# Sawi Decomposition Method for Volterra Integral Equation with Application

Authors: M. Higazy; Sudhanshu Aggarwal; Taher A. Nofal
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, authors present a new method “Sawi decomposition method” for determining the primitive of Volterra integral equation (V.I.E.) with application. Sawi decomposition method is the combination of Sawi transformation and decomposition method. Some numerical problems have been considered and solved sequentially for explaining the complete methodology. For the practical application of the Sawi decomposition method, an advance problem of medical science for determining the blood glucose concentration during an intravenous injection has been considered and solved sequentially applying this method. Results of numerical problems depict that the Sawi decomposition method is a very effective decomposition method for determining the primitives of V.I.E.


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## Body

## 1. Introduction

Many complex problems of mathematics, chemistry, biology, astrophysics, and mechanics such as problem of radiative energy transfer, oscillation problems of string and membrane, and problem of momentum representation in quantum mechanics can be expressed in the terms of Volterra integral equation. Aggarwal et al. [1] and Chauhan and Aggarwal [2] used different integral transformations for obtaining the solutions of V.I.E. of second kind. Abdelrahim [3] solved constant coefficient linear differential equations by defining Sawi transformation. Singh and Aggarwal [4] applied Sawi transformation for determining the solutions of biological problems of growth and decay. Aggarwal and Gupta [5] established duality relations between Sawi and other advanced integral transformations. Wang [6] gave the reliable mechanical algorithm for obtaining the numerical solution of famous Volterra integral equation. Maleknejad and Aghazadeh [7] used the Taylor-series expansion method and determined the numerical primitives of second kind V.I.E. with convolution kernel.Rashidinia and Zarebnia [8] solved the Volterra integral equation by the Sinc-collocation method. Babolian and Davari [9] gave the numerical implementation of the Adomian decomposition method for linear Volterra integral equations of the second kind. Lin et al. [10] used extrapolation of the iterated-collocation method for integral equations of the second kind. Zhang et al. [11] applied Galerkin methods for determining the numerical solution for second-kind Volterra integral equations. Shoukralla et al. [12] used the Barycentric-Maclaurin interpolation method for solving Volterra integral equations of the second kind. Isaacson and Kirby [13] gave the numerical solution of linear Volterra integral equations of the second kind with sharp gradients. The Adomian decomposition method of Volterra integral equation of second kind was given by Abaoud et al. [14].Aggarwal et al. [15] applied Mahgoub transform for solving linear Volterra integral equations. Aggarwal et al. [16] gave a new application of Shehu transform for handling Volterra integral equations of first kind. Solution of linear volterra integral equations of second kind using mohand transform was given by Aggarwal et al. [17]. Aggarwal et al. [18] used Aboodh transform for solving linear Volterra integral equations of first kind. Duality relations of Kamal transform with Laplace, Laplace–Carson, Aboodh, Sumudu, Elzaki, Mohand, and Sawi transforms were given by Aggarwal et al. [19]. Aggarwal and Bhatnagar [20] defined dualities between Laplace transform and some useful integral transforms.Chauhan et al. [21] gave the dualities between Laplace–Carson transform and some useful integral transforms. Aggarwal and Gupta [22] defined the dualities between Mohand transform and some useful integral transforms. Dualities between Elzaki transform and some useful integral transforms were given by Aggarwal et al. [23]. Chaudhary et al. [24] defined the connections between Aboodh transform and some useful integral transforms. Aggarwal et al. [25] applied Mahgoub transform for solving linear Volterra integral equations of first kind. Application of Elzaki transform for solving linear Volterra integral equations of first kind was given by Aggarwal et al. [26]. Aggarwal and Sharma [27] used Laplace transform for the solution of first kind linear Volterra integral equation. Mishra et al. [28] defined the relationship between Sumudu and some efficient integral transforms. Aggarwal et al. [29] discussed the exact solutions for a class of Wick-type stochastic (3 + 1)-dimensional modified Benjamin–Bona–Mahony equations. Cesarano [30] used generalized special functions in the description of fractional diffusive equations. Dattoli et al. [31] discussed special polynomials and gave some results in fractional calculus. Aggarwal et al. [32] gave the application of Aboodh transform for solving linear Volterra integral equations of first kind. Application of Kamal transform for solving linear volterra integral equations of first kind was given by Aggarwal et al. [33].The objective of the present paper is to determine the solutions of V.I.Es by applying the Sawi decomposition method on them and determining the solution of the problem of sugar level (blood glucose concentration) of a patient for explaining the applicability of this method in the field of medical science.


