minimization. With this enhancement,
MOHAM delivers first–order analytical approximations that are nearly identical to the exact so-
lutions, achieving higher accuracy, faster convergence, and greater computational efficiency than
the conventional OHAM even when the latter is extended to three terms of approximation.
2020 Mathematics Subject Classifications: 26A33, 34A08, 65L10
Key Words and Phrases: MOHAM, singular boundary value problems, residual minimization,
analytical approximation, convergence efficiency

1. Introduction

Differential equations constitute a central pillar in applied mathematics, as they pro-
vide a powerful framework to explain and predict various real-world phenomena. They
are widely used in modeling processes in physics, engineering, biology, chemistry, and eco-
nomics. Two-point boundary value problems (BVPs) are particularly important in this

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7128

Email addresses: nanakira@su.edu.om (N. Anakira),
s.bawaneh@jadara.edu.jo (S. Bawaneh), Areen.k@jadara.edu.jo (A. Al-khateeb),
AAmourah@su.edu.om (A. Amourah), t_sasa@asu.edu.jo (T. Sasa)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 2 of 11

regard, as they occur when conditions are imposed at more than one boundary point. Sin-
gular two-point BVPs, in which the governing equations contain singularities at the end
points of the domain, frequently arise in many physical situations, such as thin film flows,
boundary layer problems in fluid mechanics, reaction–diffusion processes, chemical kinet-
ics, and optimal control theory [1]. Singular boundary value problems (BVPs) appear in
fluid mechanics, reaction–diffusion, chemical kinetics, and optimal control. Representative
forms include

1
p

u′′(x) + 1
q(x)

u′(x) + 1
r(x)

u(x) = g(x), 0 ≤ x ≤ 1, (1)

with boundary conditions such as u(0) = α1, u(1) = β or u′(0) = α2, u(1) = β. In these
settings, differential equations provide not only a descriptive mechanism but also an ex-
planatory tool for the underlying dynamics. However, because non-linear and singular
models rarely admit closed-form solutions, the development of accurate and efficient ap-
proximate or numerical methods becomes essential. Over the past few decades, various
approaches have been introduced to deal with the difficulties posed by singular two-point
BVPs. Abu Arqub et al. [2] proposed a continuous genetic algorithm that effectively ap-
proximates the solutions of such problems, showing that evolutionary computation tech-
niques can achieve smooth and stable results. Bakodah et al. [3] developed an efficient
decomposition shooting method that combined decomposition strategies with shooting
techniques, enhancing stability and accuracy. Another direction was pursued by Cui and
Geng [4], who presented a kernel space reproducing approach and rigorously established
convergence properties for singular BVPs. Other notable contributions include numerical
schemes proposed by Anakira et al. [5], spline-based approximations [6], the differential
transformation method [7, 8], Adomian decomposition methods [9], and the variational
iteration method [10] and others [11, 12] . Together, these works reflect the importance of
finding effective strategies that can handle the singular nature of boundary value models
while providing accurate solutions.

Although these approaches have enriched the literature, they often involve complex
iterative computations, high computational cost, or difficulties in handling nonlinearities.
To overcome such limitations, researchers have increasingly turned to homotopy-based
techniques, which construct a deformation (homotopy) that continuously transforms a
simple solvable problem into the original nonlinear model. Among these, the OHAM
introduced by Marinca et al. [13], has received particular attention. Unlike perturbation
methods that require a small or large parameter, OHAM avoids this restriction and instead
introduces auxiliary convergence control parameters that are optimally determined to
accelerate convergence. Marinca et al. [13] successfully applied OHAM to thin-film flow,
demonstrating its efficiency, and since then it has been extended to a wide variety of linear
and non-linear problems in applied sciences. For example, Anakira [14] applied OHAM to
systems of ordinary differential equations, Agarwal et al. [15] solved systems of Volterra
integro-differential equations, and Jameel et al. [16] addressed fuzzy differential equations.
These studies confirm the adaptability and precision of OHAM in many applied contexts.

The strength of the OHAM algorithm lies in its ability to produce rapidly convergent



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 3 of 11

approximate solutions using relatively few terms, typically three or four, while maintaining
excellent agreement with exact or numerical results. Several studies [17–21] have shown
that OHAM can outperform traditional analytical and iterative techniques in terms of the
number of terms required to achieve the same level of accuracy. Unlike other methods that
may require more than four terms, OHAM achieves comparable results with fewer terms,
thereby reducing computational time, effort, and cost. To further improve efficiency and
accuracy, improved approaches have been developed to provide more precise or near-exact
solutions through several previous studies conducted by many researchers [22–29]. This
has led to efforts and initiatives to expand the scope of significant developments within
the OHAM framework, either by improving the auxiliary function or by restructuring the
symmetry equation to accelerate convergence and simplify the computational process. One
of the key advantages of the MOHAM algorithm is its ability to construct highly accurate
solutions using only a single-term (monomial) approximation, which significantly reduces
computational complexity while maintaining accuracy.

