We investigate the core features of the perturbation method with the help of some simple but sophisticated problems and demonstrate how much accurately it predicts the solutions of the problems. To fulfill the target, we use the method for getting the solution of differential equations with initial and boundary conditions. Then the results obtained are compared with the series solution and the exact/numerical solution by using Mathematica and Fortran Programming. The comparisons are shown graphically. Also, the perturbation series approximation and the exact or numerical solution are in good agreement. Our investigation shows that a certain number of terms of the perturbation series gives an excellent approximation than the same number of terms of the numerical solution.
Perturbation theory developed by Taiwo and Osila-gum, (2011) is widely used for the simplification of complex mathematical problems. The use of per-turbation theory will allow approximate solutions to determine the problems that cannot be solved by traditional analytical methods. However, in many cases, real-life situations can require much more difficult mathematical models, such as non-linear differential equations.
Perturbation theory is a powerful mathematical tech-nique used to approximate solutions to problems that are difficult to solve exactly. It is particularly useful in physics, chemistry, engineering, and applied mathe-matics, where many complex systems can be repre-sented by differential equations that are challenging to solve analytically. Perturbation theory provides a systematic way to obtain approximate solutions by breaking down the problem into simpler, more solvable parts which was investigated by Debwan and Hasan, (2020). The basic idea behind perturbation theory is to start with a known, easily solvable problem (the unperturbed problem) and then introduce small corrections or perturbations to it. These perturb-bations are typically represented by small parameters that quantify the deviation from the idealized, unperturbed system. By treating these perturbations as small deviations from the known solution, one can develop a series expansion in terms of these param-eters, which can then be used to derive increasingly accurate approximations to the true solution. There are two main types of perturbation theory: regular perturbation theory and singular perturbation theory. Regular perturbation theory developed by Dehghan and Shakeri, (2008) in which the perturbations are assumed to be small across the entire domain of interest. This allows for the development of a systematic series expansion, such as a power series, in terms of the small parameter. Each term in the series represents a higher-order correction to the solution, and the series can often be truncated at a certain order to obtain a sufficiently accurate approximation. The other is singular perturbation theory; singular perturb-bation theory is used when the perturbations are not small throughout the entire domain, but rather become significant in certain regions or under certain condi-tions. In such cases, a straightforward series expansion may not converge, and alternative techniques, such as matched asymptotic expansions or boundary layer analysis, are employed to obtain accurate solutions near the points of interest. Once the equation is non-dimensionalized, perturbation theory requires taking advantage of a “small” parameter that appears in an equation was explained by (Shakeri and Dehghon 2008; Elrazeg et al., 2022). This parameter, usually denoted “ε” is on the order of 0 < ε << 1.
Perturbation theory has its roots in early celestial mechanics, where the theory of epicycles was used to make small corrections to the predicted paths of planets. Perturbation methods are powerful techniques used in mathematics and physics to approximate the results to differential equations, especially when exact solutions are difficult or impossible to find. These methods are particularly useful for problems where a small parameter exists, allowing for an expansion around a known solution. Perturbation theory relies on the existence of a small parameter, often denoted as ϵ, which quantifies the deviation from a simpler problem. This parameter could represent a physical quantity like mass ratio, coupling constant or geometric scale. The central idea of perturbation theory is to approximate the solution to a complex problem by iteratively correcting a known solution to a simpler problem. This known solution is usually obtained by setting the small parameter to zero. Perturbation theory often involves analyzing the behavior of the solution as the small parameter approaches zero. Asymptotic behavior provides insights into the dominant terms contributing to the solution. The accuracy and validity of the appro-ximate solution are validated by comparing it with exact solutions (if available), numerical simulations, or experimental data. Perturbation theory finds appli-cations in diverse fields such as celestial mechanics, quantum mechanics, fluid dynamics, population dynamics, and many more. Advanced techniques such as multiple scales method, matched asymptotic expansions, and resummation methods are employed to improve the approximation problems (Nino et al., 2013; Gervais et al., 1975). Perturbation theory provides a systematic framework for tackling complex problems and gaining insights into the behavior of physical systems. Its versatility and applicability make it an indispensable tool in scientific and engineering research. The beginnings of perturbation theory can be traced to Isaac Newtons work on the gravitational interactions between celestial bodies in the 17th century. While Newton provided exact solutions for two-body problems, the interactions of more than two bodies posed significant challenges. In the late 18th and early 19th centuries, mathematicians such as Joseph-Louis Lagrange and Pierre-Simon Laplace made significant contributions to celestial mechanics. They developed perturbation methods by He, (2003) to study the effects of gravitational interactions among multiple celestial bodies. Aghakhani, (2015) intro-duced the concept of secular perturbations, which long-term effects are arising from gravitational interactions that cause slow changes in the orbits of celestial bodies over time. The development of series solutions for differential equations by mathematicians like Leonhard Euler and Joseph Fourier provided a mathematical foundation for perturbation methods. Laplace and Macgillivry, (2008) developed the met-hods for finding asymptotic expansions of integrals, which laid the groundwork for the asymptotic analysis used in perturbation theory. William Rowan Hamiltons reformulation of classical mechanics in terms of Hamiltonian mechanics provided a new framework for perturbation theory. Wilsen and Rallison, (2007) made significant contributions to perturbation theory in celestial mechanics, introducing the concept of canonical transformations and deve-loping perturbation methods to study the stability of the solar system. Quantum Mechanics: Perturbation theory found widespread application in quantum mechanics, particularly in the development of quan-tum electrodynamics (QED) by physicists such as Paul Dirac, Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga. Feynman diagrams revolutionized perturbation theory in quantum field theory, providing a graphical representation of particle inter-actions and facilitating calculations of scattering amplitudes.
