IHJPAS. 36(2)2023 

430 
This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

Abstract 

The integral transformations is a complicated function from a function space into a simple function 

in transformed space. Where the function being characterized easily and manipulated through 

integration in transformed function space. The two parametric form of SEE transformation and its 

basic characteristics have been demonstrated in this study. The transformed function of a few 

fundamental functions along with its time derivative rule is shown. It has been demonstrated how 

two parametric SEE transformations can be used to solve linear differential equations. This 

research provides a solution to population growth rate equation. One can contrast these outcomes 

with different Laplace type transformations.  

 

Keywords: Two parametric SEE transformation, Population growth rate equation, linear 

differential equations, time derivative rule. 

 

1.Introduction 

As The Transformations Have numerous applications in fields of engineering and other 

sciences. Numerous mathematicians presented the Fourier and Laplace transformations theory. In 

[1] authors used this technique on delay differential equations. The modified Laplace series 

findings from the Adomian polynomial decomposition method were used to solve the differential-

integral equations can be seen in [2]. The non-linearity was circumvented by using standard 

Adomian polynomials. The Laplace decomposition approach and the pade approximation were 

put to practise in order to determine the numerically approximated results of the system of partial 

on linear differential equations in [3]. The Laplace decomposition approach, which was improved 

utilised in [4] for differential equalities of the Emden-Lane type. The author used an analytical 

doi.org/10.30526/36.2.3251 

Article history: Received 30 January 2023, Accepted 26 February 2023, Published in April 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Solution of Population Growth Rate Linear Differential Model via Two 

Parametric SEE Transformation 

 
Ali Moazzam 

University of Agriculture Faisalabad, 

Pakistan. 

alimoazzam7309723@gmail.com 

 

Hira Aslam 
University of Agriculture, Faisalabad, 

Pakistan. 

hiraaslam053@gmail.com 

.  
 

Naila Tabassum 

University of Agriculture Faisalabad, 

Pakistan. 

nailatabassumgp@gmail.com 

 

Emad A. Kuffi 

College of Engineering, Al-Qadisiyah 

University, , Iraq. 

 emad.abbas@qu.edu.iq 
 

 

https://creativecommons.org/licenses/by/4.0/
mailto:alimoazzam7309723@gmail.com
mailto:hiraaslam053@gmail.com
mailto:nailatabassumgp@gmail.com
mailto:emad.abbas@qu.edu.iq


IHJPAS. 36(2)2023 

431 
 

approach to resolve the first and second types of non-linear Volterra-Fredholm integro differential 

equations in [5] by combining the modified Laplace Adomian substitution and Laplace Adomian 

polynomial decomposition methods (LADM). Laplace integral techniques are frequently utilised 

in the engineering and scientific fields. The use of integral transformations to correct erroneous 

integrals that contained error functions has been covered by a number of authors [6–9]. In [10] 

authors obtained the solutions to Abel's equation using the Kamal integral method. Solution to the 

error function based on the Kamal integral approach has shown in [11]. To solve linear partial 

differential equations with more than two independent variables, writers in [12] described the 

substitution approach of the Laplace transformation. In [13] authors used the Matlab programme 

to illustrate how the effects of bio-convection and activation energy on the Maxwell equation on 

nano-fluid. With the use of the Laplace technique of transformation, an analytical solution to the 

advection diffusion equation in a one-dimensional, semi-infinite media has been developed by 

authors of [14]. In [15 and 16] new transformations to solve new type differential equations with 

trigonometric coefficients was introduced in 2021 byMoazzam et. al. The solution to the 

temperature problem and some of the applications of the Kamal transformation in thermal 

engineering are shown in [17].  The SEE transformation, a new transformation that is very useful 

in the solution of differential equations and systems of differential equations, has been 

demonstrated in [18]. Some others transformations having different kernels then Laplace 

transformation are used to find the solutions of linear differential equations as Al-Zughair 

transformation is used to demonstrate the solution of moment Pareto differential equations with 

logarithmic coefficients in [19]. In paper [20], a new logarithmic kernel transformation was 

demonstrated which is useful for solving both ordinary differential equations and differential 

equations with a logarithmic coefficient. 

 

2.Definitions and Preliminaries  

 In this paper, we have introduced the two parametric SEE transformation, which is a 

generalised version of the SEE transformation and is used in this study to solve linear differential 

equations. The definition of two parametric SEE transformations is given below. 

