EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6215
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Analytic and Numerical Approaches for Solving
Nonlinear Painlevé Equations I and II

Sangar Mohammed Hasan1,∗, Sizar Abid Mohammed1, Ramadhan Abid Mohammed1

1 Department of Mathematics, College of Basic Education, University of Duhok,
Kurdistan Region, Iraq

Abstract. Nonlinear Painlevé equations play a pivotal role in various branches of mathematical
physics, integrable systems, and applied mathematics. These equations, characterized by their
complex and highly nonlinear nature, present significant challenges for analytical and numerical
investigation. In this paper, we develop and present both analytical and numerical solutions for
specific classes of nonlinear Painlevé equations. The analytical approach employs transformation
techniques, perturbative expansions, and exact solution methods where applicable, while the nu-
merical solutions are obtained using robust algorithms such as finite difference schemes, spectral
methods, and iterative solvers. We validate the numerical results by comparing them with known
analytical solutions and explore their accuracy, stability, and convergence properties. Further-
more, we discuss the implications of these solutions in physical models and highlight the intricate
structures exhibited by the solutions, such as pole dynamics and asymptotic behaviors. This
work contributes to the broader understanding of Painlevé equations and provides a framework for
tackling similar nonlinear differential equations in applied contexts.

2020 Mathematics Subject Classifications: 34M55, 34A34, 65L05, 65L06, 35Q99

Key Words and Phrases: Painlevé equations, DTM, MADM, LDM, NDM, Finite difference
method, Runge-Kutta method

1. Introduction

The Painlevé equations were first introduced between 1895 and 1910 through the work
of two French mathematicians, Paul Painlevé and Bertrand Gambier. They were identified
as second-order differential equations that play a crucial role in various fields of mathemat-
ics and physics. They have found applications in areas such as nonlinear waves, plasma
physics, statistical mechanics, fiber optics, and more [1–4]. Despite their importance,
Painlevé equations are highly nonlinear and do not admit general closed-form solutions,
and traditional methods such as separation of variables or perturbation techniques often

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6215

Email addresses: sangar.mohammedhasan@uod.ac (S. Mohammed),
sizar.mohammed@uod.ac (S. Abid) , ramadhan.hajani@uod.ac (R. Abid)

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



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 2 of 38

fail due to the equations’ complexity. As a result, researchers have focused on devel-
oping analytical and numerical approaches to approximate and analyze these equations
effectively. Numerous analytical and numerical methods have been successfully developed
to solve the Painlevé equations. For example, Differential Transform Method (DTM)
[5, 6], Modified Adomian decomposition method (MADM) [7, 8], Laplace decomposi-
tion Method (LDM) [9, 10] and Natural decomposition Method [11, 12], Finite difference
method [13, 14], Runge Kutta method [15, 16].

This paper aims to develop and compare analytical and numerical techniques for solv-
ing Painlevé I and II equations. It contributes to the broader understanding of providing
reliable tools and bridging the gap between analytical and numerical techniques for applied
scientists and engineers to study nonlinear differential equations in real-world applications.

This paper is organized as follows: Section 2 introduces the Analytical methods. Sub-
section 2.1 Differential Transform Method. In subsection 2.2. Modified Adomian decom-
position Method. Subsection 2.3. Laplace decomposition Method. In subsection 2.4.
Natural Decomposition Method. In Section 3 Numerical Methods. Subsection 3.1. Finite
difference method. In subsection 3.2. Runge Kutta method. In section 4 Comparative
Analysis of Analytical and Numerical Methods. Finally, Section 5 offers concluding re-
marks for the paper.

Painlevé equations of the first and second types will be defined by the following for-
mulas:

y′′(x) = 6y2(x) + x, 0 < x < 1 (1)

With the given initial conditions: y(0) = 0, y′(0) = 1.

y′′(x) = 2y3(x) + xy(x) + µ, 0 < x < 1 (2)

With the initial conditions: y(0) = 1, y′(0) = 0, where µ is a known parameter.

Error analysis

Since the exact solutions for Painlevé I and II are unknown, we utilized the maximal
error remainderMERn to evaluate the accuracy of the approximate solutions. TheMERn

for DTM, MADM, LDM, and NDM was computed using the software Mathematica code
over the domain [0.01, 0.1].

ERn = L(y)−N(y)− f(x). (3)

The maximal error remainder is

MERn = max
0.01≤x≤0.1

|ERn(x)|. (4)

The selection of the interval [0.01, 0.1] for the maximal error remainder (MER) analysis
is based on consideration of analytic stability and accuracy. Limiting the analysis to this
range allows for a more precise assessment of the approximation methods. Furthermore,



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 3 of 38

this interval is specifically chosen to focus on the region where analytical approximations
are most effective for accuracy evaluation, as extending the domain further could lead to
additional computational complexities without significantly impacting the overall conclu-
sions.

2. Analytical methods for solving Painlevé Equations I, II

Analytical methods provide powerful techniques for solving nonlinear ordinary differen-
tial equations. In this section, we apply the Differential Transform Method (DTM), Mod-
ified Adomian Decomposition Method (MADM), Laplace Decomposition Method (LDM),
and Natural Decomposition Method (NDM) to obtain approximate or closed-form solu-
tions. These methods simplify nonlinear equations by transforming them into more man-
ageable forms, offering advantages in accuracy, convergence, and efficiency. The objective
is to analyze and compare their effectiveness in solving Painlevé equations, highlighting
their role in nonlinear differential equation research [1, 2].

2.1. Differential Transform Method (DTM)

The Differential Transform, initially introduced by Zhou [5, 6, 17–20] is a technique
for solving differential equations. This method determines the coefficients of a Taylor
series of the function by solving a recursive equation induced by the given differential
equation. The Differential Transform Method has been effectively applied to solve initial
value problems represented by strongly ordinary differential equations [21–25].

The differential transform of a function y(x) is defined as follows:

Y (k) =
1

k!

(
dky(x)

dxk

)∣∣∣∣
x=0

. (5)

Where y(x) is the original function and Y (k) is the transformed function. Differential
inverse transform of Y (k) is defined as:

y(x) =
∞∑
k=0

Y (k)xk ≈ yn(x) =
n∑

k=0

Y (k)xk. (6)

By substituting (5) in (6) we get

y(x) =
∞∑
k=0

1

k!

(
dky(x)

dxk

)
x=0

. (7)

The differential transform verified the following properties: [26, 27]

• If u(x) = u1(x)± u2(x), then U(k) = U1(k)± U2(k).

• If u(x) = cu1(x) then U(k) = cU1(k), where c is constant.

• If u(x) = dnu1(x)
dxn , then U(k) = (k+n)!

k! U1(k + n).



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 4 of 38

• If u(x) = u1(x)u2(x), then U(k) =
∑k

r=0 U1(r)U2(k − r).

• If u(x) = u1(x)u2(x) . . . un(x), then

U(k) =
k∑

rn−1=0

rn−1∑
rn−2=0

. . .

r3∑
r2=0

r2∑
r1=0

U1(r1)U2(r2−r1) . . . Un−1(rn−1−rn−2)Un(k−rn−1).

