EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3622-3641
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Novel Analytical Approximate Solution for Strongly
Nonlinear Third-Order Jerk Equations Using a

Modified Iteration Method

Gamal M. Ismail1,2,∗, Mohammed I. Yamani1

1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
42351, Saudi Arabia
2 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt

Abstract. Nonlinear jerk equations, characterized by a third-order time derivative, play a crucial
role in modeling various physical phenomena across disciplines like mechanics, circuits, and biol-
ogy. Accurately solving these equations is essential for understanding and predicting the behavior
of such systems. However, obtaining analytical solutions for nonlinear jerk equations can be chal-
lenging, necessitating the development of robust and accurate approximation methods. This work
explores, for the first time, the application of the modified iteration approach to solve third-order
jerk equations. By comparing the obtained approximate solutions with both exact and existing an-
alytical solutions for established engineering problems, we demonstrate the superior accuracy and
rapid convergence of the proposed method. The significantly reduced error percentages highlight
the effectiveness of the modified iteration approach in providing precise solutions for nonlinear
jerk equations, paving the way for its application in a wide range of oscillation problems within
nonlinear sciences and engineering.

2020 Mathematics Subject Classifications: 34A34, 34C15, 34C25, 35L65, 37N30

Key Words and Phrases: Modified iteration method, Third-order jerk equations, Analytical
solutions, Nonlinear oscillator, Exact solution

1. Introduction

Differential equations are widely employed to describe the evolution of complex systems
in many fields, including physics and engineering. Nonlinear differential equations play
a crucial role in modeling various scientific phenomena and have many applications in
various branches of science. Researchers seek effective solutions to such problems using
analytical or numerical approaches.

Recently, there has been an increasing interest in analytical solutions for nonlinear
differential equations. The key issue in studying nonlinear differential equations is finding

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5427

Email addresses: gamalm2010@yahoo.com (G. M. Ismail), shingl6600@gmail.com (M. I. Yamani)

https://www.ejpam.com 3622 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3623

the exact solutions. On the other hand, computing the exact results can be challenging, es-
pecially in equations with a high degree of nonlinearity, mainly using traditional analytical
techniques. To overcome this problem, many novel approaches are presented to deal with
this shortage. For example, harmonic balance method [1–3], variational iteration method
[4, 5], Hamiltonian approach [6, 7], modified algebraic method [8, 9], global residue har-
monic balance method [10, 11], energy balance method [12–14]. linearizing method [15],
multiple scales method [16], non-perturbative approach [17, 18], Adomian decomposition
method [19], optimal variational iteration method [20] and Galerkin method [21]. These
analytical approaches have been extensively utilized to examine the frequency and periodic
solutions of nonlinear oscillators.

Nonlinear third-order Jerk equations are useful for analyzing structures with rotating
and translating movements, such as machine tools or robots [22]. Nonlinear third-order
jerk equations are differential equations that describe the evolution of a system’s jerk,
which is the rate of change of acceleration. In simpler terms, it represents the ”snap”
or ”jolt” felt when acceleration changes abruptly. Jerk equations are powerful tools for
modeling complex dynamical systems exhibiting rich and often unpredictable behavior.
Understanding their properties and developing effective methods for their analysis is cru-
cial for advancing our knowledge in various scientific and engineering disciplines.

Nonlinear third-order Jerk equations may explain a variety of physical issues, includ-
ing third-order mechanical oscillators [23]. Nowadays, due to the need of knowing the
analytical solutions of the nonlinear Jerk equations. Several diverse techniques have been
proposed to find the analytical solutions of like these problems such as block method
[24], harmonic balance method [25, 26], homotopy perturbation method [27], parameter
perturbation method [28], Mickens iteration method [29], Linstedt-Poincare methods [30],
residue harmonic balance method [31], multiple scales Lindstedt-Poincare method [32],
differential transform method [33], modified harmonic balance method [34], variational
iteration method [35] and homotopy asymptotic method [36] to solve the present prob-
lems. Recently, Ismail and Abu-Zinadah [37] used the global error minimization method
for solving the current problems.