In summary, singular two-point boundary value problems are of great significance in
applied mathematics, as they describe fundamental processes in fluid mechanics, reaction–
diffusion systems, chemical kinetics, optimal control, and many other fields. Differential
equations, as the language of these phenomena, require effective approximate techniques
when exact solutions are not attainable. A wide range of methods have been proposed, but
the OHAM by Marinca et al. [13] and its subsequent modifications represent particularly
powerful approaches. The MOHAM, in particular, is capable of producing accurate solu-
tions even with a single approximation term, highlighting its potential as an efficient and
reliable method for addressing nonlinear singular boundary value problems. The OHAM,
introduced by Marinca and collaborators[13], provides a convenient convergence control
via auxiliary constants chosen by minimizing the residual. OHAM has been successfully
applied to singular BVPs [5]. This work adopts a MOHAM a refined homotopy in which
the nonlinear/singular operator is uniquely represented across powers of the embedding
parameter, allowing faster convergence with fewer series terms. We follow the structure
in [5] and re–solve the same examples, comparing MOHAM with OHAM.

2. Modified Optimal Homotopy Asymptotic Method

We consider a general nonlinear boundary value problem

L(u(x)) + g(x) + N (u(x)) = 0, B
(
u, du

dx

)
= 0, (2)

where L is a linear operator, N is a nonlinear operator, g(x) is a known function, and B
denotes the boundary conditions.

According to the modified OHAM, we construct the homotopy

(1 − p)
[
L(v(x, p)) − L(u0(x))

]
= H(x, p)

[
L(v(x, p)) + g(x) + N (v(x, p))

]
, (3)

with the boundary condition

B
(
v(x, p), ∂v(x,p)

∂x

)
= 0, (4)



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 4 of 11

where p ∈ [0, 1] is the embedding parameter, u0(x) is the initial guess satisfying

L(u0(x)) = 0, B(u0, du0
dx ) = 0. (5)

Unlike standard OHAM where H(p) is a scalar function, here we generalize to

H(x, p) =
N∑

n=1

M∑
m=0

cnm xmpn, (6)

where cnm are unknown parameters to be optimally determined.
Expanding v(x, p) in a Taylor series about p = 0 gives

v(x, p) = u0(x) +
∞∑

k=1
uk(x) pk. (7)

Substituting the expansion of v(x, p) and H(x, p) into the homotopy equation and
equating like powers of p yields a hierarchy of linear subproblems:

• Zeroth-order:
L(u0(x)) = 0, B(u0) = 0. (8)

• First-order:

L(u1(x)) + g(x) =
M∑

m=0
c1mxm N0(u0(x)), B(u1) = 0. (9)

• Second-order:

L(u2(x)) − L(u1(x)) =
M∑

m=0
c2mxm N0(u0(x))

+
M∑

m=0
c1mxm

(
L(u1(x)) + N1(u0, u1)

)
, (10)

with B(u2) = 0.

• k-th order (k ≥ 2):

L(uk(x)) − L(uk−1(x)) =
M∑

m=0
ckmxm N0(u0(x))

+
k−1∑
i=1

M∑
m=0

cimxm
(
L(uk−i(x)) + Nk−i(u0, . . . , uk−i)

)
,

(11)

with B(uk) = 0.



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 5 of 11

Truncating at order m gives the m-th order approximate solution

ũ(x; {cnm}) = u0(x) +
m∑

k=1
uk(x). (12)

The residual is defined by

R(x; {cnm}) = L(ũ(x)) + g(x) + N (ũ(x)), (13)

and the optimal coefficients {cnm} are determined by minimizing

J({cnm}) =
∫ b

a
R(x; {cnm})2 dx,

∂J

∂cnm
= 0. (14)

This yields the optimal auxiliary function H(x, p) and the final MOHAM solution.

3. Numerical Examples

In this part, we apply the proposed method to two examples of singular two-point
boundary value problems to assess its efficiency and capability, compared with the stan-
dard OHAM results of three order, in producing near-exact solutions using only a first-
order approximation.