• If u(x) = axm, then U(k) = aδ(k −m), where

δ(k −m) =

{
1, if k = m
0, if k ̸= m

• If u(x) = u1(x)
du2(x)
dx , then U(k) =

∑k
r=0(k − r + 1)U1(r)U2(k − r + 1).

2.1.1. The DTM for solving the Painlevé I differential equation

Rewrite (1)
d2y

dx2
= 6y2 + x, y(0) = 0, y′(0) = 1.

Assume y(x) can be expressed as a power series:

y(x) =

∞∑
k=0

Ykx
k.

From the initial conditions:

Y0 = y(0) = 0, Y1 = y′(0) = 1.

Apply the DTM to the differential equation.
The second derivative is:

d2y

dx2
=

∞∑
k=0

(k + 2)(k + 1)Yk+2x
k. (8)

6y2 = 6

( ∞∑
k=0

Ykx
k

)2

= 6
∞∑
k=0

(
k∑

m=0

YmYk−m

)
xk. (9)

x =

∞∑
k=0

δ(k − 1)xk. (10)

Substitute (8),(9) and (10) into the equation.

∞∑
k=0

(k + 2)(k + 1)Yk+2x
k = 6

( ∞∑
k=0

(
k∑

m=0

YmYk−m

)
xk

)
+

∞∑
k=0

δ(k − 1)xk.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 5 of 38

For each k we get

(k + 2)(k + 1)Yk+2 = 6
k∑

m=0

YmYk−m + δ(k − 1). (11)

k = 0

(0 + 2)(0 + 1)Y0+2 = 6
0∑

m=0

YmY0−m + δ(0− 1).

2Y2 = 6Y0Y0 + 0 = 0.

Y2 = 0.

k = 1

(1 + 2)(1 + 1)Y1+2 = 6
1∑

m=0

YmY1−m + δ(1− 1).

6Y3 = 6(Y0Y1 + Y1Y0) + 1 = 6(0(1) + 1(0)) + 1 = 1.

Y3 =
1

6
.

k = 2

(2 + 2)(2 + 1)Y2+2 = 6

2∑
m=0

YmY2−m + δ(2− 1).

12Y4 = 6(Y0Y2 + Y1Y1 + Y2Y0) + 0 = 6(0(1) + 1(0)) = 6.

Y4 =
1

2
.

k = 3

(3 + 2)(3 + 1)Y3+2 = 6
3∑

m=0

YmY3−m + δ(3− 1).

20Y5 = 6(Y0Y3 + Y1Y2 + Y2Y1 + Y3Y0) + 0 = 6

(
0 + 1(0) + 0(1) +

1

6
(0)

)
= 0.

Y5 = 0.

...

Y0 = 0, Y1 = 1, Y2 = 0, Y3 =
1

6
, Y4 =

1

2
, Y5 = 0, Y6 =

1

15
, Y7 =

1

7
,

Y8 =
1

336
, Y9 =

1

40
, Y10 =

1

28
, . . .

y(x) = Y0 + Y1x+ Y2x
2 + Y3x

3 + Y4x
4 + Y5x

5 + Y6x
6 + Y7x

7 + Y8x
8 + Y9x

9 + . . .

y(x) = x+
x3

6
+

x4

2
+

x6

15
+

x7

7
+

x8

336
+

13x9

2160
+

x10

28
+ . . .



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 6 of 38

The error remainder function is evaluated:

ERn = y′′n(x)− 6y2n(x)− x.

And the MERn is:
MERn = max

0.01≤x≤0.1
|ERn(x)|.

In Table 1 and Figure 1 illustrate the convergence of the solution using MERn versus
n, demonstrating the convergence behavior of the Differential Transform Method (DTM)
in solving the first Painlevé equation. The error decreases exponentially as n increases,
with the most significant reduction occurring between n = 1 and n = 3. Beyond n = 3,
the error stabilizes around 10−17, indicating that further iterations do not significantly
improve accuracy due to machine precision limitations. The stabilization of error values
for n = 4 and n = 5 implies that additional iterations beyond this point may be compu-
tationally unnecessary. This suggests that DTM achieves high accuracy with a relatively
small number of iterations, making it an efficient method for solving nonlinear differential
equations.

Table 1: The maximum residual error: MERn by the DTM where n = 1, · · · , 5.

n MERn

1 3.790200× 10−7

2 1.116379× 10−10

3 1.040834× 10−16

4 4.857226× 10−17

5 4.857226× 10−17

Figure 1: The logarithmic plot of MER n (DTM, First Painlevé Equation). .



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 7 of 38

2.1.2. The DTM for solving the Painlevé II differential equation

Rewrite [2]
d2y

dx2
= 2y3 + xy + µ, y(0) = 1, y′(0) = 0.

Assume y(x) can be expressed as a power series:

y(x) =
∞∑
k=0

Ykx
k.

From the initial conditions:

Y0 = y(0) = 1, Y1 = y′(0) = 0.

Apply the DTM to the differential equation:

d2y

dx2
=

∞∑
k=0

(k + 2)(k + 1)Yk+2x
k. (12)

2y3 = 2

( ∞∑
k=0

Ykx
k

)3

= 2

∞∑
k=0

(
k∑

m=0

m∑
n=0

YnYm−nYk−m

)
xk. (13)

xy =
k∑

k=1

δ(k − 1)Yk−1x
k. (14)

µx0 = µ
∞∑
k=0

δ(k − 0)xk. (15)

Substitute (12), (13), (14) and (15) into the equation:

∞∑
k=0

(k + 2)(k + 1)Yk+2x
k = 2

∞∑
k=0

(
k∑

m=0

m∑
n=0

YnYm−nYk−m

)
xk . . .

+
∞∑
k=1

δ(k − 1)Yk−1x
k + µ

∞∑
k=0

δ(k − 0)xk.

For each k we get:

Yk+2 =
1

(k + 2)(k + 1)

(
2

k∑
m=0

m∑
n=0

YnYm−nYk−m + δ(k − 1)Yk−1 + µδ(k − 0)

)
. (16)

k = 0

Y2 =
2 + µ

2
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 8 of 38

k = 1

Y1+2 =
1

(1 + 2)(1 + 1)

(
2

1∑
m=0

m∑
n=0

YnYm−nY1−m + δ(1− 1)Y1−1 + µδ(1− 0)

)
.

Y3 =
1

6
.

k = 2

Y2+2 =
1

(2 + 2)(2 + 1)

(
2

2∑
m=0

m∑
n=0

YnYm−nY2−m + δ(2− 1)Y2−1 + µδ(2− 0)

)
.