Perturbation methods have been extended by Xia and Zhang, (2024) to study nonlinear dynamical systems, chaos theory, and bifurcation theory. Perturbation techniques are widely used in engineering disciplines such as fluid dynamics, structural mechanics, and control theory to analyze the effects of small distur-bances on system behavior. Throughout its history, perturbation theory has evolved from its origins in celestial mechanics to become a fundamental tool in various branches of science and engineering. Its development has been driven by the need to under-stand and quantify the effects of small perturbations on complex systems, leading to advancements in both theory and application. Overall, the development of basic perturbation theory by Clenshaw and Norton, (1963) involves a systematic process of approximating solutions to differential equations by expanding them in powers of a small parameter and iteratively refining the solution to higher orders of accuracy. These methods provide valuable insights into complex systems and phenomena that cannot be fully under-stood using exact analytical techniques.
The methodology of perturbation methods involves a systematic approach to approximate results to differ-ential equations by treating small deviations or perturbations from a known, easily solvable problem. Heres a general outline of the methodology. There are three steps and eight procedures of perturbation analysis. The three steps are given below;
To transform the main problem into a perturb-bation problem by taking a small parameterδ.
To consider an expression for the solution in the form of a perturbation series and determine the coefficient of that series.
To regain the solution to the main problem by adding the perturbation series for the appropriate value of δ.
Step (1): There is sometimes ambiguity because there are many ways to introduce anδ. However, it is preferable to introduce δ in such a way that the zeroth-order solution i.e. the leading term in the perturbation series is obtainable as a closed-form analytic expression.
Step (2): By settingδ =0 in the perturbation problem, a first-order solution consists of finding the first two terms in the perturbation series, and so on.
Step (3): Begin by identifying the differential equation that describes the problem of interest. This equation typically represents the behavior of a system under consideration, such as a physical system governed by the laws of physics or an engineering system described by mathematical models.
Now the eight procedures are as follows;
Introduce Perturbation Parameters
Identify small parameters that quantify the deviations or perturbations from the idealized, unperturbed system. These parameters may represent small vari-ations in system parameters, initial conditions, boun-dary conditions, or external influences.
Decompose the Solution
Decompose the solution into two parts: the outcome of the unperturbed problem and the perturbation cor-rection terms. The unperturbed solution represents the solution to a simplified version of the problem that is easily solvable, while the perturbation correction terms account for the effects of small deviations from this idealized solution.
Expand the Solution in a Series
Express the solution as a series expansion in terms of the small perturbation parameters. This series expan-sion typically takes the form of a power series, where each term represents a higher-order correction to the solution. The coefficients of the series are functions of the perturbation parameters and are determined iteratively.
Derive Equations for the Coefficients
Substitute the series expansion into the original differential equation and equate coefficients of like powers of the perturbation parameters on both sides of the equation. This yields a set of equations for the coefficients of the series expansion, which can be solved order by order to obtain increasingly accurate approximations to the solution.
Iterative Solution Procedure
Solve the equations for the coefficients iteratively, starting from the zeroth-order term corresponding to the unperturbed solution and proceeding to higher-order terms. Each successive iteration yields a more accurate approximation to the true solution by incorporating additional corrections from higher-order perturbation terms.