 

Definition 1: Two parametric SEE integral transform of the function 𝑓(𝑡) for all 𝑡 ≥ 0 is defined 

as: 

𝑆𝐸𝐸𝛼,𝛽 {𝑓(𝑡)} =
𝑒 −𝛽𝒗

(𝛼𝑣)𝑛
∫ 𝑓(𝑡)𝑒−

(𝛼𝑣)𝑡 𝑑𝑡
∞

0
= 𝐹 𝛼,𝛽 (𝑣)    

 (1) 

Where 𝛼, 𝛽 are real numbers, 𝑛 is the integer and 𝑣 is transformed variable. By choosing the values 

of parameters 𝛼, 𝛽 real numbers and integer 𝑛, we can get the different transformations as Laplace, 

Kamal, Mohand, SEE, Sumudu, Aboodh, and many other Laplace type transformations. 

 

 

 

 

 

 

 

 

 



IHJPAS. 36(2)2023 

432 
 

Table 1: Two parametric SEE transformation of some function 

Serial Numbers Function 𝒇(𝒕) 𝑺𝑬𝑬𝜶,𝜷{𝒇(𝒕)} =
𝒆−𝜷𝒗

(𝜶𝒗)𝒏
∫ 𝒇(𝒕)𝒆−(𝜶𝒗)𝒕𝒅𝒕

∞

𝟎

 

1.  𝑘 𝑒 −𝛽𝑣𝑘

(𝛼𝑣)𝑛+1
  

2.  1 𝑒 −𝛽𝑣

(𝛼𝑣)𝑛+1
 

3.  𝑡 𝑒 −𝛽𝑣

(𝛼𝑣)𝑛+2
 

4.  𝑡 2 𝑒 −𝛽𝑣 2!

(𝛼𝑣)𝑛+3
 

5.  𝑡 𝑚 𝑒 −𝛽𝑣 𝑚!

(𝛼𝑣)𝑛+𝑚+1
  

6.  𝑒 𝑢𝑡 𝑒 −𝛽𝑣

(𝛼𝑣)𝑛(𝛼𝑣 − 𝑢)
 

7.  𝑠𝑖𝑛 (𝑢𝑡) 𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
[

𝑢

(𝛼𝑣)2 + 𝑢2
] 

8.  𝑐𝑜𝑠(𝑢𝑡) 𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
[

𝛼𝑣

(𝛼𝑣)2 + 𝑢2
] 

9.  𝑐𝑜𝑠ℎ (𝑢𝑡) 𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
[

𝑢

(𝛼𝑣)2 + 𝑢2
] 

10.  𝑠𝑖𝑛ℎ (𝑢𝑡) 𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
[

𝛼𝑣

(𝛼𝑣)2 − 𝑢2
] 

 

Theorem 1: Two parametric SEE transform for first, second up-to nth order derivative can be 

described as: 

1. 𝑆𝐸𝐸𝛼,𝛽 {𝑓
′(𝑡)} = −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓(0) + (𝛼𝑣)𝐹𝛼,𝛽 (𝑣), 

2. 𝑆𝐸𝐸𝛼,𝛽 {𝑓
′′(𝑡)} = −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓 ′(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−1
𝑓(0) + (𝛼𝑣)2𝐹𝛼,𝛽 (𝑣), 

3. 𝑆𝐸𝐸𝛼,𝛽 {𝑓
′′′(𝑡)} = −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓 ′′(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−1
𝑓 ′(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−2
𝑓(0) + (𝛼𝑣)3𝐹𝛼,𝛽 (𝑣), 

4. 𝑆𝐸𝐸𝛼,𝛽 {𝑓
(𝑚)(𝑡)} = −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓 (𝑚−1)(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−1
𝑓 (𝑚−2)(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−2
𝑓 (𝑚−3)(0) − ⋯ −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−𝑚+1
𝑓(0) + (𝛼𝑣)𝑚𝑆𝐸𝐸𝛼,𝛽 {𝑓(𝑡)} 

Proof : To prove 1, let double parametric SEE transformation for 𝑓(𝑡) = 𝑓 ′(𝑡), 

𝑆𝐸𝐸𝛼,𝛽 {𝑓
′(𝑡)} =

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
∫  𝑓′(𝑡)𝑒−

(𝛼𝑣)𝑡 𝑑𝑡 
∞

0
   

=
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓(𝑡)|0

∞ + (𝛼𝑣) ∫  𝑓(𝑡)𝑒−
(𝛼𝑣)𝑡 𝑑𝑡 

∞

0
  

= −
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓(0) + (𝛼𝑣)𝐹𝛼,𝛽 (𝑣)         (2) 

Let double parametric SEE transformation for 𝑓(𝑡) = 𝑓 ′′(𝑡). 