In this study, we apply the modified iteration technique, a powerful analytical method
with high accuracy and efficiency, to obtain higher-order analytic approximations for non-
linear Jerk equations. The primary advantage of this method lies in its ability to provide
both simplicity and accuracy when solving higher-order differential equations. The nu-
merical solution is obtained using the fourth-order Runge-Kutta method. A comparison
between the analytical and numerical solutions, presented through tables and correspond-
ing figures, emphasizes the accuracy of the modified iteration technique.

2. The iteration procedure

Consider a non-linear equation

ẍ+ f(x, ẋ, ẍ) = 0, x(0) = A, ẋ(0) = 0. (1)



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3624

Rewrite Eq. (1) to be in the form:

ẍ+ ω2x = ω2x− f(x, ẋ, ẍ) ≊ F (xk, ẋk, ẍk), (2)

where ẋ and ẍ represent the first and second derivatives with respect to time, respectively,
and ω is an unknown constant.

According to Ref [38], Eq. (2) can be rewritten as

ẍk+1 + ω2
kxk+1 = F (xk, ẋk, ẍk) , k = 0, 1, 2, ..., (3)

and the imputes of starting functions are

x0(t) = A cosω0t. (4)

It is further required that for each k, the solution to Eq. (3), is to satisfy initial conditions

xk+1(0) = A, ẋk+1(0) = 0. (5)

3. Applications

In this section, we illustrate the fundamental concept of the modified iteration approach
by considering the following non-linear differential equation:

...
x + f (x, ẋ, ẍ) , x(0) = 0, ẋ(0) = A, ẍ(0) = 0. (6)

Following [3], the general non-linear third order Jerk equation has the form

...
x = −γ ẋ− αẋ3 − βx2ẋ+ δxẋẍ− εẋ ẍ2, (7)

with

x(0) = 0, ẋ(0) = A, ẍ(0) = 0. (8)

where the parameters α, β, γ, δ and ε are constants.

3.1. Jerk function containing time’s velocity times acceleration and ve-
locity

In this case at γ = δ = 1, α = β = ε = 0, in Eq. (7), the non-linear equation is in the
following [1]:

...
x + ẋ− xẋẍ = 0, x(0) = 0, ẋ(0) = A, ẍ(0) = 0. (9)

Eq. (9) can be rewritten in the form

ẋ = y, ẏ = ẍ, (10)



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3625

then Eq. (10) becomes

ÿ + y − ȳyẏ = 0, y(0) = A, ẏ(0) = 0. (11)

where ȳ is the integration of x

ÿ + ω2y = ω2y − y + ȳyẏ. (12)

The iteration technique according to Eq. (3) is

ÿk+1 + ω2
kyk+1 = ω2

kyk − yk + ȳkykẏk. (13)

3.1.1. First iteration solution

For the first iteration, at k = 0, we get

ÿ1 + ω2
0y1 = ω2

0y0 − y0 + ȳ0y0ẏ0. (14)

According to Eq. (4), we have
y0(t) = A cosω0t. (15)

Inserting Eq. (15) into Eq. (14), to obtain

ÿ1 + ω2
0y1 =

1

4

(
−4A−A3 + 4Aω2

0

)
cos(ω0t)−

A3

4
cos(3ω0t). (16)

A secular term is a term in the solution that grows linearly or polynomially with time.
Secular terms often arise when using naive perturbation methods on systems with natural
frequencies that are close to being resonant with the perturbation frequency. Secular terms
are undesirable for several reasons, for example, breakdown of perturbation theory and
loss of periodicity:

To avoid secular term from equation (16), we obtain

ω0 =
1

2

√
4 +A2. (17)

Solving Eq. (16) with initial conditions (5), the first approximate solution y1 of is obtained
as

y1 =

(
A+

A3

32ω2
0

)
cos(ω1t)−

A3

32ω2
0

cos(3ω1t). (18)

3.1.2. Second iteration solution

For the second level of iteration continuing to k = 1. Substituting Eq. (18) into Eq. (13),
to obtain