Truncate the Series Expansion
Truncate the series expansion at a certain order based on the desired level of accuracy or practical considerations. In many cases, only a few terms of the series are needed to obtain sufficiently accurate approximations, especially if the perturbation para-meters are small.
Analyze Convergence and Validity
Analyze the convergence properties of the series expansion and assess the validity of the perturbation approach. Ensure that the perturbation parameters are indeed small and that the series converges to the true solution within the desired range of validity.
Interpretation and Application
Interpret the results into the domain and apply the approximate solution to analyze the behavior of the system, make predictions, or design engineering solutions. Understand the obstacles of the perturbation approach and validate the results through comparisons with numerical simulations or experimental data when possible. By following these steps and procedures, perturbation methods provide a systematic and powerful framework for approximating solutions to differential equations in a wide range of scientific and engineering applications, allowing researchers and engineers to tackle complex problems that are otherwise difficult to solve analytically.
Non-singular Perturbation Theory with First-order
First-order non-singular perturbation theory introduced by Nayfeh, (1981) is a technique used to approximate solutions to differential equations where perturbations become significant at certain points or under specific conditions, but are not small throughout the entire domain. This method was employed by Kumar and Parul, (2011) for the straightforward application of regular perturbation theory, which assumes small perturbations across the entire domain, is inadequate. Now, we have to solve the following differential equation,
, (1)
Where, D implies differential operator, and is an eigenvalue. Now it can be written as,
, (2)
Where, is very small, and operator are known. That is, one has a set of solutions labeled by index n, such that
. (3)
Furthermore, one assumes an orthonormal set,
, (4)
Where, implies the Kronecker delta. Now, for the unperturbed solutions . That is,
g(x)=f_n^((0) ) (x)+Ҩ(ε). (5)
And =〖 〗_n^((0))++Ҩ(ε), (6)
Where, Ҩ denotes by big-O pattern, of the perturbation. We consider the linear combination : , (7)
for = Ҩ(ε)except for n, where = Ҩ(1) by orthogonality condition,
, (8)
Where
, (9)
where the matrix elements are given by
. (10)
The trivial solution,
. (11)
By order Ҩ(ε^2 ), we get
. (12)
So that
, (13)
gives an equation for ,
. (14)
To give . (15)
(Salem and Thanoon, 2021; Liu and Chang, 2022) interpreted these results in the domain which can be applied the approximate solution to analyze the behavior of the system, make predictions, or design engineering solutions. Understand the limitations of the first-order non-singular perturbation approach and validate the results through comparisons with numerical simulations or experimental data when possible. By following these steps, first-order non-singular perturbation theory provides a systematic approach to approximate solutions to differential equations in situations where perturbations become significant in specific regions or under certain condi-tions. This method allows researchers and engineers to tackle complex problems that cannot be explained using regular perturbation theory alone.
Approximate Solution of an Initial-value Problem
The initial value problem was chosen by Fowkes, (1968) and the boundary value problem was chosen by Baltaveva and Agarwal, (2018), which gives the following,
y"=f(x)y, y(0) = 1, y (0) =1, (16)
Where, f(x) is continuous, this problem has no closed-form solution except for very special choices for f(x).
First, we introduced an ε as,
y"= ε f(x)y, y(0) =1, y(0) =1. (17)
Secondly, we take y(x) as,
y(x)=∑_(n=0)^∞▒〖ε^n y_n (x)〗. (18)
Where, and (n ³ 1)
The zeroth-order problem y"=0 is obtained by setting ε=0. The nth-order problem (n ³1) is obtained by Eq. (18) into Eq. (17) and setting the coefficient of εn (n ³1) equal to zero. The result is
. (19)
The solution to (19) is
y_n=∫_0^x▒〖dt∫_0^t▒〖f(s) y_(n-1) (s)ds, n≥1〗〗. (20)
Now, the successive terms in the parturition series (3):
… (21)
The nth term in this series is by with upper bound K for |f(t)| in the internal 0 £|t|£x|.
Neumann Boundary Conditions
Neumann boundary conditions developed by Kadum and Abdul-Hassan, (2023) are commonly encountered in partial differential equations (PDEs). They specify the function at the boundary of a domain rather than specifying the function itself. Neumann boundary condition concept arise in various physical and mathe-matical contexts, particularly in problems involving diffusive processes, heat transfer, fluid dynamics, and electromagnetism was given by Saltzman, (1962).