## 2. Definition of Sawi Transformation

The Sawi transformation of the functionωt,t≥0is given by Abdelrahim [3]:(1)Sωt=1σ2∫0∞ωte−1/σtdt=Tσ,σ>0.Remark 1.
In the above equation, the function1/σ2e−1/σt is called the kernel of the Sawi transformation.Remark 2.
The Sawi transformation of the functionωt,t≥0exists if the integral 1/σ2∫0∞ωte−1/σtdt exists.Remark 3.
The integral1/σ2∫0∞ωte−1/σtdt exists if ωt is a function of exponential order and piecewise continuous in the interval 0≤t<∞.


## 3. Some Important Properties of Sawi Transformation

Authors present important characteristics of Sawi transformation in this part of the paper.


### 3.1. Linearity Property of Sawi Transformation

If Sawi transform of functionsω1t and ω2t are T1σ and T2σ, respectively, then Sawi transform of lω1t+mω2t is given by lT1σ+mT2σ, where landm are arbitrary constants.Proof.
Using (1), we obtain(2)Sωt=1σ2∫0∞ωte−1/σtdt⇒Slω1t+mω2t=1σ2∫0∞lω1t+mω2te−1/σtdt⇒Slω1t+mω2t=l1σ2∫0∞ω1te−1/σtdt+m1σ2∫0∞ω2te−1/σtdt⇒Slω1t+mω2t=lSω1t+mSω2t⇒Slω1t+mω2t=lT1σ+mT2σ,where landm are arbitrary constants.Remark 4.
One immediate consequence of the above property is that ifωt=∑i=1kaiωit, where ai are arbitrary constants, then Sωt=S∑i=1kaiωit=∑i=1kaiSωit.


### 3.2. Scaling Property of Sawi Transformation

IfSωt=Tσ, then Sωkt=kTkσ.Proof.
Using (1), we obtain(3)Sωt=1σ2∫0∞ωte−1/σtdt⇒Sωkt=1σ2∫0∞ωkte−1/σtdt.
Puttingkt=p⇒kdt=dp in the above equation, we have(4)Sωkt=1σ21k∫0∞ωpe−1/σp/kdp⇒Sωkt=1k1σ2∫0∞ωpe−1/σp/kdp=kk2σ2∫0∞ωpe−1/kσpdp⇒Sωkt=kTkσ.


### 3.3. Translation Property of Sawi Transformation

IfSωt=Tσ, then Sektωt=1/1−kσ2Tσ/1−kσ.Proof.
Using (1), we obtain(5)Sωt=1σ2∫0∞ωte−1/σtdt⇒Sektωt=1σ2∫0∞ektωte−1/σtdt⇒Sektωt=1σ2∫0∞ωte−1/σ−ktdt=1σ2∫0∞ωte−1−kσ/σtdt=11−kσ21−kσ2σ2∫0∞ωte−1−kσ/σtdt=11−kσ2Tσ1−kσ.


## 3.1. Linearity Property of Sawi Transformation

If Sawi transform of functionsω1t and ω2t are T1σ and T2σ, respectively, then Sawi transform of lω1t+mω2t is given by lT1σ+mT2σ, where landm are arbitrary constants.Proof.
Using (1), we obtain(2)Sωt=1σ2∫0∞ωte−1/σtdt⇒Slω1t+mω2t=1σ2∫0∞lω1t+mω2te−1/σtdt⇒Slω1t+mω2t=l1σ2∫0∞ω1te−1/σtdt+m1σ2∫0∞ω2te−1/σtdt⇒Slω1t+mω2t=lSω1t+mSω2t⇒Slω1t+mω2t=lT1σ+mT2σ,where landm are arbitrary constants.Remark 4.
One immediate consequence of the above property is that ifωt=∑i=1kaiωit, where ai are arbitrary constants, then Sωt=S∑i=1kaiωit=∑i=1kaiSωit.


## 3.2. Scaling Property of Sawi Transformation

IfSωt=Tσ, then Sωkt=kTkσ.Proof.
Using (1), we obtain(3)Sωt=1σ2∫0∞ωte−1/σtdt⇒Sωkt=1σ2∫0∞ωkte−1/σtdt.
Puttingkt=p⇒kdt=dp in the above equation, we have(4)Sωkt=1σ21k∫0∞ωpe−1/σp/kdp⇒Sωkt=1k1σ2∫0∞ωpe−1/σp/kdp=kk2σ2∫0∞ωpe−1/kσpdp⇒Sωkt=kTkσ.