Y4 =
2 + µ

4
,

Y5 =
1

20
,

Y6 =
1

20
(2 + µ+ (2 + µ)2),

Y7 =
3 + µ

84
,

...

y(x) = Y0 + Y1x+ Y2x
2 + Y3x

3 + Y4x
4 + Y5x

5 + Y6x
6 + Y7x

7 + · · ·

y(x) = 1 +
2 + µ

2
x2 +

1

6
x3 +

2 + µ

4
x4 +

1

20
x5 +

1

20
(2 + µ+ (2 + µ)2)x6 +

3 + µ

84
x7 + · · ·

The error remainder function is evaluated:

ERn = y′′n(x)− 2y3n(x)− xyn(x)− µ.

And the MERn is:
MERn = max

0.01≤x≤0.1
|ERn(x)|.

In Table 2 and Figure 2 show the convergence of the solution using MERn versus n
demonstrates the convergence behavior of the Differential Transform Method (DTM) in
solving the second Painlevé equation. The error decreases significantly from n = 1 to
n = 2, showing a rapid initial improvement in accuracy. However, from n = 3 onward,
the reduction in error becomes much smaller, indicating a slower rate of convergence. The
stabilization of MERn at approximately 0.001537 for n = 4 and n = 5 suggests that
further iterations do not lead to a significant decrease in error, likely due to numerical
limitations or the inherent accuracy of the method. Overall, the DTM achieves rapid initial
convergence but reaches a saturation point where further improvements are marginal.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 9 of 38

Table 2: The maximum residual error: MERn by the DTM where n = 1, · · · , 5.

n MERn

1 0.09390383
2 0.00337824
3 0.00156204
4 0.00153786
5 0.00153758

Figure 2: The logarithmic plot of MERn (DTM, Second Painlevé Equation).

2.2. Modified Adomian decomposition method (MADM)

The Adomian Decomposition Method (ADM), introduced by George Adomian in the
1980s, [7], revolutionized the solution of nonlinear differential equations by providing an ef-
ficient decomposition approach without linearization or perturbation [8]. Over the decades,
ADM has been widely applied in various fields of applied mathematics and physics due to
its ability to handle complex nonlinearities. Building upon this foundation, the Modified
Adomian Decomposition Method (MADM) enhances ADM by improving convergence and
accuracy, making it particularly effective for solving highly nonlinear problems. In this
subsection, we focus on the application of MADM to nonlinear Painlevé equations I and
II [28–34].

2.2.1. The MADM for solving Painlevé I Equation

Rewrite (1)
y′′(x) = 6y2(x) + x, y(0) = 0, y′(0) = 1. (17)

Rewrite in the form Ly = g(x)− F (y):

Ly = x+ 6y2.

The differential operator L is defined by:

L = e−
∫
p(x) dx d

dx

(
e
∫
p(x) dx d

dx

)
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 10 of 38

L =
d2

dx2
.

Applying the inverse operator L−1, we get:

y(x) = φ(x) + L−1(x) + L−1(6y2).

Where satisfies Lφ(x) = 0 from initial conditions.

φ(x) = y(0) + y′(0)x = x.

y(x) = x+ L−1(x) + L−1(6y2).

Recall that the MADM introduces the solution y(x) and the nonlinear function F (y) by
infinite series:

y(x) =

∞∑
n=0

yn(x).

F (y) = y2.

F (y) =

∞∑
n=0

An.

∞∑
n=0

yn(x) = x+ L−1(x) + 6L−1
∞∑
n=0

An.

y0(x) = x+ L−1(x).

yn+1(x) = 6L−1[An].

y0(x) = x+

∫ x

0

∫ x

0
x dx dx.

y0(x) = x+
x3

6
.

y1(x) = 6L−1[A0].

Where
A0 = y20.

A0 =

(
x+

x3

6

)2

.

Thus

A0 = x2 +
x4

3
+

x6

36
.

Now compute L−1[A0].

y1(x) = 6L−1[A0] = 6

∫ x

0

∫ x

0

(
x2 +

x4

3
+

x6

36

)
dx dx.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 11 of 38

Thus

y1(x) =
x4

2
+

x6

15
+

x8

336
.

A1 = 2y0y1.

A1 = 2

(
x+

x3

6

)(
x4

2
+

x6

15
+

x8

336

)
.

A1 = x5 +
3x7

10
+

71x9

2520
+

x11

1008
.

y2(x) = 6L−1[A1] = 6

∫ x

0

∫ x

0
(A1) dx dx.

y2(x) = 6L−1[A1] = 6

∫ x

0

∫ x

0

(
x5 +

3x7

10
+

71x9

2520
+

x11

1008

)
dx dx.

y2(x) =
x7

7
+

x9

40
+

71x11

46200
+

x13

26208
.

...

Then

y(x) =
∞∑
n=0

yn(x).

y(x) = x+
x3

6
+

x4

2
+

x6

15
+

x8

336
+

x7

7
+

x9

40
+

71x11

46200
+

x13

26208
+ · · ·

The remainder function of error is evaluated:

ERn = y′′n(x)− 6y2n(x)− x. (18)

And the MERn is:
MERn = max

0.01≤x≤0.1
|ERn(x)|. (19)

In Table 3 and Figure 3 show the convergence of the solution using MERn versus
n demonstrates the rapid convergence of the Modified Adomian Decomposition Method
(MADM) for solving the first Painlevé equation, the most significant error reduction oc-
curs between n = 1 and n = 3, after which the error drops to near n = 5. The error
decreases exponentially with each iteration, indicating that MADM provides highly accu-
rate results with relatively few terms. The consistent decrease in error on the logarithmic
scale confirms the efficiency and robustness of MADM in solving nonlinear differential
equations.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 12 of 38

Table 3: The maximum residual error: MERn by the MADM where n = 1, · · · , 5.

n MERn

1 0.0000601952
2 3.22501× 10−8

3 1.29031× 10−11

4 4.40620× 10−15

5 6.93889× 10−18

Figure 3: The logarithmic plot of MERn. (MADM, First Painlevé Equation).

2.2.2. The MADM for solving Painlevé II Equation

Rewrite (2)
y′′(x) = 2y3 + xy + µ. (20)

The initial conditions:
y(0) = 1, y′(0) = 0.

Rewrote (20) in the form: Ly = g(x)− F (x, y).

y′′(x)− 2y3 − xy = µ.

Ly = µ− (−2y3 − xy).

Ly = µ+ 2y3 + xy.

The differential operator L is defined by

L = e−
∫
p(x) dx d

dx

(
e
∫
p(x) dx d

dx

)
.

L =
d2

dx2
.

Applying the inverse operator L−1, we get:

y(x) = φ(x) + L−1(µ) + L−1(2y3 + xy).



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 13 of 38

Where satisfies Lφ(x) = 0 from initial conductions:

φ(x) = y(0) + y′(0)x = 1.

y(x) = 1 + L−1(µ) + L−1(2y3 + xy).

Recall that the ADM introduces the solution y(x) and the nonlinear function F (x, y) by
infinite series:

y(x) =
∞∑
n=0

yn(x).

F (x, y) = 2y3 + xy.

F (x, y) =
∞∑
n=0

An.

∞∑
n=0

yn(x) = 1 + L−1(µ) + L−1
∞∑
n=0

An.

y0(x) = 1 + L−1(µ).

y0(x) = 1 +

∫ x

0

∫ x

0
µdx dx.

y0(x) = 1 +
x2µ

2
.

y1(x) = L−1[A0].