Consider the differential equation,
y^+y=0. (22)
On the interval [a,b] take the form:
y^ (a)=α and y^ (b)=β. (23)
Where, α and β are given numbers. Now, we have
∇^2 y+y=0. (24)
Where, ∇^2 denotes the given conditions on a domain Ω⊂R^n take the form:
∂y/∂n (x)=f(x) ∀x∈∂Ω. (25)
The normal derivative which shows up on the left-hand side is defined as:
∂y/∂n (x)=∇y(x).n(x). (26)
Where, ∇ is the gradient (vector) and the dot is the inner product.
The Perturbation Process employ to the outcome of an Algebraic Equation
The Perturbation Method Expanded with pro-posed equation
Consider the following equation. First, the equ-ation is introduced as seen below.
x^2 + ε x – 1 = 0, 0<εε<<1.
The Perturbation Technique for Solving Alge-braic Equation
Leading Order Solutions
The approximation process to x^2+ε x-1=0 is to set ε =0.
This reduces to:
x^2=1. (27)
Or,
x=±1. (28)
First-order solution
The approximation with second order,
x=±1+d (x), (29)
Where, d(x) is some connection factor.
Inserting x=±1 into x2+ε x-1=0 yields;
(1+δ(x))(1+δ(x))+ε(1+δ(x))-1=0. (30)
Expanding (30) Þ
δ〖(x)〗^2+2δ(x)+1+ε+εδ(x)-1=0. (31)
Solving the remainder for (31) yields,
. (32)
Substitution of d(x) back into Eq. (29) for the positive roots yields:
. (33)
Second-order solution
The approximation with third result is,
(34)
Putting (34) into yields:
. (35)
Expanding (35) Þ
. (36)
The equation (36) is employ for b(x),
. (37)
Substitution of b(x) back into (34) for yields the third positive root approximation:
. (38)
Analytic Result
Here, gusting the solution by Shampine, (1968),
(39)
and Substituting equation y=eRx into the following equation.
Now,
ε R2 + 2R + 2 = 0. (40)
For roots R1 and R2,
. (41)
and . (42)
Now,
. (43)
Substitution of equations (41) and (42) into equation (43) yields the following:
. (44)
Enforcement of the initial condition seen in equation y(0)=0 to Eq. (44) yields:
C1=C2. (45)
Substituting Eq. (45) into (44) Eq. (44) yields,
=1 (46)
Solving Eq. (46) for C2 and then using Eq. (45) to solve for C1 yields:
(47)
and (48)
Substitution of equations (47) and (48) into equation (44) which gives,
Comparison of Perturbation Approximation to the Analytical Solution
Let, ε=0.01, valves determined from equations (41), (42), (47) and (48) the given table,
Table 1: Analytical results obtained for the ordinary differential equation.
Approximate Solutions of an IVP using the Perturbation Method
The most important and excellent example of perturbation in this paper is example 1. It can be taken as a notable example of the perturbation solution of given equation. It shows how much good the perturbation solution is!
Example 1
If ε≪1, obtain the perturbed equation from
y^-εxy=0,y(0)=1,y^ (0)=1
Solution: Given,
y^-εxy=0,y(0)=1,y^ (0)=1. (50)
Let y(x)=y_0 (x)+εy_1 (x)+ε^2 y_2 (x). (51)
Therefore,y^ (x)=y_0^ (x)+εy_1^ (x)+ε^2 y_2^ (x). (52)
Since, y(0)=y_0 (0)+εy_1 (0)+ε^2 y_2 (0)+⋯………………………………..
∴1=y_0 (0)+εy_1 (0)+ε^2 y_2 (0)+⋯………………………………..
y_0 (0)=1 and y_n (0)=0,n≥1
Also, y^ (0)=y_0^ (0)+εy_1^ (0)+ε^2 y_2^ (0)+⋯…………………..
∴1=y_0^ (0)+εy_1^ (0)+ε^2 y_2^ (0)+⋯…………………..
y_0^ (0)=1 and y_n^ (0)=0,n≥1
Putting (51) and (52) in (50) we get
y_0^+εy_1^+ε^2 y_2^+..……=εx[y_0+εy_1+ε^2 y_2+⋯]. (53)
Eq. (53) is an identity. It is true only when the coefficients of the like powers of ε from both sides are equal.
y_0^=0,y_0 (0)=1,y_0^ (0)=1
y_1^=xy_(0 , ) y_1 (0)=0,y_1^ (0)=0
y_2^=xy_(1 , ) y_2 (0)=0,y_2^ (0)=0
Proceeding in this way, we have
y_n^=xy_(n-1 , ) y_n (0)=0,y_n(0)=0.