## 3.3. Translation Property of Sawi Transformation

IfSωt=Tσ, then Sektωt=1/1−kσ2Tσ/1−kσ.Proof.
Using (1), we obtain(5)Sωt=1σ2∫0∞ωte−1/σtdt⇒Sektωt=1σ2∫0∞ektωte−1/σtdt⇒Sektωt=1σ2∫0∞ωte−1/σ−ktdt=1σ2∫0∞ωte−1−kσ/σtdt=11−kσ21−kσ2σ2∫0∞ωte−1−kσ/σtdt=11−kσ2Tσ1−kσ.


## 4. Sawi Transformation of Derivatives

IfSωt=Tσ, then(6)Sω′t=1σTσ−1σ2ω0,Sω″t=1σ2Tσ−1σ3ω0−1σ2ω′0,Sωρt=1σρTσ−∑k=0ρ−11σρ−k−1ωk0.Proof.
(a)
Using (1), we obtain(7)Sωt=1σ2∫0∞ωte−1/σtdt⇒Sω′t=1σ2∫0∞ω′te−1/σtdt=1σ2ωte−1/σt0∞−1σ2∫0∞−1σωte−1/σtdt=1σ2limt⟶∞ωte−1/σt−1σ2ω0+1σ3∫0∞ωte−1/σtdt=0−1σ2ω0+1σSωt⇒Sω′t=1σTσ−1σ2ω0.(b)
We haveSω′t=1/σTσ−1/σ2ω0=1/σSωt−1/σ2ω0,(8)⇒Sω″t=1σSω′t−1σ2ω′0=1σ1σSωt−1σ2ω0−1σ2ω′0,⇒Sω″t=1σ2Sωt−1σ3ω0−1σ2ω′0,⇒Sω″t=1σ2Tσ−1σ3ω0−1σ2ω′0.(c)
We haveSω″t=1/σ2Tσ−1/σ3ω0−1/σ2ω′0,(9)=1σ2Sωt−1σ3ω0−1σ2ω′0⇒Sω‴t=1σ2Sω′t−1σ3ω′0−1σ2ω″0=1σ21σSωt−1σ2ω0−1σ3ω′0−1σ2ω″0⇒Sω‴t=1σ3Sωt−1σ4ω0−1σ3ω′0−1σ2ω″0=1σ3Tσ−1σ4ω0−1σ3ω′0−1σ2ω″0.
In general, we have(10)Sωρt=1σρSωt−1σρ+1ω0−1σρω′0−⋯−1σρ−1ωρ−10⇒Sωρt=1σρTσ−1σρ+1ω0−1σρω′0−⋯−1σρ−1ωρ−10⇒Sωρt=1σρTσ−∑k=0ρ−11σρ−k−1ωk0.Remark 5.
The results of this section are very useful and can be used for solving initial value problems.


## 5. Faltung Theorem for Sawi Transformation

Faltung theorem of integral transforms has many applications in the solution of differential equations and integral equations of Faltung form. IfSω1t=T1σ and Sω2t=T2σ, then Sω1t∗ω2t=Sω1tSω2t=σ2T1σT2σ, where Faltung of ω1t and ω2t is denoted by ω1t∗ω2t, and it is defined by (11)ω1t∗ω2t=∫0tω1t−uω2udu=∫0tω1uω2t−udu.Proof.
Using (1), we obtain(12)Sωt=1σ2∫0∞ωte−1/σtdt⇒Sω1t∗ω2t=1σ2∫0∞e−1/σtω1t∗ω2tdt⇒Sω1t∗ω2t=1σ2∫0∞e−1/σt∫0tω1t−uω2ududt.
After reversing the order of integration, we obtain(13)Sω1t∗ω2t=1σ2∫0∞ω2u∫u∞ω1t−ue−1/σtdtdu.
Puttingt−u=v so that dt=dv in the above equation, we have(14)Sω1t∗ω2t=1σ2∫0∞ω2u∫0∞ω1ve−1/σv+udvdu⇒Sω1t∗ω2t=1σ2∫0∞ω2ue−1/σu∫0∞ω1ve−1/σvdvdu⇒Sω1t∗ω2t=σ21σ2∫0∞ω2ue−1/σu1σ2∫0∞ω1ve−1/σvdvdu⇒Sω1t∗ω2t=σ2T1σT2σ.Remark 6.
The above result can be used to obtain a connection between ak-fold Faltung of k functions and the product of the transforms of these functions.Remark 7.
In the above definition, the termω1t−u or ω2t−u is called the influence function.