A0 = 2y30 + xy0.

A0 = 2 + x+ 3x2µ+
3x4µ2

2
+

x6µ3

4
+

x3µ

2
.

Now compute L−1[A0]:

y1(x) = L−1[A0] =

∫ x

0

∫ x

0

(
2 + x+ 3x2µ+

3x4µ2

2
+

x6µ3

4
+

x3µ

2

)
dx dx.

Performing the integration:

y1(x) = x2 +
x3

6
+

x4µ

4
+

x5µ

40
+

x6µ2

20
+

x8µ3

224
.

A1 = y1(6y
2
0 + x).

A1 = 2x3 +
x4

2
+

x5

30
+

3x5µ

2
+

7x6µ

30
+

x7µ

280
+

33x7µ2

70
. . .

+
9x8µ2

160
+

131x9µ3

1680
+

47x10µ3

11200
+

57x11µ4

6160
+

3x13µ5

5824
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 14 of 38

y2(x) = L−1[A1] =

∫ x

0

∫ x

0
(A1) dx dx.

y2(x) =
x4

2
+

x5

10
+

x6

180
+

x6µ

4
+

x7µ

30
+

x8µ

2240
+

33x8µ2

560
. . .

+
x9µ2

160
+

131x10µ3

16800
+

47x11µ3

123200
+

19x12µ4

24640
+

3x14µ5

81536
.

...

yn+1(x) = L−1[An].

y(x) =
∞∑
n=0

yn(x).

y(x) = y0(x) + y1(x) + y2(x) + · · ·

y(x) = 1 +
x2µ

2
+ x2 +

x3

6
+

x4µ

4
+

x5µ

40
+

x6µ2

20
+

x8µ3

224
+

x4

2
. . .

+
x5

10
+

x6

180
+

x6µ

4
+

x7µ

30
+

x8µ

2240
+

33x8µ2

560
+

x9µ2

160
. . .

+
131x10µ3

16800
+

47x11µ3

123200
+

19x12µ4

24640
+

3x14µ5

81536
+ · · ·

The remainder function of error is evaluated:

ERn = y′′n(x)− 2y3n(x)− xyn(x)− µ.

And the MERn is:
MERn = max

0.01≤x≤0.1
|ERn(x)|.

In Table 4 and Figure 4 show the convergence of the solution using MERn, the error
decreases with increasing n. The results indicate a rapid reduction in error, with the
most significant improvement occurring between n = 1 and n = 3. By n = 5, the error
reaches an extremely low magnitude, around 10−10, confirming the high accuracy of the
method. This behavior highlights the computational efficiency of MADM, as it provides
highly accurate solutions to nonlinear differential equations with minimal computational
effort.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 15 of 38

Table 4: The maximum residual error: MERn by the MADM, where n = 1, . . . , 5.

n MERn

1 0.06341250000
2 0.00095060500
3 0.00001041000
4 9.77952× 10−8

5 8.40300× 10−10

Figure 4: The logarithmic plot of MERn. (MADM, Second Painlevé Equation).

2.3. Laplace decomposition Method (LDM)

The Laplace Decomposition Method (LDM) is an analytical technique that combines
the Laplace transform and the Adomian Decomposition Method (ADM) to solve nonlin-
ear differential equations efficiently. The Laplace transform, introduced by Pierre-Simon
Laplace in the 18th century, [9, 10] is widely used for transforming differential equations
into algebraic equations, simplifying their solutions. Integrating Adomian decomposition
methods and Laplace transform offers a reliable approach to solving nonlinear ordinary
differential equations, particularly those encountered in mathematical physics and engi-
neering applications [35–38].

In this subsection, we apply LDM to obtain analytical solutions for Painlevé equations,
by leveraging the strengths of both the Laplace transform and the Adomian Decomposition
Method. LDM provides a systematic and effective framework for deriving solutions in a
series form with rapid convergence. We aim to explore the effectiveness of LDM in solving
these equations and contribute to the broader understanding of analytical techniques for
nonlinear differential equations.

2.3.1. The LDM for solving Painlevé equation I

Rewrite (1)
y′′(x) = 6y2 + x. (21)

With the initial conditions:

y(0) = 0, y′(0) = 1. (22)



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 16 of 38

Now, apply the Laplace transform on both sides of (21).
With initial conditions:

L[y′′] = L[6y2] + L[x].

s2Y (s) + y(0)s− y′(0) = L[6y2] + L[x]. (23)

Substituting (22) into (23):

Y (s) =
1

s2
+

1

s4
+

6

s2
L[y2]. (24)

Taking the inverse of the Laplace transform

L−1[Y (s)] = L−1

[
1

s2
+

1

s4
+

6

s2
L[y2]

]
. (25)

y(x) = x+
x3

3!
+ L−1

[
6

s2
L[y2]

]
. (26)

Equation (26) can be written as

y(x) = y0(x) + L−1

[
6

s2
L[An]

]
, n ≥ 0 (27)

y0(x) = x+
x3

3!
.

From (27), we conclude that

yn+1(x) = L−1

[
6

s2
L[An]

]
, n ≥ 0 (28)

The terms becomes

y1(x) = L−1

[
6

s2
L[A0]

]
.

y2(x) = L−1

[
6

s2
L[A1]

]
.

y3(x) = L−1

[
6

s2
L[A2]

]
.

...

Therefore, from (28), the other remaining terms of the function y(x) can be easily
calculated as follows:

y0(x) = x+
x3

3!
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 17 of 38

y1(x) =
x4

2
+

x6

15
+

x8

336
.

y2(x) =
x7

7
+

x9

40
+

71x11

46200
+

x13

26208
.

y3(x) =
x10

28
+

23x12

3080
+

5219x14

8408400
+

3551x16

1441440000
+

95x18

224550144
.

y4(x) =
3x13

364
+

131x15

64680
+

19867x17

95295200
+

163469x19

14378364000
+

163451x21

491203440000
+

31x23

7101398304
.

...

y(x) =

∞∑
n=0

yn(x). (29)

y(x) = x+
x3

6
+

x4

2
+

x6

15
+

x8

336
+

x7

7
+

x9

40
+

71x11

46200
+

x13

26208
+

x10

28
. . .

+
23x12

3080
+

5219x14

8408400
+

3551x16

1441440000
+

95x18

224550144
+

3x13

364
+

131x15

64680
. . .

+
19867x17

95295200
+

163469x19

14378364000
+

163451x21

491203440000
+

31x23

7101398304
+ · · ·

The remainder function of error is evaluated:

ERn = y′′n(x)− 6y2n(x)− x.

MERn = max
0.01≤x≤0.1

|ERn(x)|.

In Table 5 and Figure 5 show the convergence of the solution using MERn against
n for the first Painlevé equation by LDM, the plot demonstrates the rapid decrease in
error with increasing n, confirming the convergence of the Laplace Decomposition Method
(LDM). The nearly linear trend in the logarithmic scale indicates an exponential decay of
the maximum residual error, showcasing the method’s high accuracy and efficiency.