Example 1
As a first example, we examine the singular two-point boundary value problem con-

sidered by Kanth, A. R., & Reddy, Y. N. (2005)[30].

y′′(x) + 2
x

y′(x) − 4y(x) = −2, 0 < x ≤ 1, (15)

y′(0) = 0, y(1) = 5.5. (16)

Following the MOHAM procedure outlined in the previous section and based on the flex-
ibility of standard OHAM in selecting the initial guess, we define the linear operator the
nonlinear operator as follows

L[v(x, p)] = x
d2v(x, p)

dx2 + 2x (17)

N [v(x, p)] = x
d2v(x, p)

dx2 + dv(x, p)
dx

− 4xv(x, p) + 2x. (18)

Then, we construct the following family of homtopy equations h(u(x; p) : R− > [0, 1]

(1 − q) L
(
v(x; q) − y0(x)

)
= H(q)

(
L[v(x; q)] + N [v(x; q)]

)
, (19)



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 6 of 11

with boundary conditions

(1 − q)
(
v′(0; q) − y′

0(0)
)

= H(q)
(
v′(0; q) − 0

)
, (20)

(1 − q)
(
v(1; q) − y0(1)

)
= H(q)

(
v(1; q) − 5.5

)
. (21)

Thus, as p varies from 0 to 1, the solution approach from u0(x) to u(x), where u0(x) is
the zeroth -order problem that can be obtained

u′′
0(x) = −2, u′

0(0) = 0, u(1) = 5.5. (22)

Then, by expanding u(x; p; ci), i = 1 in the Taylor series on p, we have

u(x; p; ci) =
m∑

k=0
uk(x, C1, C2, . . . , C7) pk. (23)

Substituting Eq.(23) into Eq. (19) and by equating the coefficient of similar power p,
yields

The first-order deformation problem

u′
1(x) = 1.33333C7x10 + 1.33333C6x9 + 1.33333C5x8 − 4.C7x8 + 1.33333C4x7 − 4.C6x7

−21.3333C7x7 + 1.33333C3x6 − 4.C5x6 − 21.3333C6x6 + 1.33333C2x5 − 4.C4x5

−21.3333C5x5 + 1.33333C1x4 − 4.C3x4 − 21.3333C4x4 + 1.33333C0x3 − 4.C2x3

−21.3333C3x3 − 4.C1x2 − 21.3333C2x2 − 4.C0x − 21.3333C1x − 21.3333C0 + 2 (24)

By solving Eqs. (22) and (24) and substituting them into Eq.(23), the first-order
approximate solution by MOHAM becomes:

u1(x) = 0.333333
(
17.5 − 1.x3

)
+ 0.010101c7x12 + 0.0121212c6x11 + 0.0148148c5x10

−0.0444444c7x10 + 0.0185185c4x9 − 0.0555556c6x9 − 0.296296c7x9 + 0.0238095c3x8

−0.0714286c5x8 − 0.380952c6x8 + 0.031746c2x7 − 0.0952381c4x7 − 0.507937c5x7

−0.133333c3x6 − 0.711111c4x6 − 0.2c2x5 − 1.06667c3x5 − 1.77778c2x4

+c1
(
0.0444444x6 − 0.333333x4 − 3.55556x3 + 3.84444

)
+c0

(
0.0666667x5 − 0.666667x3 − 10.6667x2 + 11.2667

)
+ 1.94603c2

+1.17619c3 + 0.787831c4 + 0.56455c5 + 0.424387c6 + 0.33064c7 + 1.x2 − 1. (25)

By using the proposed method of section 2 on [0, 1], we use the residual error,

R = xũ′′ (x, C1, · · · , C7) + 2ũ′ (x, C1, · · · , C7)
−4xũ (x, C1, · · · , C7) + 2x. (26)

The Less Square error can be formed as

J(∂J(C1, · · · , C7)) =
∫ 1

0
R2dx, (27)



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 7 of 11

and

∂J(C1, · · · , C7)
∂C1

= ∂J(C1, · · · , C7)
∂C2

= · · · = ∂J(C1, · · · , C7)
∂C7

= 0. (28)

Thus, the following optimal values of Ci’s are obtained:

C0 = −0.078574, C1 = −0.0790727, C2 = −0.191309 C3 = 0.0274625.

C4 = −0.0495492, C5 = −0.0175115, C6 = 0.0112912 C7 = −0.00830839.

Upon substituting those values into equation (25), the first-order approximate solution is
obtained.

ũ(x) = 3.25721 + 1.83812x2 + 0.000196623x3 + 0.366462x4 + 0.0037302x5 + 0.028059x6

+0.00754041x7 − 0.00239673x8 + 0.000916876x9 + 0.000109832x10

+0.000136863x11 − 0.0000839231x12 (29)

The numerical comparison presented in Table 1 clearly demonstrates the superiority of
the modified OHAM (MOHAM) with only one order of approximation over the standard
OHAM using three terms. At x = 0.2, the OHAM absolute error is of order 10−4, whereas
MOHAM reduces the error dramatically to the order of 10−9, showing an improvement of
nearly five orders of magnitude. A similar trend is observed at x = 0.4, 0.6, and 0.8, where
OHAM produces errors between 10−4 and 10−5, while MOHAM consistently achieves
errors in the range of 10−9–10−10. At the endpoint x = 1, both methods attain machine-
precision accuracy, but overall the MOHAM approach achieves substantially higher accu-
racy with fewer approximation terms, confirming its efficiency and effectiveness.

