Sawi Decomposition Method for Volterra Integral Equation with Application

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.


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. e 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]. e 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.

Remark 2.
e Sawi transformation of the function is a function of exponential order and piecewise continuous in the interval 0 ≤ t < ∞.

Some Important Properties of Sawi Transformation
Authors present important characteristics of Sawi transformation in this part of the paper.
Proof. Using (1), we obtain where l and m are arbitrary constants.
□ Remark 4. One immediate consequence of the above property is that if

Scaling Property of Sawi Transformation
Proof. Using (1), we obtain Putting kt � p ⇒ kdt � dp in the above equation, we have 2 Journal of Mathematics Proof. Using (1), we obtain

Sawi Transformation of Derivatives
Proof (a) Using (1), we obtain In general, we have □ Remark 5. e results of this section are very useful and can be used for solving initial value problems.

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.
, and it is defined by Proof. Using (1), we obtain After reversing the order of integration, we obtain Putting t − u � v so that dt � dv in the above equation, we have □ Remark 6. e above result can be used to obtain a connection between a k-fold Faltung of k functions and the product of the transforms of these functions.

Remark 7.
In the above definition, the term ω 1 (t − u) or ω 2 (t − u) is called the influence function.

Linearity Property of Inverse Sawi Transformation. If
where l and m are arbitrary constants.

Sawi Decomposition Method for Volterra Integral Equation
is section contains Sawi decomposition method for the primitive of linear Faltung-type second kind V.I.E. e general form of second kind linear Faltung-type Volterra integral equation is given by [2,[34][35][36] where 4 Journal of Mathematics Operating Sawi transformation on both sides of (16), we have Applying Faltung theorem of Sawi transformation on (18), we obtain Operating inverse Sawi transformation on both sides of (19), we have e Sawi decomposition method assumes the solutionω(t) of (16) is analytic, so ω(t) can be expressed in terms of infinite series as Using (21) in (20), we obtain In general, the recursive relations are given by Using (23), we can find the values of ω 0 (t), ω 1 (t), ω 2 (t), ω 3 (t), . . .easily. After finding these values, we use (21) for required solution of (16).

Remark 8.
e 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 Tables 1 and 2, respectively. Example 1. Consider following linear second kind Faltungtype V.I.E.: Operating Sawi transformation on both sides of (24), we have Applying Faltung theorem of Sawi transformation on (25), we obtain Operating inverse Sawi transformation on both sides of (26), we have e Sawi decomposition method assumes the solution ω(t)of (24) is analytic so ω(t) can be expressed in terms of infinite series as Using (8) in (27), we obtain In general, the recursive relations are given by S.N.
cosh lt with the help of (30), we obtain , , and so on. Using (28), the series solution of (24) is given by Example 2. Consider the following linear second kind Faltung-type V.I.E.: Operating Sawi transformation on both sides of (33), we have Applying Faltung theorem of Sawi transformation on (34), we obtain Operating inverse Sawi transformation on both sides of (35), we have e Sawi decomposition method assumes the solutionω(t) of (33) is analytic, so ω(t) can be expressed in terms of infinite series as Using (37) in (36), we obtain In general, the recursive relations are given by With the help of (39), we obtain , and so on. Using (37), the series solution of (33) is given by

Application
is part of the paper contains an application from the field of medical science during an intravenous injection (continuous) for determining blood glucose concentration C(t) of a patient at any particular time t. is concentration C(t) is determined by the following linear Volterra integral equation: where k: constant velocity of elimination α: the rate of infusion V: volume in which glucose is distributed C i : initial concentration of glucose in the blood Operating Sawi transformation on both sides of (42), we have (44) Applying Faltung theorem of Sawi transformation on (44), we obtain Operating inverse Sawi transformation on both sides of (45), we have e Sawi decomposition method assumes the solution C(t)of (42) is analytic, so it can be represent in power series (Taylor's series) as (47) In general, the recursive relations are given by Using (49), we obtain Journal of Mathematics 8 Journal of Mathematics and so on. Using (47), the series solution of (42) is given by e values of blood glucose concentration C(t) are obtained for different values of initial concentration of glucose C i , 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. e normal range of blood glucose mentioned by American Diabetes Association [37] is 79 to 110 mg/dL. From Table 3, it can be concluded that, as time t increases from 0 to 90 min, blood glucose concentration C(t) decreases for all the five combinations, namely, V � 45 dL, α � (280 mg/min), k � 0.058 min − 1 , α � (280 mg/min), k � 0.058 min − 1 , C i � 326 mg/dL V � 45 dL, α � (280 mg/min), k � 0.058 min − 1 , C i � 328 mg/dL}. From this table, it is also clear that the normal blood glucose concentration is achieved in 90 min for all five combinations. e graph plotted in Figure 1 supports the results of Table 3.

Conclusions
In the present paper, authors fruitfully discussed the Sawi decomposition method for V.I.E. and complete        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. e 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.
Data Availability e authors declare that the datasets used to support the finding of this paper are available from the corresponding author upon request.

Conflicts of Interest
e authors have no conflicts of interest regarding the publication of the paper.