## 6. Inverse Sawi Transformation

The inverse Sawi transformation ofTσ, designated by S−1Tσ, is another function ωt having the property that Sωt=Tσ.


### 6.1. Linearity Property of Inverse Sawi Transformation

IfS−1T1σ=ω1t and S−1T2σ=ω2t, then(15)S−1lT1σ+mT2σ=lS−1T1σ+mS−1T2σ⇒S−1lT1σ+mT2σ=lω1t+mω2t,where landm are arbitrary constants.


## 6.1. Linearity Property of Inverse Sawi Transformation

IfS−1T1σ=ω1t and S−1T2σ=ω2t, then(15)S−1lT1σ+mT2σ=lS−1T1σ+mS−1T2σ⇒S−1lT1σ+mT2σ=lω1t+mω2t,where landm are arbitrary constants.


## 7. Sawi Decomposition Method for Volterra Integral Equation

This section contains Sawi decomposition method for the primitive of linear Faltung-type second kind V.I.E. The general form of second kind linear Faltung-type Volterra integral equation is given by [2, 34–36](16)ωt=gt+μ∫0tKt−uωudu,where(17)ωt=unknown functiongt=known functionperturbation functionμ=non−zero parameterKt−u=faltung type kernel.Operating Sawi transformation on both sides of (16), we have(18)Sωt=Sgt+μS∫0tKt−uωudu.Applying Faltung theorem of Sawi transformation on (18), we obtain(19)Sωt=Sgt+μσ2SKtSωt.Operating inverse Sawi transformation on both sides of (19), we have(20)ωt=gt+μS−1σ2SKtSωt.The Sawi decomposition method assumes the solutionωt of (16) is analytic, so ωt can be expressed in terms of infinite series as(21)ωt=∑m=0∞ωmt.Using (21) in (20), we obtain(22)∑m=0∞ωmt=gt+μS−1σ2SKtS∑m=0∞ωmt.In general, the recursive relations are given by(23)ω0t=gtωm+1t=μS−1σ2SKtS∑m=0∞ωmt,m≥0.Using (23), we can find the values of ω0t,ω1t,ω2t,ω3t,…easily. After finding these values, we use (21) for required solution of (16).Remark 8.
The present scheme can be used for determining the primitives of system of linear Faltung-type second kind Volterra integral equations in future.
Sawi and inverse Sawi transformations of frequently used functions are given in Tables1 and 2, respectively.Table 1
Sawi transformation of frequently used functions [4,5].
S.N.ωtSωt=Tσ1.11/σ2.t13.t22σ4.tρ,ρ∈Nρ!σρ−15.tρ,ρ>−1Γρ+1σρ−16.elt1/σ1−lσ7.sinltl/1+l2σ28.coslt1/σ1+l2σ29.sinhltl/1−l2σ210.coshlt1/σ1−l2σ2Table 2