Now,
y_0^=0
y_0^=c_1
∴y_0=c_1 x+c_2
y_0 (0)=c_1.0+c_2
∴c_2=1
y_0^=c_1
y_0^ (0)=c_1
∴c_1=1
∴y_0=x+1
Again, y_1^=x
∴y_1=x^4/12+x^3/6+c_1 x+c_2
y_1 (0)=c_2
∴c_2=0
And
y_1^ (0)=c_1
∴c_1=0
∴y_1=x^4/12+x^3/6
Again, y_2=xy_1
y_2=x^5/12+x^4/6
〖∴y〗_2=x^6/72+x^5/30 +c_1
〖∴y〗_2=x^7/504+x^6/180 +c_1 x+c_2
y_2 (0)=c_2
∴c_2=0
And,
y_2^ (0)=c_1
∴c_1=0
〖∴y〗_2=x^7/504+x^6/180
∴ The required perturbation solution is
y(x)=1+x+ε(x^3/6+x^4/12)+ε^2 (x^6/180+x^7/504)+⋯
Example 2
y^=-e^(-x) y , y(0)=y^ (0)=1
We want to find out its solution by different methods and show a comparison among those solutions.
Solution:
The perfect solution given,
y^=-e^(-x) y , y(0)=y^ (0)=1. (54)
Using the substitution,
z=2e^(-x/2) . (55)
In the equation, we take the equation of the form,
z (d^2 y)/〖dz〗^2 +dy/dz+zy=0 . (56)
Two linearly independent solutions of Eq. (56) are J_0 (z), and Y_0 (z) . Therefore, the optimum output is given by,
y(x)=c_1 J_0 (z)+〖c_2 Y〗_0 (z), Where, z=2e^(-x/2) . (57)
This gives,
y^ (x)={c_1 J(z)}(〖-e〗^(-x/2) ) . (58)
For c1 and c2, in (57) and (58) finally we get,
c_1=(〖Y^〗_0 (2)+Y_0 (2))/(J_0 (2) 〖Y^〗_0 (2)-〖J^〗_0 (2)Y_0 (2)). (59)
c_2=-(〖J^〗_0 (2)+J_0 (2))/(J_0 (2) 〖Y^〗_0 (2)-〖J^〗_0 (2)Y_0 (2)) . (60)
Putting these values in Eq. (57), we obtain the exact solution
y(x)=({〖Y^〗_0 (2)+Y_0 (2)} J_0 (2e^(-x/2) )-{〖J^〗_0 (2)+J_0 (2)} Y_0 (2e^(-x/2) ))/(J_0 (2) 〖Y^〗_0 (2)-〖J^〗_0 (2) Y_0 (2)) . (61)
The perturbation Solution:
We change the problem into a perturbation problem by introducing the parameter ε in such a way that the unperturbed problem is solvable,
Y"=-εe^(-x) y , y(0)=y(0)=1 . (62)
We take a perturbation expansion for y(x) of the form,
. (63)
Hence
. (64)
Now with the use of (63) and (64), (62) takes the form
∑_(n=0)^∞▒ε^n y_n^ (x)=-εe^(-x) ∑_(n=0)^∞▒ε^n y_n (x)
i.e. y_0^+εy_1^+ε^2 y_2^+ε^3 y_3^+⋯+ε^n y_n^…=-〖εe〗^(-x) y_0-〖ε^2 e〗^(-x) y_1-…-ε^n e^(-x) y_(n-1)-… (65)
This gives a sequence,
y_0^=0,y_o (0)=1,y_0^ (0)=1. (66)
y_n^=e^(-x) y_(n-1), y_n (0)=0,y_n^ (0)=0, n≥1. (67)
Solving (67) we get, y_0=1+x.
Now for n=1, y_1^=e^(-x) y_0 ,〖 y〗_1 (0)=0, y_1^ (0)=0, whose solution is,
y_1=∫_0^x▒〖dt∫_0^t▒〖(1+s)(〗〗 〖-e〗^(-s))ds=3-2x-(3+x)e^(-x).