Table 5: The maximum residual error: MERn by the LDM, where n = 1, . . . , 5.

n MERn

1 0.0000601952
2 3.22501× 10−8

3 1.29031× 10−11

4 4.4062× 10−15

5 6.93889× 10−18



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 18 of 38

Figure 5: logarithmic plot of MERn (LDM, First Painlevé Equation).

2.3.2. The LDM for solving Painlevé equation II

Rewrite (2)
y′′(x) = 2y3 + xy + µ. (30)

The initial conditions:
y(0) = 1, y′(0) = 0. (31)

Now, apply Laplace transform on both sides of (30).
With initial conditions:

L[y′′] = L[2y3 + xy + µ].

y′′ = s2Y (s)− sy(0)− y′(0). (32)

Substituting (31) into (32), we have:

s2Y (s)− s = L[2y3] + L[xy] + L[µ]. (33)

Y (s) =
1

s
+

1

s2
L[2y3] + 1

s2
L[xy] + µ

s3
. (34)

Equation (34) becomes:

L−1[Y (s)] = L−1

[
1

s
+

1

s2
L[2y3] + 1

s2
L[xy] + µ

s3

]
.

L−1[Y (s)] = L−1

[
1

s
+

µ

s3
+

1

s2
L[2y3 + xy]

]
. (35)

y(x) = 1 +
x2µ

2
+ L−1

[
1

s2
L[An]

]
. (36)

y0(x) = 1 +
x2µ

2
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 19 of 38

From (36) we conclude that:

yn+1(x) = L−1

[
1

s2
L[An]

]
, n ≥ 0.

Then

y1(x) = L−1

[
1

s2
L[A0]

]
.

A0 = F (y0).

F (y) = 2y3 + xy.

y1(x) = L−1

[
1

s2
L[2y30 + xy0]

]
.

y1(x) = L−1

[
2

s3
+

1

s4
+

3(2!)µ

s5
+

3!µ

2 s6
+

3(4!)µ2

2 s7
+

6!µ3

4 s9

]
.

y1(x) = x2 +
x3

6
+

x4µ

4
+

x5µ

40
+

x6µ2

20
+

x8µ3

224
.

A1 = 2x3 +
x4

2
+

x5

30
+

3x5µ

4
+

7x6µ

30
+

x7µ

280
+

33x7µ2

70
. . .

+
9x8µ2

160
+

131x9µ3

1680
+

47x10µ3

11200
+

57x11µ4

6160
+

3x13µ5

5824
.

y2(x) = L−1

[
1

s2
L[A1]

]
.

y2(x) =
x4

2
+

x5

10
+

x6

180
+

x6µ

4
+

x7µ

30
+

x8µ

2240
+

33x8µ2

560
. . .

+
x9µ2

160
+

131x10µ3

16800
+

47x11µ3

123200
+

19x12µ4

24640
+

3x14µ5

81536
.

...

y(x) =
∞∑
n=0

yn(x).

y(x) = 1 +
x2µ

2
+ x2 +

x3

6
+

x4µ

4
+

x5µ

40
+

x6µ2

20
+

x8µ3

224
+

x4

2
. . .

+
x5

10
+

x6

180
+

x6µ

4
+

x7µ

30
+

x8µ

2240
+

33x8µ2

560
+

x9µ2

160
. . .

+
131x10µ3

16800
+

47x11µ3

123200
+

19x12µ4

24640
+

3x14µ5

81536
+ · · ·



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 20 of 38

The remainder function of error is evaluated:

ERn = y′′n(x)− 2y3n(x)− xyn(x)− µ.

And the MERn is:
MERn = max

0.01≤x≤0.1
|ERn(x)|.

In Table 6 and Figure 6 show the convergence of the solution using MERn against n
for the second Painlevé equation by LDM. The data shows a clear trend of rapid error
reduction as n increases. Initially, at n = 1, the error is relatively large at 0.0634125.
However, as n increases to 2 and 3, the error drops significantly, and at n = 5, it reaches
an extremely small value of 8.40300× 10−10. This exponential decrease in error suggests
that the LDM exhibits fast convergence, making it a highly effective method for solving
the second Painlevé equation with high accuracy.

Table 6: The maximum residual error: MERn by the LDM, where n = 1, . . . , 5.

n MERn

1 0.0634125
2 0.000950605
3 0.00001041
4 9.77952× 10−8

5 8.40300× 10−10

Figure 6: logarithmic plot of MERn (LDM, Second Painlevé Equation).

2.4. Natural Decomposition Method (NDM)

The Natural Decomposition Method (NDM) was developed in the early 21st century
as an advancement in analytical techniques for solving nonlinear differential equations.
Research on NDM began appearing in mathematical literature in the 2000s and 2010s
as an improvement over existing decomposition methods [11, 12]. It was introduced to
enhance the accuracy and convergence of solutions for highly nonlinear problems without
requiring linearization or perturbation. Since its introduction, NDM has been applied to



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 21 of 38

various fields, including fluid dynamics, mathematical physics, and engineering. In this
subsection we use NDM to solving Painlevé equations I and II. [39–41].

2.4.1. The NDM for solving the Painlevé I differential equation

Rewrite [1]
y′′(x) = 6y2 + x. (37)

y(0) = 0, y′(0) = 1. (38)

By applying the N-transform on both sides of (37):

N+
[
y′′(x)

]
= N+

[
6y2 + x

]
. (39)

Using the properties of N-transform we obtain that

s2

t2
y(s, t)− s

t2
y(0)− 1

t
y′(0) = N+[6y2] +

t

s2
. (40)

Substituting (38) into (40), we have

y(s, t) =
t

s2
+

t3

s4
+

t2

s2
N+[6y2]. (41)

Now, applying the inverse of N-transform, we have

N−1[y(s, t)] = N−1

[
t

s2
+

t3

s4
+

6t2

s2
N+[y2]

]
. (42)

Equation (42) becomes

y(x) = x+
x3

3!
+N−1

[
6t2

s2
N+[y2]

]
. (43)

Equation (43) can be written as

y(x) = x+
x3

3!
+N−1

[
6t2

s2
N+[An]

]
. (44)

y0(x) = x+
x3

3!
.

From (44), we can conclude that

yn+1(x) = N−1

[
t2

s2
N+[An]

]
, n ≥ 0. (45)

The terms become:

y1(x) = N−1

[
t2

s2
N+[A0]

]
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 22 of 38

y2(x) = N−1

[
t2

s2
N+[A1]

]
.

...

Therefore, from (45), the other remaining terms of the function y(x) can be easily calcu-
lated as follows:

y1(x) =
1

2
x4 +

1

15
x6 +

1

336
x8.

y2(x) =
1

7
x7 +

1

40
x9 +

71

46200
x11 +

1

26200
x13.

y3(x) =
1

28
x10 +

23

3080
x12 +

5219

8408400
x14 +

3551

144144000
x16 +

95

224550144
x18.

y4(x) =
3

364
x13 +

131

64680
x15 +

19867

95295200
x17 +

163469

14378364000
x19 . . .