Inverse Sawi transformation of frequently used functions [4].
S.N.Tσωt=S−1Tσ1.1/σ12.1t3.σt2/2!4.σρ−1, ρ∈Ntρ/ρ!5.σρ−1,ρ>−1tρ/Γρ+16.1/σ1−lσelt7.1/1+l2σ2sinlt/l8.1/σ1+l2σ2coslt9.1/1−l2σ2sinhlt/l10.1/σ1−l2σ2coshltExample 1.
Consider following linear second kind Faltung-type V.I.E.:(24)ωt=t+∫0tωudu.
Operating Sawi transformation on both sides of (24), we have(25)Sωt=St+S∫0tωudu⇒Sωt=1+S∫0tωudu.
Applying Faltung theorem of Sawi transformation on (25), we obtain(26)Sωt=1+σ2S1Sωt⇒Sωt=1+σ21σSωt.
Operating inverse Sawi transformation on both sides of (26), we have(27)ωt=t+S−1σSωt.
The Sawi decomposition method assumes the solutionωtof (24) is analytic so ωt can be expressed in terms of infinite series as(28)ωt=∑m=0∞ωmt.
Using (8) in (27), we obtain(29)∑m=0∞ωmt=t+S−1σS∑m=0∞ωmt.
In general, the recursive relations are given by(30)ω0t=t,ωm+1t=S−1σS∑m=0∞ωmt,m≥0,with the help of (30), we obtain(31)ω1t=S−1σSω0t=S−1σSt=S−1σ⋅1=S−1σ=t22,ω2t=S−1σSω1t=S−1σSt22=S−1σ⋅σ=S−1σ2=t36,ω3t=S−1σSω2t=S−1σSt36=S−1σ⋅σ2=S−1σ3=t424,ω4t=S−1σSω3t=S−1σSt424=S−1σ⋅σ3=S−1σ4=t5120,and so on. Using (28), the series solution of (24) is given by(32)ωt=∑m=0∞ωmt,⇒ωt=t+t22+t36+t424+t5120+⋯=et−1.Example 2.
Consider the following linear second kind Faltung-type V.I.E.:(33)ωt=t−∫0tωudu.
Operating Sawi transformation on both sides of (33), we have(34)Sωt=St−S∫0tωudu,⇒Sωt=1−S∫0tωudu.
Applying Faltung theorem of Sawi transformation on (34), we obtain(35)Sωt=1−σ2S1Sωt,⇒Sωt=1−σ21σSωt.
Operating inverse Sawi transformation on both sides of (35), we have(36)ωt=t−S−1σSωt.
The Sawi decomposition method assumes the solutionωt of (33) is analytic, so ωt can be expressed in terms of infinite series as(37)ωt=∑m=0∞ωmt.
Using (37) in (36), we obtain(38)∑m=0∞ωmt=t−S−1σS∑m=0∞ωmt.
In general, the recursive relations are given by(39)ω0t=t,ωm+1t=−S−1σS∑m=0∞ωmt,m≥0.
With the help of (39), we obtain(40)ω1t=−S−1σSω0t=−S−1σSt=−S−1σ⋅1=−S−1σ=−t22,ω2t=−S−1σSω1t=−S−1σS−t22=S−1σ⋅σ=S−1σ2=t36,ω3t=−S−1σSω2t=−S−1σSt36=−S−1σ⋅σ2=−S−1σ3=−t424,ω4t=−S−1σSω3t=−S−1σS−t424=S−1σ⋅σ3=S−1σ4=t5120,and so on. Using (37), the series solution of (33) is given by(41)ωt=∑m=0∞ωmt=t−t22+t36−t424+t5120−⋯=1−e−t.


