SAWI TRANSFORMATION FOR SOLVING A SYSTEM OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS

ABSTRACT


Sawi Transformation; Inverse Sawi Transformation; System of Ordinary Differential Equations;
First-Order Chemical Reaction; Mass-Spring System.

INTRODUCTION
The integral transform method is a convenient mathematical tool.It can quickly deal with the first choice of researchers for solving advanced problems and solutions to critical issues in various sciences, for example, astronomy, economics, telecommunications, statistics, thermal sciences, marine sciences, biology, chemistry, physics, mathematics, medicine, aerodynamics, civil engineering, control theory, cardiology, and mechanics.Some real problems in the world can be expressed in multiple forms of mathematical modelling.The modellings are in the form of equations or a system of equations such as the ordinary differential equation.An ordinary differential equation is a differential equation that contains the ordinary derivative of one or more dependent variables concerning one independent variable (see [1]).In contrast, a system of differential equations is an equation consisting of two or more related equations.Systems of differential equations are classified into two systems: linear and nonlinear differential equations [2].To solve a systems of differential equations can be used a matrix Laplace transformation method.Especially, we can use it to solve initial value problems of second-order homogenous linear systems of differential equations with constant coefficients [3].
The Sawi transformation method is a form of integral transformation obtained from the Fourier integral so that the Sawi transformation and its basic properties are obtained.This method was introduced by Mahgoub in 2019 and effectively solves problems in ordinary and partial differential equations [4].Systems in ordinary differential equations are widely applied to various problems, such as deflection problems in curve beams, problems in three-layered beams, aircraft control problems in cosmic space, problems in chemical reaction circuits, and others [5].There are quite several studies on the Sawi transformation, but there are still few studies on the application of the Sawi transformation in various problems in various fields.The study of the Sawi transformation in various problems was carried out by Aggarwal (2019), successfully applying the transform to the problem of population growth and decline [6].In 2021, Sahoo applied the Sawi transformation to solve problems in Newton's law of cooling [7].Higazy and Aggarwal in 2021 succeeded in applying the Sawi transformation to find a concentration solution from a first-order chemical reaction by considering four numerical problems [8].Aggarwal  The development of research in the last decade on Sawi transformation has produced many new things.For example, the Sawi decomposition method combines the transformation and Sawi decomposition methods.This method can be used to determine the primitive Volterra integral equation (V.I.E.) with applications [11]- [15].This article contains the following: in part 2, a review is given containing the definitions and characteristics of the Sawi transformation, then section 3 contains two issues that have been defined, namely the chemical reaction system of three certain chemical substances to determine their concentration and the forced motion mass-spring system.to determine the spring displacement of the two masses.Then it is described the completion of the analytic solutions of the two problems step by step using the Sawi transformation so that the analytic solutions are presented in the form of numerical tables and graphs, and section 4 contains the conclusions of the research results.

RESEARCH METHODS
This article was obtained based on the research initiated by observing some literature regarding Sawi's transformation theory and its research developments.A few definitions and theorems are given in this section to support the research.Besides that, the literature study provides information that one of the tools to solve the initial value problem has a similar solution procedure to the Laplace transform, known as the Sawi transformation.To solve the initial value problem in the form of a system of linear ordinary differential equations, we show that the Sawi transformation can be used if several additional modifications and identity forms appropriated to the problem are given.We provide it in this section with examples of applying the resulting theory.Two examples (a first-order chemical reaction system with three specific chemicals and a mass-spring system with forced motion) are also involved.For numerical work, we focused on the concentration of chemical reactants to solve a problem that can be interpreted and easily understood for several different initial values and reaction rate constants.Compared to other methods (Laplace transformation), the results obtained from using the Sawi transformation for the cases discussed show that the analytical solutions for the selected initial values have similar solutions.