+
163451

491203440000
x21 +

131

7101398304
x23.

In this way, we obtained the approximate result of the function y(x), which is given as

y(x) =
∞∑
n=0

yn(x). (46)

y(x) = x+
x3

6
+

1

2
x4 +

1

15
x6 +

1

336
x8 +

1

7
x7 +

1

40
x9 +

71

64200
x11 +

1

26208
x13 . . .

+
1

28
x10 +

23

3080
x12 +

5219

8408400
x14 +

3551

144144000
x16 +

95

224550144
x18 . . .

+
3

364
x13 +

131

64680
x15 +

19867

95295200
x17 +

163469

14378364000
x19 . . .

+
163451

491203440000
x21 +

131

7101398304
x23 + · · ·

The remainder function of error is evaluated:

ERn = y′′n(x)− 6y2n(x)− x.

MERn = max
0.01≤x≤0.1

|ERn(x)|.

In Table 7 and Figure 7 show the convergence of the solution using MERn from n = 1 to
n = 5 using the NDM. A significant reduction in error as n increases. Initially, for n = 1,
the error is relatively large, but it decreases drastically for n = 2 and continues to drop to
6.93889× 10−18 for n = 5. This exponential decay in error indicates that each additional
term in the LDM approximation significantly improves the accuracy of the solution. The
results confirm that LDM provides a highly effective approach for solving the second
Painlevé equation, with higher-order approximations yielding more precise solutions.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 23 of 38

Table 7: The maximum residual error: MERn by the LDM, where n = 1, . . . , 5.

n MERn

1 0.0000601952
2 3.22501× 10−8

3 1.29031× 10−11

4 4.4062× 10−15

5 6.93889× 10−18

Figure 7: The logarithmic plot of MERn (NDM. First Painlevé Equation).

2.4.2. The NDM for solving the Painlevé II differential equation

Rewrite (2)
y′′(x) = 2y3 + xy + µ. (47)

y(0) = 1, y′(0) = 0. (48)

By apply N-transform on both sides of (47):

N+[y′′(x)] = N+[2y3 + xy + µ]. (49)

By using the properties of N-transform we obtain:

s2

t2
y(s, t)− s

t2
y(0)− 1

t
y′(0) = N+[2y3 + xy] +

µ

s2
. (50)

Substituting (48) into (50), we have

y(s, t) =
1

s
+

µt2

s4
+

t2

s2
N+[2y3 + xy]. (51)

Now, apply the inverse of N-transform we have

N−1[y(s, t)] = N−1

[
1

s
+

µt2

s4
+

t2

s2
N+[2y3 + xy]

]
. (52)

y(x) = 1 +
µx2

2
+N−1

[
t2

s2
N+[2y3 + xy]

]
. (53)



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 24 of 38

Equation (53) can be written as

y(x) = 1 +
µx2

2
+N−1

[
t2

s2
N+[An]

]
, n ≥ 0 (54)

y0(x) = 1 +
µx2

2
.

From (54) we can conclude that

yn+1(x) = N−1

[
t2

s2
N+[An]

]
, n ≥ 0. (55)

The terms becomes

y1(x) = N−1

[
t2

s2
N+[A0]

]
.

y2(x) = N−1

[
t2

s2
N+[A1]

]
.

Therefore, from (55) the other remaining terms of the function y(x) can be easily calculated
as follows:

A0 = 2 + 3x2µ+
3

2
x4µ2 +

x6µ3

4
+ x+

x3µ

2
.

y1(x) = N−1

[
t2

s2
N+[A0]

]
.

y1(x) = x2 +
x4µ

4
+

x6µ2

20
+

x8µ3

224
+

x3

6
+

x5µ

40
.

A1 = 2x3 +
x4

2
+

x5

30
+

3x5µ

2
+

7x6µ

30
+

x7µ

280
+

33x7µ2

70
. . .

+
9x8µ2

160
+

131x9µ3

1680
+

47x10µ3

11200
+

57x11µ4

6160
+

3x13µ5

5824
.

y2(x) = N−1

[
t2

s2
N+[A1]

]
.

y2(x) = N−1
{

t2

s2
N+

[
2x3 + x4

2 + x5

30 + 3x5µ
2 + 7x6µ

30 + x7µ
280 . . .

+33x7µ2

70 + 9x8µ2

160 + 131x9µ3

1680 + 47x10µ3

11200 + 57x11µ4

6160 + 3x13µ5

5824

]}
.

y2(x) =
x4

2
+

x5

10
+

x6

180
+

x6µ

4
+

x7µ

30
+

x8µ

2240
+

33x8µ2

560
. . .



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 25 of 38

+
x9µ2

160
+

131x10µ3

16800
+

47x11µ3

123200
+

19x12µ4

24640
+

3x14µ5

81536
.

...

y(x) =
∞∑
n=0

yn(x).

y(x) = 1 +
x2µ

2
+ x2 +

x3

6
+

x4µ

4
+

x5µ

40
+

x6µ2

20
+

x8µ3

224
. . .

+
x4

2
+

x5

10
+

x6

180
+

x6µ

4
+

x7µ

30
+

x8µ

2240
+

33x8µ2

560
. . .

+
x9µ2

160
+

131x10µ3

16800
+

47x11µ3

123200
+

19x12µ4

24640
+

3x14µ5

81536
+ · · ·

The remainder function of error is evaluated:

ERn = y′′n(x)− 2y3n(x)− xyn(x)− µ.

And the MERn is:
MERn = max

0.01≤x≤0.1
|ERn(x)|.

In Table 8 and Figure 8 show the convergence of the solution using MERn for the sec-
ond Painlevé equation by NDM. Shows the exponential decay of error and nearly linear
downward trend on the logarithmic scale. The values indicate a rapid reduction in error
as n increases, demonstrating improved accuracy with higher-order approximations. Ini-
tially, for n = 1, the error is relatively large. However, as n increases, the error decreases
significantly, reaching a much smaller value at n = 5. This trend suggests that the NDM
effectively enhances the accuracy of the solution.

Table 8: The maximum residual error: MERn by the LDM, where n = 1, . . . , 5.

n MERn

1 0.0634125
2 0.000950605
3 0.00001041
4 9.77952× 10−8

5 8.40300× 10−10



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 26 of 38

Figure 8: The logarithmic plot of MERn (NDM. Second Painlevé Equation).

3. Numerical methods for solving Painlevé Equations I, II

The numerical methods provide practical means for approximating solutions where
analytical methods fall short. The Finite Difference Method discretizes the continuous
domain and approximates derivatives using difference quotients, which is particularly use-
ful for handling boundary value problems. [13, 14] The RK4 method, on the other hand, is
well-known for its robustness and high accuracy in solving initial value problems, making
it an excellent choice for exploring the dynamical behavior of these nonlinear equations
[42].