## 8. Application

This part of the paper contains an application from the field of medical science during an intravenous injection (continuous) for determining blood glucose concentrationCt of a patient at any particular time t. This concentrationCt is determined by the following linear Volterra integral equation:(42)Ct=Ci+αVt−k∫0tCxdx,where(43)k:constant velocity  of eliminationα:the rate  of infusionV:volume in which glucose is distributedCi:initial concentration of glucose  in the blood.Operating Sawi transformation on both sides of (42), we have(44)SCt=CiS1+αVSt−kS∫0tCxdx,⇒SCt=Ci1σ+αV⋅1−kS∫0tCxdx.Applying Faltung theorem of Sawi transformation on (44), we obtain(45)SCt=Ci1σ+αV−kσ2S1SCt,⇒SCt=Ci1σ+αV−kσ21σSCt.Operating inverse Sawi transformation on both sides of (45), we have(46)Ct=Ci+αVt−kS−1σSCt.The Sawi decomposition method assumes the solutionCtof (42) is analytic, so it can be represent in power series (Taylor’s series) as(47)Ct=∑m=0∞Cmt.Using (47) in (46), we obtain(48)∑m=0∞Cmt=Ci+αVt−kS−1σS∑m=0∞Cmt.In general, the recursive relations are given by(49)C0t=Ci+αVtCm+1t=−kS−1σS∑m=0∞Cmt,m≥0.Using (49), we obtain(50)C1t=−kS−1σSC0t=−kS−1σSCi+αVt=−kS−1σSCi+SαVt=−CikS−1σS1−αVkS−1σSt=−CikS−1σ⋅1σ−kαVS−1σ⋅1=−CikS−11−kαVS−1σ=−kCit+αVt22!,C2t=−kS−1σSC1t=−kS−1σS−kCit+αVt22!=k2S−1σSCit+SαVt22!=k2CiS−1σSt+αVS−1σSt22!=k2CiS−1σ⋅1+αVS−1σ⋅σ=k2CiS−1σ+αVS−1σ2=k2Cit22!+αVt33!,C3t=−kS−1σSC2t=−kS−1σSk2Cit22!+αVt33!=−k3S−1σSCit22!+SαVt33!=−k3CiS−1σSt22!+αVS−1σSt33!=−k3CiS−1σ⋅σ+αVS−1σ⋅σ2=−k3CiS−1σ2+αVS−1σ3=−k3Cit33!+αVt44!,C4t=−kS−1σSC3t=−kS−1σS−k3Cit33!+αVt44!=k4S−1σSCit33!+SαVt44!=k4CiS−1σSt33!+αVS−1σSt44!=k4CiS−1σ⋅σ2+αVS−1σ⋅σ3=k4CiS−1σ3+αVS−1σ4=k4Cit44!+αVt55!,and so on. Using (47), the series solution of (42) is given by(51)Ct=∑m=0∞Cmt=Ci+αVt−kCit+αVt22!+k2Cit22!+αVt33!−k3Cit33!+αVt44!+k4Cit44!+αVt55!+⋯=Ci1−kt+k22!t2−k33!t3+k44!t4−⋯+αVt−k2!t2+k23!t3−k34!t4+k45!t5−⋯=Cie−kt+αVkkt−k2t2+k26t3−k324t4+k4120t5−⋯=Cie−kt+αVkkt−k22t2+k36t3−k424t4+k5120t5−⋯=Cie−kt+αVk1−1+kt−k22t2+k36t3−k424t4+k5120t5−⋯=Cie−kt+αVk1−1−kt+k22t2−k36t3+k424t4−k5120t5+⋯,Ct=Cie−kt+αVk1−e−kt.The values of blood glucose concentrationCt are obtained for different values of initial concentration of glucose Ci, volume in which glucose is distributed V, the rate of infusion α, constant velocity of elimination k, and time t. All these results are presented in Tables 3–6.Table 3
Blood glucose concentrationCt for different values of initial concentration of glucose Ci and time t with V=45dL,α=280mg/min,andk=0.058min−1, see Figure 1.
tminV=45dL,α=280mg/min,k=0.058min−1Ci=320mg/dLCi=322mg/dLCi=324mg/dLCi=326mg/dLCi=328mg/dL032032232432632815196.40197.24198.08198.91199.7530144.62144.97145.32145.67146.0245122.92123.07123.22123.36123.5160113.83113.89113.96114.02114.0875110.03110.05110.08110.10110.1390108.43108.44108.45108.46108.47Table 4
Blood glucose concentrationCt for different values of volume in which glucose is distributed V and time t with Ci=325mg/dL,α=280mg/min,andk=0.058min−1, see Figure 2.
tminCi=325mg/dL,α=280mg/min,k=0.058min−1V=45dLV=46dLV=47dLV=48dLV=49dL032532532532532515198.49197.14195.84194.60193.4130145.49143.57141.73139.97138.2745123.29121.13119.06117.08115.1860113.99111.73109.56107.49105.5075110.09107.79105.58103.47101.4590108.46106.14103.92101.7999.75Table 5
Blood glucose concentrationCt for different values of the rate of infusion α and time t with V=45dL,Ci=320mg/dL,andk=0.058min−1, see Figure 3.
tminV=45dL,Ci=320mg/dL,k=0.058min−1α=280mg/minα=281mg/minα=282mg/minα=283mg/minα=284mg/min032032032032032015196.40196.62196.84197.07197.2930144.62144.93145.25145.56145.8845122.92123.28123.63123.99124.3460113.83114.20114.58114.95115.3275110.03110.40110.78111.16111.5490108.43108.81109.19109.57109.95Table 6
Blood glucose concentrationCt for different values of constant velocity of elimination k and time t with V=45dL,α=280mg/min,andCi=320mg/dL, see Figure 4.