Definition of Sawi Transformation
Mahgoub (2019) define that if given a set with its members is an exponential function with power so that it can be written [4], With M is a finite number and 12 , kk is a finite or infinite number.Then the Sawi transformation is ( ) ft denoted by the operator ( )

Properties of the Sawi Transformation
The Sawi transformation has properties with the following definitions (see [8] and [16] for detail).
1.The linearity property of the Sawi transformation is defined that if Where , mn is an arbitrary constants.
2. The scaling properties of the Sawi transformation are defined as follows a.If () T  is the Sawi transformation of () ft then () b.The translation properties of the Sawi transformation are defined that if the Sawi transformation () of the Sawi transform of the function () kt e f t

The Derivative Function of the Sawi Transformation
The derivative function of the Sawi transformation is defined as if () T  Sawi transform of () ft, then 1.
is the Sawi transformation of the derivative function '( ) ft.

3.
( is the Sawi transformation of the derivative function .

The Duality of the Laplace-Sawi Transformation
In 2019, the duality of the Laplace-Sawi transformation is defined by Aggarwal and Gupta.If the Laplace transformation of ( ) Z  is ( ) F  and the Sawi transformation of ( ) Z  is ( )

The Inverse of the Sawi Transformation
The inverse Sawi transformation is the inverse process of the Sawi transformation which is defined as follows: Suppose that ( ) T  a function from the Sawi transform ( ) , then ( ) ft is called the inverse Sawi transformation of ( ) T  .This relationship can be seen in Table 1.

Numerical Example: System of Chemical Reactions of Three Specific Chemical Substances
So that the chemical reaction system modeled in the form of the following system of ordinary differential equations.− , 12 ,0 kk ( ) System ( 5) can be written in the form Applying the Sawi transformation to the system of Equation ( 5) By applying the linearity property of the Sawi transformation to Equation ( 7), it is obtained Use the derivative function of the Sawi transformation in Equation ( 8), where 11 00 11 00 S C = , then the system of Equation ( 10) can be written as 11 00 11 00 Simplify the form of the system of Equation ( 10) and substitute the initial conditions (6) into the system of Equation ( 10) The system of Equation ( 10) can be written in the form AB = where , and   ( ) ( ) ( ) can be decomposed into a partial fraction by using partial fraction decomposition, so By using partial fraction decomposition and polynomial long division, ( )( ) can be decomposed into partial fractions, so Equation ( 16) becomes With the inverse Sawi transformation and its properties applied to Equation (13), Equation ( 15), and Equation (17), the solutions 12 ,, CC and 3 C obtained, respectively ( ) ( )           3 for 1  = and Table 4 for 2  = .3 and Table 4 show that the greater the value  3 and Table 4.        Ct concentration of a chemical N increases during time t .In addition, Table 5 and Table 6 also illustrate that the value of the rate constant 2 k increases from 1.5 to 2 Furthermore, Table 5 and Table 6 show that the greater k , the greater the concentration of ( ) Ct in a chemical N produced.The graphical sketch in Figure 3 presents the same results as in Table 5 and Table 6.lies on a rough plane so that when it moves, it produces a frictional force of 3sin 2t Newton.Table 7 shows that the displacement of the springs ( )  We have shown that the Sawi transformation can solve a system of linear ordinary differential equations accompanied by initial values (the initial value problem).To obtain the solution, we do two main steps.Firstly, it was changing a system of linear first-order differential equations into a system of linear first-order differential equations involving the Sawi operator.Secondly, using the linear identity property, the derivative of the Sawi transformation function, the Sawi inverse transformation, and the required initial value replacement will give the expected analytical solution.
and Gupta in 2019 established duality relations between some useful integral transformations namely Laplace transformation, Kamal transformation, Elzaki transformation, Aboodh transformation, Sumudu transformation, Mahgoub (Laplace-Carson) transformation and Mohand transformation with Sawi transformation [9].Sawi decomposition method for the primitive of linear Faltung-type second kind linear Faltung-type Volterra integral equation is given by [10].