𝑆𝐸𝐸𝛼,𝛽 {𝑓
′′(𝑡)} =

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
∫  𝑓′′(𝑡)𝑒−

(𝛼𝑣)𝑡 𝑑𝑡 
∞

0
  

=
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓′(𝑡)|0

∞ + (𝛼𝑣) ∫  𝑓′(𝑡)𝑒−
(𝛼𝑣)𝑡 𝑑𝑡 

∞

0
  

= −
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓′(0) + (𝛼𝑣)𝑆𝐸𝐸𝛼,𝛽 {𝑓

′(𝑡)}. 



IHJPAS. 36(2)2023 

433 
 

By the help of (2) we can get: 

= −
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓′(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−1
𝑓(0) + (𝛼𝑣)2𝐹𝛼,𝛽 (𝑣)      (3) 

Let double parametric SEE transformation for 𝑓(𝑡) = 𝑓 ′′′(𝑡). 

𝑆𝐸𝐸𝛼,𝛽 {𝑓
′′′(𝑡)} =

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
∫  𝑓′′′(𝑡)𝑒−

(𝛼𝑣)𝑡 𝑑𝑡 
∞

0
  

=
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓′′(𝑡)|0

∞ + (𝛼𝑣) ∫  𝑓′′(𝑡)𝑒−
(𝛼𝑣)𝑡 𝑑𝑡 

∞

0
  

= −
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓′′(0) + (𝛼𝑣)𝑆𝐸𝐸𝛼,𝛽 {𝑓

′′(𝑡)}. 

By the help of (3) we can get: 

 = −
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓′′(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−1
𝑓′(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−2
𝑓(0) + (𝛼𝑣)3𝐹𝛼,𝛽 (𝑣)   (4) 

Similarly we can find the result form the statement 4 mentioned in theorem: 

𝑆𝐸𝐸𝛼,𝛽 {𝑓
(𝑚)(𝑡)} = −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑓 (𝑚−1)(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−1
𝑓 (𝑚−2)(0) −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−2
𝑓 (𝑚−3)(0) − ⋯ −

𝑒 −𝛽𝑣

(𝛼𝑣)𝑛−𝑚+1
𝑓(0) + (𝛼𝑣)𝑚𝑆𝐸𝐸𝛼,𝛽 {𝑓(𝑡)}        (5) 

Linear Property: 

Let 𝑓(𝑡) and 𝑔(𝑡) be two functions then linearity property of two parametric extensions of 

SEE transformation define as: 

𝑆𝐸𝐸𝛼,𝛽 {𝑐1𝑓(𝑡) ± 𝑐2𝑔(𝑡)} = 𝑐1𝑆𝐸𝐸𝛼,𝛽 𝑓(𝑡) ± 𝑐2𝑆𝐸𝐸𝛼,𝛽 𝑔(𝑡)    (6) 

Where 𝑐1 and 𝑐2 are constants. 

Change of Scale Property: 

Let 𝑓(𝑡) be a function and 𝑆𝐸𝐸𝛼,𝛽 of 𝑓(𝑡) is 𝐹𝛼,𝛽 (𝑣) then change of scale property of two 

parametric SEE transformation can be described as: 

𝑆𝐸𝐸𝛼,𝛽 {𝑓(𝑐𝑡)} = (
1

𝑐𝑛−1
) 𝐹𝛼,𝛽 (

𝑣

𝑐
)        (7) 

Here 𝑐 is a constant value. 

Shifting Property: 

Let 𝑓(𝑡) be a function and 𝑆𝐸𝐸𝛼,𝛽 of 𝑓(𝑡) is 𝐹𝛼,𝛽 (𝑣) then shifting property of two 

parametric SEE transformation can be described as: 

 𝑆𝐸𝐸𝛼,𝛽 {𝑒
𝑐𝑡 𝑓(𝑡)} = 𝐹𝛼,𝛽 (𝑣 − 𝑐)         (8) 

Theorem 2. 