ÿ2 + ω2
1y2 = −

(
A(32+9A2)(6144+4608A2+1136A4+93A6−384(4+A2)2ω2

1
3072(4+A2)3

)
cos(ω1t)

+
(
A3(9216+7936A2+2296A4+223A6−64(4+A2)2ω2

1)
512(4+A2)3

)
cos(3ω1t)

+
(
A5(3328+1928A2+279A4)

1536(4+A2)3

)
cos(5ω1t) +

(
13A7(32+9A2)
6144(4+A2)3

)
cos(7ω1t)

−
(

A9

2048(4+A2)3

)
cos(9ω1t).

. (19)



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3626

To avoid dominating terms, we obtain

ω1 =

√
6144 + 4608A2 + 1136A4 + 93A6

6144 + 3072A2 + 384A4
. (20)

After solving Eq. (19) with the initial conditions, we have the second approximate solution.

y2 =
(
A3(103680+85120A2+23485A4+2178A6)+720A(4+A2)2(256+63A2)ω2

1

46080(4+A2)33ω2
1

)
cos(ω2t)

+
(
A3(103680+85120A2+23485A4+2178A6)+720A(4+A2)2(256+63A2)ω2

1

4096(4+A2)3ω2
1

)
cos(3ω2t)

+
(
A5(3328+1928A2+279A4)

36864(4+A2)3ω2
1

)
cos(5ω2t) +

(
13A7(32+9A2)

294912(4+A2)3ω2
1

)
cos(7ω2t)

+
(

A9

163840(4+A2)3ω2
1

)
cos(9ω2t).

(21)

3.1.3. Third iteration solution

For the third level of iteration, continuing to k = 2, we substituting y2 from Eq. (21) into
the right-hand side of Eq. (13), we have.

ÿ3 + ω2
2y3 = ω2

2y2 − y2 + ȳ2y2ẏ2. (22)

Solving Equation (22), and avoiding secular terms, we can obtain ω2 by using Mathematica
command software program. The findings found for y3, need too much space and cannot
be shown here. However, the numerical values will be shown in the findings and discussion
part.

ω2 = (
√
(1/(4 +A2)9A(1470839609502185029632000 + 4826192468679044628480000A2

+7296583778952951103488000A4 + 6729655286686996758528000A6

+4224062453812034745139200A8 + 1905190794429295126118400A10

+635148579378340823040000A12 + 158430275820381536256000A14

+29559820082388423147520A16 + 4072969446127757557760A18

+402650199557972162560A20 + 27028701991773424000A22

+1103351520402981540A24 + 20664999056050911A26)))
/(3840

√
7
√
( 1
(4+A2)7

A(6144 + 4608A2 + 1136A4 + 93A6)2(23592960

+24330240A2 + 9397760A4 + 1618520A6 + 105309A8))).
(23)

Integrating Eq. (21), we obtain the analytical solution of Eq. (9) in the form:

x =

((
4A(23592960+24330240A2+9397760A4+1618520A6+105309A8)

ω2

)
sin(ω2t)

−
(

20A3(49152+43008A2+12640A4+1245A6)
ω2

)
sin(3ω2t)

+
(
26624A5+15424A7+2232A9

ω2

)
sin(5ω2t)−

(
2080A7+585A9

7ω2

)
sin(7ω2t)

+
(
A9

ω2

)
sin(9ω2t)

)
/
(
3840(4 +A2)(6144 + 4608A2 + 1136A4 + 93A6)

)
(24)



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3627

3.2. Jerk function containing velocity times acceleration-squared, and ve-
locity

Another case of the Jerk equation is considered by putting γ = ε = 1 , α = β = δ = 0,
in Eq. (7) to obtain [1]:

...
x + ẋ+ ẋẍ2 = 0, x(0) = 0, ẋ(0) = A, ẍ(0) = 0. (25)

Similar to application (1), Eq. (25) can be rewritten in the form

ÿ + y + yẏ2 = 0 y(0) = A, ẏ(0) = 0. (26)