is 1 C 3 C
Consider three specific chemical substances in a row: chemicals ,, LM and N .The concentration of chemical L , the concentration of a chemical M is 2 C , and the concentration of a chemical N is .The chemical reaction assisted by mixing a chemical substance that reacts with each reactant ,, LM and product N and reacts at a rate 12 , kk which assumes that 12 ,0 kk .The chemical reaction illustrated as in the chemical reaction below [5].
The concentration of a chemical L at time t == The concentration of a chemical M at time t == The concentration of a chemical N at time t the Equation (9) obtained by Cramer's method to get    

2 ,, CC and 3 CL
Equation (18), Equation (19), and Equation (20) are analytical solutions of the system of Equation (5) with initial conditions(6).Equation (18), Equation (19), and Equation (20) give the concentrations of 1 required for a chemical reaction, respectively.The concentration values of , the variation in value determines as follows.

1 Ct concentration of a chemical substance L for all variations in  and 1 k
values that determine can see in

Figure 2 . 1 kFigure 2 1 Ct 3 1 1 Ct 3 1 2 k 2 Ct 3 Ct
Figure 2. Graph of Solution ( ) 1 Ct, Which Depicts The Concentration of a chemical L during time t with a Variety Of Different Values  and 1 k

2 k 2 Ct 2 Ct 2 k
determined previously obtained the value of the ( ) concentration of a chemical substance M .The results obtained for the value of ( ) concentration of a chemical substance M with all variations in values determined can see in Table

( ) 2 Ct 2 2 Ct 1 Ct) 3 Ct
concentration of a chemical substance M increases but decreases with increasing time t from 0 to 6 seconds.The ( ) Ct concentration value of a chemical substance M decreases during time t.The ( ) concentration of chemical M initially increases due to a reduced ( ) concentration of chemical L. It decreases due to an increase in the ( concentration of chemical N. Furthermore, Table

2 Ct
in the chemical substance M produced.The graph in Figure3illustrates the same results as Table

3 Ct 3 Ct 2 Ct
concentration of a chemical N increases over time t .The ( ) concentration of chemical N increases due to the reduced ( ) concentration of chemical M when a chemical reaction occurs.

.
The position of the object 1x when 0 t = is in its equilibrium position, while 2x when 0 t = shifts to the right by 1 meter.For clarity, Figure4depicts the mass-spring system [3].

Figure 4 .
Figure 4. Mass-spring system with some grainy textureWhere can the mass-spring system in Figure1be represented in the form of the following system of linear ordinary differential equations.
linearity property of the Sawi transformation to Equation (23), it is obtained the system of Equation (25) can be written as

2 xt
move right first, then springs ( ) 1 xt and ( ) 2 xt move directly and left in time t , both in the same direction and in the antipodes.Until the spring system stops when the spring velocity slows down with time t .The graph in Figure 5 illustrates the displacement of springs ( )

Figure 5 . 2 xt
Figure 5.The Graphs Of The Solutions ( ) 1 xt and ( ) 2 xt Depict The Horizontal Displacement Of The Spring Associated With Masses 1 m And 2 m During Time t

Table 2 .
( ) 1 Ct concentration of a chemical substance L during time t with different Value variations of  and 1 k t (sec)

Table 2
shows that when the time t 1 Ct of a chemical substance L continues to decrease due to a chemical reaction that produces a new substance, namely a chemical substance M 1 Ct in a chemical substance L produced.The graph supports the results obtained in the table (see Figure 2).

Table 3 . The Concentration of
( ) 2 Ct of a Chemical Substance M During Time t with

Table 3 and
Table 4 illustrate that initially, the value of the

Table 5 . The Concentration Of
3Ct

Table 6 .
The Concentration of ( ) 3Ct of a Chemical Substance N During Time t with

Table 5 and
Table 6 illustrate that the value of the rate constant 1

Equation (33) and Equation (34) are analytical solutions of the system of Equation (21) with initial conditions (22). Equation (33) and Equation (34) are
) the displacements of the springs ( ) 2xt can be seen in Table7below.