tminV=45dL,α=280mg/min,Ci=320mg/dLk=0.058min−1k=0.059min−1k=0.060min−1k=0.061min−1k=0.062min−1032032032032032015196.40194191.64189.31187.0230144.62142139.46136.97134.5545122.92120.54118.24116.01113.8560113.83111.69109.61107.61105.6875110.03108.03106.11104.25102.4690108.43106.52104.68102.90101.19The normal range of blood glucose mentioned by American Diabetes Association [37] is 79 to 110mg/dL.From Table3, it can be concluded that, as time t increases from 0 to 90min, blood glucose concentration Ct decreases for all the five combinations, namely, V=45dL,α=280mg/min,k=0.058min−1,Ci=320mg/dLV=45dL,α=280mg/min,k=0.058min−1,Ci=322mg/dLV=45dL,α=280mg/min,k=0.058min−1,Ci=324mg/dLV=45dL,α=280mg/min,k=0.058min−1,Ci=326mg/dLV=45dL,α=280mg/min,k=0.058min−1,Ci=328mg/dL. From this table, it is also clear that the normal blood glucose concentration is achieved in 90min for all five combinations. The graph plotted in Figure 1 supports the results of Table 3.Figure 1
Blood glucose concentrationCt for different values of initial concentration of glucose Ci and time t with. V=45dL,α=280mg/min, andk=0.058min−1.From Table4, it can be concluded that, as time t increases from 0 to 90min, blood glucose concentration Ct decreases for all the five combinations, namely, Ci=325mg/dL,α=280mg/min,k=0.058min−1,V=45dLCi=325mg/dL,α=280mg/min,k=0.058min−1,V=46dLCi=325mg/dL,α=280mg/min,k=0.058min−1,V=47dLCi=325mg/dL,α=280mg/min,k=0.058min−1,V=48dLCi=325mg/dL,α=280mg/min,k=0.058min−1,V=49dL. From this table, it is also clear that the time required for achieving normal blood glucose concentration reduces as volume in which glucose is distributed V increases. The results of Table 4 are supported by the graph, which is plotted in Figure 2.Figure 2
Blood glucose concentrationCt for different values of volume in which glucose is distributed V and time t with. Ci=325mg/dL,α=280mg/min,andk=0.058min−1.Table5 shows that, as time t increases from 0 to 90min, blood glucose concentration Ct decreases for all the five combinations, namely, V=45dL,Ci=320mg/dL,k=0.058min−1,α=280mg/minV=45dL,Ci=320mg/dL,k=0.058min−1,α=281mg/minV=45dL,Ci=320mg/dL,k=0.058min−1,α=282mg/minV=45dL,Ci=320mg/dL,k=0.058min−1,α=283mg/minV=45dL,Ci=320mg/dL,k=0.058min−1,α=284mg/min. From this table, it is also clear that the time required for achieving normal blood glucose concentration increases as the rate of infusion α increases. The graph of Figure 3 supports the results of Table 5.Figure 3
Blood glucose concentrationCt for different values of the rate of infusion α and time t with. V=45dL,Ci=320mg/dL,andk=0.058min−1.Table6 shows that, as time t increases from 0 to 90 min, blood glucose concentration Ct decreases for all the five combinations, namely, V=45dL,α=280mg/min,Ci=320mg/dL,k=0.058min−1V=45dL,α=280mg/min,Ci=320mg/dL,k=0.059min−1V=45dL,α=280mg/min,Ci=320mg/dL,k=0.060min−1V=45dL,α=280mg/min,Ci=320mg/dL,k=0.061min−1V=45dL,α=280mg/min,Ci=320mg/dL,k=0.062min−1. From this table, it is also clear that the time required for achieving normal blood glucose concentration decreases as the constant velocity of elimination k increases. The results of Table 6 are supported by the graph which is plotted in Figure 4.Figure 4
Blood glucose concentrationCt for different values of constant velocity of elimination k and time t with V=45dL,a=280mg/min,andCi=320mg/dL.


## 9. Conclusions

In the present paper, authors fruitfully discussed the Sawi decomposition method for V.I.E. and complete methodology explained by taking numerical examples with application in the field of medical science during an intravenous injection (continuous) for determining blood glucose concentration of a patient at any particular time. Results of numerical examples depict that the Sawi decomposition method is a very fast and effective decomposition method for determining the solution of V.I.E. Furthermore, the Sawi decomposition method gives the solution of problem of blood glucose concentration and provides the information of required time to achieve normal blood glucose concentration, which is very useful for sugar patients and at the time of operation. The Sawi decomposition method will be useful for determining the primitives of system of V.I.E. and other problems of medical science, engineering, physical chemistry such as determination of tumor growth, counting the total number of infected cells, determining the concentration of viral particles in plasma during HIV-1 infections, examining the temperature effect on the vibration of skew plates, and determining the concentration of chemical substances of the chemical chain reaction in future.



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