Following the iteration scheme (3), we have

ÿk+1 + ω2
k+1yk+1 = ω2

kyk − yk − ykẏ
2
k. (27)

3.2.1. First iteration solution

For first iteration at k = 0, we get

ÿ1 + ω2
0y1 =

1

4

(
−4ẏ + 4ẏω2

0 −A3ω2
0

)
cos(ω0t) +

A3ω2
0

4
cos(3ω0t). (28)

To avoid secular term from equation (28), we obtain

ω0 =
2√

4−A2
. (29)

Solving Eq. (28) with initial conditions (5), the first approximate solution y1 is obtained
as

y1 =

(
A+

A3

32

)
cos(ω1t)−

A3

32
cos(3ω1t). (30)

3.2.2. Second iteration solution

For the second level of iteration continuing to k = 1 gives,

ÿ2 + ω2
1y2 =

(
−A(32+A2)(2048+(−2048+512A2−48A4+7A6)ω2

1)
65536

)
cos(ω1t)

+
(
A3(1024+(7168+1280A2+56A4+3A6)ω2

1)
32768

)
cos(3ω1t)

−.
(
A5(1792+88A2+A4)ω2

1
32768

)
cos(5ω1t)

+
(
15A7(32+A2)ω2

1
131072

)
cos(7ω1t)

−
(
9A9ω2

1)
131072

)
cos(9ω1t).

. (31)

To avoid dominating terms in Eq (31), we obtain

ω1 =
32
√
2
√
32 +A2

√
65536− 14336A2 + 1024A4 − 176A6 − 7A8

, (32)



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3628

After solving Eq. (31) with the initial conditions, we have the second approximate solution.

y2 =
(
A+ 7A3

256 + A5

384 + 35A7

196608 + 23A9

1966080 + A3

256ω2
1

)
cos(ω2t)

−
(
A3(1024+(7168+1280A2+56A4+3A6)ω2

1)

262144ω2
1

)
cos(3ω2t)

+
(
A5(1792+88A2+A4)

786432

)
cos(5ω2t)−

(
5A7(32+A2)

2097152

)
cos(7ω2t)

+
(

9A9

10485760

)
cos(9ω2t).

(33)

3.2.3. Third iteration solution

For the third level of iteration continuing to k = 2. Substituting y2 from Eq. (32) into
the right hand side of Eq. (13), we have

ÿ3 + ω2
2y3 = ω2

2y2 − y2 − y2ẏ
2
2. (34)

Solving Eq. (34), and avoiding secular terms, we can obtain ω2 in the same manner
as application (1). The numerical data will be presented in the findings and discussion
section.

ω2 =
(
10485760

√
23592960A+737280A3+38400A5+6360A7−39A9

2048−512A2+48A4−7A6

)
/(√

((−2594073385365405696000A+ 567453553048682496000A3

−44754521296994304000A5 − 3232564185661440000A7

+172513374398054400A9 + 248228493865779200A11

+19274524734259200A13 + 2109308770713600A15

+185788373401600A17 + 8293138432000A19 + 200006092800A21

+2252995200A23 + 174151100A25 − 997773A27)/(−2048 + 512A2

−48A4 + 7A6)))

. (35)

Integrating Eq. (33), we obtain the analytical solution of Eq. (25) in the form:

x = 1
220200960 ω2

(
−28A

(
−7864320− 245760A2 − 12800A4 − 2120A6 + 13A8

)
sin(ω2t)

+140A3
(
−16384− 2048A2 − 160A4 +A6

)
sin(3ω2t)

+(100352A5 + 4928A7 + 56A9) sin(5ω2t)
− (2400A7 + 75A9) sin(7ω2t) + 21A9 sin(9ω2t)

)
.

.