 Let 𝑓(𝑡) and 𝑔(𝑡) be two functions and 𝑆𝐸𝐸𝛼,𝛽 of 𝑓(𝑡) is 𝐹𝛼,𝛽 (𝑣), 𝑆𝐸𝐸𝛼,𝛽 of ℎ(𝑡) is 

𝐻𝛼,𝛽 (𝑣) then the convolution of two parametric 𝑆𝐸𝐸 transform can be described as: 

 𝑆𝐸𝐸𝛼,𝛽 {𝑓(𝑡) ∗ ℎ(𝑡)} = −(𝛼𝑣)
𝑛𝐹𝛼,𝛽 (𝑣). 𝐻𝛼,𝛽 (𝑣)      (9) 

 

3.APPLICATION 

Population Growth Rate Problem: Exponential growth is linked to the process of population 

growth rate equation decline. Authors in [21] have mathematically described the decay problems 

of a substance with the help of first order ordinary linear differential equation: 

 
𝑑𝑃

𝑑𝑡
= −𝜇𝑃          (10) 

With the initial condition: 

 𝑃(0) = 𝑃0          (11) 



IHJPAS. 36(2)2023 

434 
 

Where 𝜇 is a real positive number, 𝑃 is the value of population at 𝑡-time and 𝑃0 is the initial 

value of population at initial time 𝑡0. 

Sol: By applying the two parametric SEE transform on (10) and by using the rule mentioned in 

(2) we will get: 

 −
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑃(0) + (𝛼𝑣)𝑃𝛼,𝛽 (𝑣) = −𝜇𝑃𝛼,𝛽 (𝑣). 

By using initial condition in (11) we get: 

 −
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑃0 + (𝛼𝑣)𝑃𝛼,𝛽 (𝑣) = −𝜇𝑃𝛼,𝛽 (𝑣)  

 [(𝛼𝑣) + 𝜇]𝑃𝛼,𝛽 (𝑣) =
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛
𝑃0 

 𝑃𝛼,𝛽 (𝑣) =
𝑒 −𝛽𝑣

(𝛼𝑣)𝑛[(𝛼𝑣)+𝜇]
𝑃0        (12) 

By using the definition in (1) and table values we get: 

 𝑃(𝑡) = 𝑃0𝑒
−𝜇𝑡         (13) 

Equation 13 provides the solution of population growth rate problem. 

 

4.Conclusion 

SEE transformation with two parameter has been defined in this research. The solution of 

population growth rate equation has been described to show the efficiency of this proposed 

technique which can be compared with other Laplace type transformation. It can be seen that it is 

helpful in solving linear differential models. The proposed transformation can turned into ordinary 

SEE transformation by adjusting the values of parameters. 

 

References: 

1.Evans D.J.; Raslan K.R. The Adomian Decomposition Methhod for Solving Delay Differential 

Equation. International Journal of Computer Mathematics, 2005, 82(1), 49-54. 

2.Shah, K.; Khalil, H.; Khan, RA. A Generalized Scheme Based on Shifted Jacobi Polynomials 

for Numerical Simulation of Coupled Systems of Multi-term Fractional-order Partial Differential 

Equations. LMS Journal of Computation and Mathematics, 2017, 20(1), 11-29. 

3.Abassy, T.A.; El-Tawil, M.A.; El-Zoheiry, H. Exact Solutions of Some Nonlinear Partial 

Differential Equations Using the Variational Iteration Method Linked with Laplace Transforms 

and the Padé Technique. Computers and Mathematics with Applications. 2007, 54(7-8), 940-54. 

4.Yindoula, J.B.; Youssouf, P.; Bissanga, G.; Bassono, F. Some B. Application of the Adomian 

Decomposition Method and Laplace Transform Method to Solving the Convection Diffusion-

Dissipation Equation. International Journal of Applied Mathematics Research. 2014, 3(1), 30-35. 

5.Hamoud, A.A.; Ghadle, K.P. The Combined Modified Laplace with Adomian Decomposition 

Method for Solving the Nonlinear Volterra-Fredholm Integro Differential Equations. Journal of 

the Korean Society for Industrial and Applied Mathematics, 2017, 21(1), 17-28. 