In this section, we focus exclusively on numerical approaches, specifically detailing the
implementation of the Finite Difference Method (FDM) and the fourth-order Runge-Kutta
Method (RK4) for solving the first and second Painlevé equations. We aim to provide clear
insights into their strengths and limitations when applied to the Painlevé equations I and
II.

3.1. Finite difference method (FDM)

The finite difference method (FDM) is a numerical approach for solving ordinary differ-
ential equations (ODEs), originally studied by Richardson in 1910. FDM became widely
used in the mid-20th century for solving engineering and physics problems. With ad-
vancements in computing, it remains a popular method due to its simplicity, efficiency,
and accuracy. [15, 16] The finite difference method works by discretizing the domain and
approximating derivatives using finite differences. This transforms the ODE into a system
of algebraic equations, making it easier to solve numerically. In this subsection, we use
FDM to solve Painlevé equations [43, 44].

3.1.1. The FDM method for solving the Painlevé I differential equation

Rewrite (1)
y′′(x) = 6y2 + x. (56)

y(0) = 0, y′(0) = 1.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 27 of 38

Divide the interval (0, 1) into four sub-intervals such that h = 1
4 and the pivot points

are at

x0 = 0, x1 =
1

4
, x2 =

1

2
, x3 =

3

4
, and x4 = 1.

Then the second differential equation is approximated as

y′′(x) =
yi+1 − 2yi + yi−1

h2
.

Substitute into (56)
1

h2
[yi+1 − 2yi + yi−1] = 6y2i + xi.

yi+1 − 2yi + yi−1 = h2(6y2i + xi). (57)

By using a forward difference approximation for the derivative to compute the initial
value for y1.

y′(x) =
1

h
[y(x+ h)− y(x)] .

Since
y0 = y(x0) = y(0).

y1 = y(x1) = y

(
1

4

)
.

y′(0) =
y(h)− y0

h
.

1 =
y1 − 0

h
.

y1 = h.

Thus we have

y0 = 0, y1 =
1

4
.

yi+1 − 2yi + yi−1 = h2(6y2i + xi).

For i = 1,

y2 − 2y1 + y0 =

(
1

4

)2

(6y21 + x1).

y2 =
69

128
.

For i = 2,

y3 − 2y2 + y1 =
3y22
8

+
x2
16

.

y3 =
126923

131072
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 28 of 38

Their solution gives

y1 =
1

4
, y2 =

69

128
, y3 =

126923

131072
.

...

3.1.2. The FDM for solving the Painlevé II differential equation

Rewrite (2)
y′′(t) = 2y3 + xy + µ. (58)

y(0) = 1, y′(0) = 0.

Divide the interval (0, 1) into four sub-intervals such that h = 1
4 and the pivot points

are at

x0 = 0, x1 =
1

4
, x2 =

1

2
, x3 =

3

4
, and x4 = 1.

Then the second differential equation is approximated as

y′′(x) =
yi+1 − 2yi + yi−1

h2
.

Substitute into (58):

1

h2
[yi+1 − 2yi + yi−1] = 2y3i + xiyi + µ,

yi+1 − 2yi + yi−1 = h2
(
2y3i + xiyi + µ

)
.

By using a forward difference approximation for the derivative to compute the initial
value for y1,

y′(x) =
1

h
[y(x+ h)− y(x)] .

y′(0) =
y(h)− y0

h
.

0 =
y1 − 1

h
.

y1 = 1.

Thus we have
yi+1 − 2yi + yi−1 = h2(2y3i + xiyi + µ).

For i = 1
y2 − 2y1 + y0 = h2(2y31 + x1y1 + µ).

y2 =
73

64
+

µ

16
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 29 of 38

For i = 2
y3 − 2y2 + y1 = h2(2y32 + x2y2 + µ).

y3 − 2

(
73

64
+

µ

16

)
+ 1 =

(
1

4

)(
2

(
73

64
+

µ

16

)3

+
1

2

(
73

64
+

µ

16

)
+ µ

)
.

y3 =
1135513

524288
+

66163µ

131072
+

219µ2

32768
+

µ3

8192
.

Their solution gives:

y1 = 1, y2 =
73

64
+

µ

16
, y3 =

1135513

524288
+

66163µ

131072
+

219µ2

32768
+

µ3

8192
.

3.2. Runge-Kutta Method (RK4)

The Runge-Kutta method is a powerful numerical technique for solving ordinary differ-
ential equations (ODEs), including nonlinear equations like the Painlevé equations. Devel-
oped by Carl Runge and Wilhelm Kutta in 1990, [45, 46], it provides higher accuracy than
the Euler method without requiring higher-order derivatives. The fourth-order Runge-
Kutta method (RK4) is particularly popular due to its balance between computational
efficiency and precision. By computing intermediate slopes at each step, RK4 improves
stability and accuracy; in this subsection, we use it to solve the Painlevé equations [47–50].

3.2.1. The RK4 for solving the Painlevé I differential equation

Rewrite (1)
y′′(x) = 6y2 + x. (59)

with the initial conditions:

y(0) = 0, y′(0) = 1. (60)

Rewrite the second-order equation as a system of first-order equations:

dy

dx
= z = f(x, y, z).

y′ = z,

z′ = y′′ = 6y2 + x = ϕ(x, y, z).

Here, f(x, y, z) represents the first derivative of y, while ϕ(x, y, z) defines the right-
hand side of the second equation. With initial conditions at x0 = 0, y0 = 0, z0 = 1,
we proceed. Use step size h = 1

4 and calculate k1, k2, k3, k4 and l1, l2, l3, l4. Runge-Kutta
formulas for y, we use:

k1 = hf(x0, y0, z0) =
1

4
.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 30 of 38

k2 = hf

(
x0 +

h

2
, y0 +

k1
2
, z0 +

l1
2

)
=

1

4
,

k3 = hf

(
x0 +

h

2
, y0 +

k2
2
, z0 +

l2
2

)
=

263

1024
,

k4 = hf(x0 + h, y0 + k3, z0 + l3) =
135

512
.

k =
1

6
(k1 + 2k2 + 2k3 + k4) =

391

1536
.

l1 = hϕ(x0, y0, z0) = 0.

l2 = hϕ

(
x0 +

h

2
, y0 +

k1
2
, z0 +

l1
2

)
=

7

128
.

l3 = hϕ

(
x0 +

h

2
, y0 +

k2
2
, z0 +

l2
2

)
=

7

128
.

l4 = hϕ(x0 + h, y0 + k3, z0 + l3) =
338579

2097152
.

l =
1

6
(l1 + 2l2 + 2l3 + l4) =

797331

8388608
.

Hence,

y = y0 + k =
391

1536
≈ 0.25456.

y′ = z = z0 + l = 1 +
797331

8388608
≈ 1.06337.

3.2.2. The RK4 for solving the Painlevé II differential equation

Rewrite (2)
y′′(t) = 2y3 + xy + µ. (61)

With the initial conditions:

y(0) = 1, y′(0) = 0. (62)

Rewrite the second-order equation as a system of first-order equations:

dy

dx
= z = f(x, y, z), y′ = z.

z′ = y′′ = 2y3 + xy + µ = φ(x, y, z).