For n=2 ,y_2^=e^(-x) y_1 ,〖 y〗_2 (0)=0,y_2^ (0)=0, whose solution is,
y_2= ∫_0^x▒〖dt∫_0^t▒(〗 〖-e〗^(-u))[y_1 (u)]du=-2+3/4 x+(1+2x) e^(-x)+(1+x/4)e^(-2x).
We find,
y_3=41/108-1/9 x+(1/2-3x/4) e^(-x)-(3/4+x/2) e^(-2x)-(7/54+x/36)e^(-3x).
and
y(x)=y_0+εy_1+〖ε^2 y〗_2+〖ε^3 y〗_3.
When ε=1, we get the approximate solution up to fourth-term as,
y(x)=1+x+3-2x-(3+x) e^(-x)-2+3/4 x+(1+2x) e^(-x)+(1+x/4) e^(-2x)+41/108-1/9 x+(1/2-3x/4) e^(-x)-(3/4+x/2) e^(-2x)-(7/54+x/36) e^(-3x) (68)
Example 3
An approximation solution of the formidable-looking non-linear two-point boundary value problem -
. (69)
May be readily obtained using perturbation theory.
Since it is not possible to find the exact solution to this problem, it can also be solved numerically using nonlinear shooting with Newtons Method.
Perturbation solution
First, we introduce ε in such that the under-handed problem is solvable:
.
Or, (x+y^2 )(y^+y)=εcosx. (A)
Secondly, let us assume that a perturbation expansion for y(x) of the form
Þ(3 + y.y) (y" + y) = εCosx
Þ = εcosx
Þ
Again,
Þ
.
Equating the coefficient of various powers of ε, we get.
y_0^+y_0=0,y_0 (0)=2,y_0 (π/2)=2. (70)
y_1^+y_1=cosx/(3+y_0^2 ),y_1 (0)=0,y_1 (π/2)=0. (71)
The A.E of (70) is, m2 + 1 = 0 Þ m = ± i
yc = c1Cosx + c2Sinx. (72)
y0 (0) = c1.1+c2.0 = 2 Þ c1=2
and
(72) Þ .
(71) Þ
y_1^+y_1=cosx/(7+4sin2x). (73)
We assume that, y_p (x)=v_1 (x)sinx+v_2 (x)cosx. (74)
Þ
We impose the condition,
v_1 (x)sinx+v_2 (x)cosx=0. (75)
y_p^ (x)=-v_1 (x)sinx-v_2 (x)cosx+v_1 (x)cosx-v_2 (x)sinx. (76)
Putting these values from (76) and (74) in (7) Þ
Þ v_1 (x)cosx-v_2 (x)sinx=cosx/(7+4sin2x). (77)
Now, (75) and (77) Þ
and .
Þ .
The solution of Eq. (73) can be given in the following form:
For, then, and .
A comparison of perturbation series approxi-mation and Numerical solution of the boun-dary value problem in (A) is presented in the following figure. The graphs are one-term perturbation series approximation and two-term perturbation series approximation (das-hed line) of the form in (C).
The present paper deals with the perturbation method and implies how much effective it is to solve a nonlinear differential equation involving initial as well as boundary conditions. Numerical solutions have been used to fulfill the investigation. Compari-sons between numerical and perturbation solutions have been performed using Mathematica and For-tran Programming. Perturbation methods offer a valuable approach for approximating solutions to differential equations encountered in diverse scien-tific and engineering contexts. By systematically treating small deviations from known solutions, perturbation methods provide insights into system behavior, facilitate analytical predictions, and aid in the design of engineering systems. Through theo-retical developments, numerical techniques, and practical applications, perturbation methods con-tinue to play a vital role in advancing our under-standing of complex dynamical systems.
Firstly, I want to thank Almighty Allah. Then I like to thank my family for their support over the course of my graduate study. It is a matter of great pleasure to me expresses my heartiest grati-tude and profound respect to my supervisor Assistant Professor Bishnu Pada Ghosh, Dept. of Mathematics, Jagannath University, Dhaka. Also, my deepest thanks go to Kbd Md. Ashraful Kabir for spending time mentoring me during this research .
The authors declare that there are no conflicts of interest.
Academic Editor
Dr. Toansakul Tony Santiboon, Professor, Curtin University of Technology, Bentley, Australia.
Khatun S., and Iqbal MA. (2024). Application of Perturbation method to approximate the solutions of differential equation, Int. J. Mat. Math. Sci., 6(3), 62-77. https://doi.org/10.34104/ijmms.024.062077