Table 1: Numerical results resulted from example 1.

x Exact OHAM MOHAM
Solution Absolute Error Absolute Error

0.2 3.331321581291895 6.90 × 10−4 2.15 × 10−9

0.4 3.560863537324634 5.60 × 10−5 3.7 × 10−9

0.6 3.9682461451285476 1.50 × 10−4 2.59 × 10−9

0.8 4.593705860688229 4.74 × 10−4 5.55 × 10−10

1.0 5.500000000000000 8.88 × 10−16 8.86 × 10−16

Example 2
As a second example, we examine the singular two-point boundary value problem

considered by Anakira et al. (2013).

u′′ + 1
x

u′ + u = g(x), 0 ≤ x ≤ 1, u′(0) = 0, u(1) = 17
16 , (30)



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 8 of 11

g(x) = 5
4 + x2

16 , uexact(x) = 1 + x2

16 . (31)

By applying the same procedure as in the previous example, we determine the correspond-
ing optimal values of Ci.

C0 = −0.666669, C1 = 0.0000909446, C2 = 0.221205 C3 = 0.00521847.

C4 = −0.0880682, C5 = 0.0194006, C6 = 0.0146485 C7 = −0.00582658.

Utilizing these values, the first–order MOHAM solution is constructed, which demon-
strates both efficiency and reliability in approximating the given problem.

ũ(x) = 1. + 0.0624983x2 + 0.0000213151x3 − 0.000119261x4 + 0.000369055x5

−0.000671864x6 + 0.000707815x7 − 0.000369332x8 + 0.0000125056x9

+0.0000762941x10 − 0.0000248292x11 (32)

Table 2: Comparison between Exact and MOHAM solutions with absolute error.

x Exact OHAM MOHAM
Solution Absolute Error Absolute Error

0.2 1.0025 4.76×10−7 3.46 × 10−9

0.4 1.0100 249×10−7 1.746 × 10−9

0.6 1.0225 1.01×10−7 1.37 × 10−9

0.8 1.0400 6.14×10−8 4.70 × 10−10

1.0 1.0625 0.00 2.22 × 10−16

The results in Table 2 demonstrate a remarkable finding when comparing OHAM
and MOHAM. It is important to note that the OHAM results were obtained using a
third–order approximation, whereas MOHAM achieved its results using only a first–order
approximation. Despite this significant difference in approximation order, MOHAM pro-
vides much smaller absolute errors at all tested values of x. For instance, at x = 0.2, the
error of OHAM is 4.76 × 10−7, while MOHAM reduces this to 3.46 × 10−9, representing
an improvement of nearly two orders of magnitude. A similar pattern is observed across
all points: at x = 0.6, OHAM yields an error of 1.01 × 10−7, whereas MOHAM achieves
1.37×10−9. Even at x = 0.8, where the error of OHAM is already small (6.14×10−8), MO-
HAM still provides a lower error (4.70×10−10). This comparison highlights that MOHAM
achieves higher accuracy with fewer approximation terms, thereby reducing computational
effort while ensuring stable and reliable convergence throughout the domain. Thus, the
modified formulation of MOHAM outperforms OHAM not only in precision but also in
efficiency, making it a superior method for solving such boundary value problems.



N. Anakira et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7128 9 of 11

Conclusion

In this work, we have successfully applied MOHAM to singular two–point boundary
value problems. The modification introduced in the auxiliary function and the homo-
topy construction has demonstrated significant improvements over the standard OHAM.
Numerical experiments clearly show that MOHAM achieves higher accuracy even with a
first–order approximation, whereas OHAM requires up to three terms to reach a compa-
rable level of precision. This reduction in approximation order highlights the efficiency of
MOHAM in minimizing computational effort while maintaining excellent reliability. The
residual minimization strategy used ensures uniformly small errors in the domain, which
confirms the robustness of the method. Moreover, MOHAM preserves the flexibility and
simplicity of OHAM while enhancing convergence speed and stability. These results sug-
gest that MOHAM provides an effective analytical tool for handling this type of equations
that are otherwise challenging to solve. With its strong balance of accuracy and efficiency,
MOHAM can serve as a powerful alternative to existing analytical and numerical methods.
Future work may extend MOHAM to multistage frameworks and more complex nonlinear
systems.

Acknowledgements

The author gratefully acknowledges Jadara University for supporting this research and
for encouraging scientific work.

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