(36)

4. Discussions

In this section, the approximate analytical solutions of Eqs. (9) and (25), obtained
using the modified iteration approach as shown in Eqs. (24) and (36), are compared
with those obtained from fourth-order Runge-Kutta numerical solutions and other known
analytical methods from the literature. This comparison is presented in Figures 1 − 4



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3629

and Tables 1− 6. The graphical representations clearly demonstrate that the results ob-
tained from the present approach are in excellent agreement with those obtained using the
fourth-order Runge-Kutta method. Moreover, the present approach accurately predicts
the periodic behavior of the equations over a wide range.

To demonstrate the exceptional accuracy of the modified iteration technique, we ex-
amine two cases of non-linear third-order jerk equations. We compare the approximate
results obtained using this technique with existing analytical solutions from the literature
and with numerical integration results to validate the accuracy of the solutions derived.

For the same cases discussed using the block method [24], harmonic balance method
[25], homotopy perturbation method [27], Linstedt-Poincare methods [30], residue har-
monic balance method [31], multiple scales Lindstedt-Poincare [32], modified harmonic
balance method [34], differential transform method [33], and global error minimization
method [37], the present results were compared with those obtained by the modified itera-
tion technique, as shown in Tables 1− 6. The numerical results show excellent agreement
with the third-order approximate analytical solutions obtained in this study using the
modified iteration technique.

A comparison between the higher-order approximate solution, the differential trans-
form method [33], and the modified global error minimization method [37] with the cor-
responding numerical solution is presented in Figures 1 − 4 for A = 0.3 and A = 1. It is
clear that the approximation of the solution using the modified iteration technique agrees
with the differential transform method, the modified global error minimization method,
and the numerical solution. Furthermore, the approximate frequencies agree well with the
corresponding exact solutions, implying that using the modified iteration technique with
higher orders produces realistic results.

We looked at the percentage error (%) by the definition to confirm the accuracy.

Error =

∣∣∣∣Te − TApp

Te

∣∣∣∣× 100%.

where the various approximate periods obtained by TApp and Te represents the correspond-
ing exact period of the oscillator.



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3630

Table 1: Comparison of the approximate and exact solutions for Eq. (9).

A Te T3current T3 [37] T3 [32] T3 [31] T3 [27] T3 [30] T3 [25] T3 [34]

0.1 6.275347 6.275347 6.275347 6.2753468 6.275346837 6.27534684 6.275348 6.275346 6.275346837

0 0 3.19e
−6

2.59e
−6

2.55e
−6

1.59e
−5

1.59e
−5

2.59e
−6

0.2 6.252016 6.252016 6.252016 6.2520158 6.25201599 6.25201599 6.252028 6.252003 6.25201599

0 0 3.19e
−6

1.59e
−7

1.59e
−7

1.92e
−4

2.08e
−4

1.59e
−7

0.5 6.096061 6.096061 6.096061 6.0960246 6.09606050 6.09605904 6.096491 6.095585 6.096060516

0 0 5.97e
−4

8.20e
−6

3.12e
−5

0.00706 0.00781 7.93e
−7

1 5.626007 5.62587 5.62602 5.6245487 5.62599289 5.62579479 5.630343 5.619852 5.62599937

2.43e
−5

2.31e
−4

0.02592 2.51e
−4

0.00377 0.07707 0.10940 1.36e
−4

2 4.491214 4.48492 4.47661 4.4664554 4.49012538 4.48208113 4.509311 4.442883 4.49112308

0.14014 0.47234 0.55127 0.02424 0.20335 0.40294 1.07612 0.00202

Table 2: Comparison between the numerical solution and analytical solutions at A = 0.2 for Eq.

(9)

t Block Method MDTM [4/4] and Present Solution Numerical Solution

[24] [5/5] [33]