6.Aggarwal, S.; Gupta, A.R.; Kumar, Mohand, D. Transform of Error Function. International 

Journal of Research in Advent Technology, 2019, 7(5), 224-231. 

7.Aggarwal, S.; Sharma, S.D. Sumudu Transform of Error Function. Journal of Applied Science 

and Computations. 2019, 6(6), 1222-1231. 



IHJPAS. 36(2)2023 

435 
 

8.Aggarwal, S.; Bhatnagar, K.; Dua, A. Dualities Between Elzaki Transform and Some Useful 

Integral Transforms. International Journal of Innovative Technology and Exploring Engineering, 

2019, 8(12), 4312-4318. 

9.Aggarwal, S.; Gupta, A.R.; Sharma, S.D.; Chauhan, R.; Sharma, N. Mahgoub Transform 

(Laplace-Carson Transform) of Error Function. International Journal of Latest Technology in 

Engineering, Management & Applied Science, 2019, 8(4), 92-98. 

10.Aggarwal, S.; Sharma, S.D. Application of Kamal Transform for Solving Abel’s Integral 

Equation. Global Journal of Engineering Science and Researches, 2019, 6(3), 82-90. 

11.Aggarwal, S.; Singh, G.P. Kamal Transform of Error Function. Journal of Applied Science and 

Computations, 2019, 6(5), 2223-2235. 

12.Fariha, M.; Muhammad, Z.I.; Moazzam, A.; Usman, A.; Muhammad, U.N. Subtituition Method 

Using the Laplace Transformation for Solving Partial Differential Equations Involving More than 

Two Independent Variables. Bulletin of Mathematics and Statistics Research. 2021, 9(3), 104-116. 

13Muhammad, I.K.; Khurrem, S.; Muhammad, M.U.R.; Maria; Moazzam, A. Influence of 

Bioconvection and Activation Energy on Maxwell Flow of Nano-fluid in the Existence of Motile 

Microorganisms Over a Cylinder. International Journal of Multidisciplinary Research and 

Growth Evaluation. 2021, 2(5), 135-144. 

14.Ammara, A.; Madiha, G.; Amina, A.; Moazzam, A. Laplace Transforms Techniques on 

Equation of Advection-Diffussion in One-Dimensional with Semi-Infinite Medium to Find the 

Analytical Solution. Bulletin of Mathematics and Statistics Research. 2021, 9(3), 86-96. 

15.Muhammad, K.; Khurrem, S.; Moazzam, A.; Ammara, A.; Mariyam, B. New Transformation 

‘AMK-Transformation’ to Solve Ordinary Linear Differential Equation of Moment Pareto 

Distribution. International Journal of Multidisciplinary Research and Growth Evaluation. 2021, 

2(5), 125-134. 

16.Moazzam, A.; Kashif, M.; Amjed, U.; Khawar, M.I. Development of A new transformation to 

Solve a New Type of Ordinary Linear Differential Equation. Bulletin of Mathematics and Statistics 

Research (Bomsr). 2021, 9(3), 56-60. 

17.Muhammad, W.; Khurrem, S.; Moazzam, A.; Alizay, B. Applications of Kamal Transformation 

in Temperature Problems. Scholar Journal Engineering and Technology. 2022, 10(2), 5-8.  

18.Eman, A.; Mansour; Emad, A. K.; Sadiq, A.; Mehdi. On the SEE Transform and System of 

Ordinary Differential Equations. Periodicals of engineering and natural sciences, 2021, 9(3), 277-

281.  

19.Moazzam, A.; Muhammad, Z.I. ;Al-Zughair Transformations on Linear Differential Equations 

of Moment Pareto Distribution. Proc. 19th International Conference on Statistical Science, 2022, 

36, 199-206. 

20.Moazzam, A.; Muhammad, Z.l. A new Integral Transform "Ali And Zafar" Transformation and 

It's Application in Nuclear Physics. Proc. 19th International Conference on Statistical Sciences, 

2022, 36, 177-182. 

21.Aggarwal, S.; Gupta, A.R.; Singh, D.P.; Asthana, N.; Kumar, N. Application of Laplace 

Transform for Solving Population Growth and Decay Problems. International Journal of Latest 

Technology in Engineering, Management & Applied Science. 2018, 7(9), 141-145.