Here, f(x, y, z) represents the first derivative of y, while φ(x, y, z) defines the right-hand
side of the second equation. With initial conditions at x0 = 0, y0 = 1, and z0 = 0.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 31 of 38

Use step size h = 1
4 and calculate k1, k2, k3, k4 and l1, l2, l3, l4. Range Kutta formulas

for y, we use:
k1 = hf(x0, y0, z0) = 0.

k2 = hf

(
x0 +

h

2
, y0 +

k1
2
, z0 +

l1
2

)
=

2 + µ

32
.

k3 = hf

(
x0 +

h

2
, y0 +

k2
2
, z0 +

l2
2

)
=

1 + 256µ

1024
.

k4 = hf (x0 + h, y0 + k3, z0 + l3) =
966280 + 160140µ+ 294µ2 + µ3

524288
.

k =
1

6
(k1 + 2k2 + 2k3 + k4) =

1032840 + 455052µ+ 294µ2 + µ3

3145728
.

l1 = hφ(x0, y0, z0) =
2 + µ

4
.

l2 = hφ

(
x0 +

h

2
, y0 +

k1
2
, z0 +

l1
2

)
=

1 + 256µ

1024
.

l3 = hφ

(
x0 +

h

2
, y0 +

k2
2
, z0 +

l2
2

)
=

966280 + 160140µ+ 294µ2 + µ3

524288
.

l4 = hφ (x0 + h, y0 + k3, z0 + l3) =
1025

16384
+

1

2

(
1025

1024
+

µ

4

)3

+
17µ

64
.

l =
1

6
(l1 + 2l2 + 2l3 + l4) =

10204941 + 4299784960µ+ 203931648µ2 + 16785408µ3

12884901888
.

Hence,

y = y0 + k =
4178568 + 455052µ+ 294µ2 + µ3

3145728
,

y′ = z = z0 + l =
10204941 + 4299784960µ+ 203931648µ2 + 16785408µ3

12884901888
.

4. Comparative Analysis of Analytical and Numerical Methods

The study of nonlinear differential equations, particularly Painlevé equations, requires
robust analytical and numerical approaches to obtain accurate solutions. Analytical meth-
ods such as the Differential Transform Method (DTM), Modified Adomian Decomposition
Method (MADM), Laplace Decomposition Method (LDM), and Natural Decomposition
Method (NDM) provide series solutions, while numerical techniques such as the Finite
Difference Method (FDM) and Runge-Kutta Method (RK4) offer direct computational
approximations. This section presents a rigorous comparison of these methods [1, 2].



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 32 of 38

4.1. Comparison between analytical methods

To assess the accuracy and convergence of DTM, MADM, LDM, and NDM, we com-
pute approximate solutions for the Painlevé I and II equations at selected points x =
0, 0.25, 0.5, 0.75, 1. This is presented for Painlevé equation I in Table 9 and for Painlevé
equation II in Table 10. A graphical comparison of the approximate solutions obtained
from DTM, MADM, LDM, and NDM for Painlevé equation I is presented in Figure 9,
and for Painlevé equation II in Figure 10.

Table 9: Provides the computed values of y(x) at selected points for Painlevé equation I

x DTM MADM LDM NDM

0.00 0.000000 0.000000 0.000000 0.000000
0.25 0.254582309975 0.2545823823838 0.2545823823838 0.2545823823838
0.50 0.5542582826348 0.5542898995536 0.5542898995536 0.5542898995536
0.75 1.009901884624 1.0113271032061 1.0113271032061 1.0113271032061
1.00 1.882208994709 1.9011904761905 1.9011904761905 1.9011904761905

Figure 9: Convergence methods for Painlevé Equation I.

Table 10: Provides the computed values of y(x) at selected points for Painlevé equation II.

x DTM MADM LDM NDM

0.00 1.0000 1.0000 1.0000 1.0000
0.25 1.099332682292 1.097355143229 1.097355143229 1.097355143229
0.50 1.444270833333 1.412239583333 1.412239583333 1.412239583333
0.75 2.163232421875 1.999096679688 1.999096679688 1.999096679688
1.00 3.466666666667 2.941666666667 2.941666666667 2.941666666667

4.2. Comparison between numerical methods

In this subsection the numerical methods considered include the Finite Difference
Method (FDM) and the Runge-Kutta Method (RK4) [47].



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 33 of 38

Figure 10: Convergence behavior of these methods for Painlevé equation II.

The computed values for Painlevé I at selected points are presented in Table 11, which
compares the approximate solutions obtained using FDM and RK4. The graphical com-
parison of these numerical methods is further illustrated in Figure 11. For Painlevé II,
the computed values using FDM and RK4 are presented in Table 12, and the graphical
comparison is shown in Figure 12.

Table 11: Comparison of the approximate solution by FDM and RK4, for Painlevé equation I at selected points.

x FDM RK4

0.00 0.00000 0.00000
0.25 0.250000000000 0.254455682891
0.50 0.539062500000 0.554123359580
0.75 0.968345642090 1.013010257333
1.00 1.955078125000 1.955172884313

Figure 11: The accuracy of numerical methods for Painlevé equation I.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 34 of 38

Table 12: Comparison of the approximate solution by FDM and RK4, for Painlevé equation II at selected points.

x FDM RK4

0.00 1.00000 1.00000
0.25 1.00000 1.099484364191691
0.50 1.203125000000000 1.458011474879827
0.75 2.677408218383789 2.377499005219746
1.00 6.738824991141589 5.909391830792607

Figure 12: The accuracy of numerical methods for Painlevé equation II.

5. Conclusion

This paper explored various analytical and numerical methods for solving the Painlevé
equations I and II, which are significant in the field of nonlinear differential equations due
to their integrability and applications in mathematical physics. The analytical methods
examined, including the Differential Transform Method, the Modified Adomian Decom-
position Method, the Laplace Decomposition Method, and the Natural Decomposition
Method, showcased their potential for handling the inherent complexities of nonlinear
equations with high precision.

On the other hand, numerical methods, specifically the Finite Difference Method and
the fourth-order Runge-Kutta Method (RK4), demonstrated their effectiveness in provid-
ing approximate solutions when exact solutions were challenging to derive.

Comparing the two approaches, analytical methods are advantageous for gaining in-
sights into the structural properties of solutions, while numerical methods are more ver-
satile for practical computations. This combination of techniques forms a comprehensive
toolbox for tackling nonlinear ordinary differential equations like the Painlevé equations.
Future work will focus on analytic and numerical methods’ approaches for solving systems
of differential equations.

Acknowledgements

The author thanks the Editor and referees for the valuable comments and suggestions that
helped him improve the quality and readability of the paper, which led to a significant
improvement of the paper.



S. Mohammed, S. Abid, R. Abid / Eur. J. Pure Appl. Math, 18 (3) (2025), 6215 35 of 38

Conflict of Interest

The authors declare that there is no conflict of interest to disclose.

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