0 0 0 0 0
0.125 0.024934 0.024951214 0.024934943 0.024935034
0.25 0.049480 0.049511005 0.049480664 0.049480891
0.375 0.073253 0.073293861 0.073253553 0.073253560
0.5 0.095881 0.095926024 0.095881202 0.095881465
0.625 0.117008 0.117051228 0.117007962 0.117008021
0.75 0.1363 0.136336241 0.136300404 0.136300551
0.875 0.153453 0.153476143 0.153452636 0.153452785
1 0.168191 0.168199251 0.168191377 0.168191348
1.125 0.180281 0.180271603 0.180280677 0.180280833
1.25 0.180281 0.180271603 0.180280677 0.180280833
1.375 0.195778 0.195739784 0.195778569 0.195778587
1.5 0.198937 0.198888450 0.198936744 0.198936684
1.625 0.198949 0.198896377 0.198949462 0.198949302
1.75 0.195816 0.195763266 0.195816515 0.195816396
1.875 0.189589 0.189539059 0.189588706 0.189588464
2 0.180367 0.180323081 0.180366838 0.180366565



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3631

Table 3: Comparison between the numerical solution and analytical solutions at A = 0.4 for Eq.

(9)

t Block Method MDTM [4/4] and Present Solution Numerical Solution

[24] [5/5] [33]

0 0 0 0 0
0.125 0.049869 0.049869861 0.049869861 0.049870029
0.25 0.098960 0.098960562 0.098960558 0.098961002
0.375 0.146501 0.146501385 0.146501379 0.146501402
0.5 0.191739 0.191738916 0.191738937 0.191739420
0.625 0.233947 0.233946661 0.233946816 0.233946758
0.75 0.272436 0.272435646 0.272436179 0.272435964
0.875 0.306566 0.306566039 0.306567314 0.306566352
1 0.33576 0.335759596 0.335761911 0.335759593
1.125 0.359513 0.359512477 0.359515583 0.359513002
1.25 0.377408 0.377407791 0.377410047 0.377407810
1.375 0.389127 0.389126999 0.389124206 0.389127373
1.5 0.39446 0.394459308 0.394443406 0.394459515
1.625 0.393308 0.393308207 0.393266152 0.393308138
1.75 0.385695 0.385694562 0.385607769 0.385694560
1.875 0.371756 0.371755941 0.371755564 0.371600658
2 0.351742 0.351742248 0.351491092 0.351741644



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3632

Figure 1: Comparison between higher order analytical solution (black line), MGEMM [37] (red line), and the
numerical solution (blue line) at A = 1, for Eq. (9).



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3633

Figure 2: Comparison between higher order analytical solution (black line), MDTM [33] (red line), and the
numerical solution (blue line) at A = 0.3, for Eq. (9).



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3634

Table 4: Comparison of the approximate and exact solutions for Eq. (25).

A Te T3current T3 [37] T3 [32] T3 [31] T3 [27] T3 [30] T3 [25] T3 [34]

0.1 6.27533378 6.27533378 6.27533378 6.27533378 6.2753338 6.27533378 6.275329 6.2753264 6.27533378

0 0 0 0 0 7.61e
−5

1.18e
−4

0

0.2 6.251809 6.25180898 6.25180898 6.25180884 6.25180911 6.25182078 6.251740 6.251690 6.2518089

3.19e
−7

3.19e
−7

2.56e
−6

1.76e
−6

1.88e
−4

0.00110 0.00190 1.59e
−6

0.5 6.088449 6.08845097 6.08845017 6.08841902 6.08848374 6.08815979 6.085649 6.083668 6.088450

3.24e
−5

1.92e
−5

4.92e
−4

5.71e
−4

0.00475 0.04599 0.07853 1.64e
−5

1 5.527200 5.527510790 5.527656919 5.52576588 5.52994105 5.50818960 5.477174 5.441398 5.527497

0.00562 0.00827 2.59e
−4

0.04959 0.343943 0.90509 1.55236 0.00537

2 4.690247 4.6831871 4.7771790 4.68572454 4.72603111 4.44735707 4.412733 4.155936 4.683269

0.15052 1.85346 0.09642 0.76295 5.17847 5.91683 11.39196 0.14878

Table 5: Comparison between the numerical solution and analytical solutions at A = 0.2 for Eq.

(25)

t Block Method MDTM [4/4] and Present Solution Numerical Solution

[24] [5/5] [33]

0 0 0 0 0
0.125 0.024934 0.024934942 0.024934943 0.024935034
0.25 0.049480 0.049480663 0.049480664 0.049480891
0.375 0.073253 0.073253551 0.073253552 0.073253560
0.5 0.095881 0.095881197 0.095881199 0.095881461
0.625 0.117008 0.117007942 0.117007949 0.117008005
0.75 0.1363 0.136300334 0.136300356 0.136300499
0.875 0.153452 0.153452432 0.153452503 0.153452647
1 0.168191 0.168190868 0.168191071 0.168191036
1.125 0.18028 0.180279542 0.180280059 0.180280209
1.25 0.189525 0.189523851 0.189525026 0.189524937
1.375 0.195777 0.195774284 0.195776702 0.195776711
1.5 0.198934 0.198929290 0.198933855 0.198933786
1.625 0.198945 0.198937278 0.198945268 0.198945100
1.75 0.195811 0.195797675 0.195810754 0.195810626
1.875 0.189581 0.189560997 0.189581165 0.189580915
2 0.180357 0.180327919 0.180357378 0.180357098



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3635

Table 6: Comparison between the numerical solution and analytical solutions at A = 0.4 for Eq.

(25)

t Block Method MDTM [4/4] and Present Solution Numerical Solution

[24] [5/5] [33]

0 0 0 0 0
0.125 0.049869 0.049869861 0.049869863 0.049870029
0.25 0.098960 0.098960556 0.098960576 0.098961001
0.375 0.146501 0.146501354 0.146501423 0.146501386
0.5 0.191739 0.191738758 0.191738937 0.191739302
0.625 0.233946 0.233945991 0.233946416 0.233946249
0.75 0.272434 0.272433351 0.272434399 0.272434276
0.875 0.306562 0.306559422 0.306562164 0.306561832
1 0.335749 0.335743004 0.33575013 0.335749291
1.125 0.359492 0.359475362 0.359492741 0.359492282
1.25 0.37737 0.377332285 0.377371297 0.377370224
1.375 0.389065 0.388985315 0.389065644 0.389064877
1.5 0.394363 0.394211444 0.394363972 0.394363009
1.625 0.393169 0.392900613 0.393169669 0.393168420
1.75 0.385503 0.385060429 0.385504547 0.385503391
1.875 0.371507 0.370817684 0.371508112 0.371506638
2 0.351432 0.350416451 0.351432939 0.351431339



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3636

Figure 3: Comparison between higher order analytical solution (black line), MGEMM [37] (red line), and the
numerical solution (blue line) at A = 1, for Eq. (25).



G. M. Ismail, M. I. Yamani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3622-3641 3637

Figure 4: Comparison between higher order analytical solution (black line), MDTM [33] (red line), and the
numerical solution (blue line) at A = 0.3, for Eq. (25).



REFERENCES 3638

5. Conclusion

In this study, a modified iteration approach was developed and employed to solve two
nonlinear third-order jerk equations with broad engineering applications. The modified
approach successfully determined approximate analytical solutions for these equations. A
comparison of the obtained results with numerical solutions demonstrated the excellent
accuracy of the proposed method. The present approach offers solutions in a readily usable
analytical form, exhibiting superior accuracy and a wider range of applicability compared
to other established analytical methods found in the literature. This technique proves to
be a powerful and effective mathematical tool for solving highly nonlinear third-order dif-
ferential equations arising in mathematical physics, applied mathematics, and engineering.
Furthermore, the iterative nature of the method allows for the computation of higher-order
approximations to achieve even greater accuracy if desired. The modified iteration tech-
nique facilitates the straightforward calculation of these higher-order terms, leading to
solutions that closely approximate the exact solutions. Consequently, the present method
demonstrates consistent and reliable performance, offering a simple yet effective approach
for obtaining novel solutions to a variety of nonlinear problems.

Future research can focus on a deeper qualitative analysis of third-order systems, ex-
ploring the existence and uniqueness of solutions, stability analysis, and bifurcation be-
havior [39, 40].

Because of the great importance of applications of nonlinear third-order Jerk equations,
more analytical and numerical methods will be developed to solve such problems in future
work.

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