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.

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 function is given by Abdelrahim [3]:

Remark 1. In the above equation, the function is called the kernel of the Sawi transformation.

Remark 2. The Sawi transformation of the function exists if the integral exists.

Remark 3. The integral exists if is a function of exponential order and piecewise continuous in the interval .

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 functions and are and , respectively, then Sawi transform of is given by where are arbitrary constants.

Proof. Using (1), we obtainwhere are arbitrary constants.

Remark 4. One immediate consequence of the above property is that if , where are arbitrary constants, then

3.2. Scaling Property of Sawi Transformation

If , then .

Proof. Using (1), we obtainPutting in the above equation, we have

3.3. Translation Property of Sawi Transformation

If , then .

Proof. Using (1), we obtain

4. Sawi Transformation of Derivatives

If , then

Proof. (a)Using (1), we obtain(b)We have ,(c)We have ,In general, we have

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. If and , then , where Faltung of and is denoted by , and it is defined by

Proof. Using (1), we obtainAfter reversing the order of integration, we obtainPutting so that in the above equation, we have

Remark 6. The 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 or is called the influence function.

6. Inverse Sawi Transformation

The inverse Sawi transformation of designated by , is another function having the property that

6.1. Linearity Property of Inverse Sawi Transformation

If and , thenwhere 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]where

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

The Sawi decomposition method assumes the solution of (16) is analytic, so 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 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 Tables 1 and 2, respectively.

Example 1. Consider following linear second kind Faltung-type V.I.E.:Operating Sawi transformation on both sides of (24), we haveApplying Faltung theorem of Sawi transformation on (25), we obtainOperating inverse Sawi transformation on both sides of (26), we haveThe Sawi decomposition method assumes the solution of (24) is analytic so can be expressed in terms of infinite series asUsing (8) in (27), we obtainIn general, the recursive relations are given bywith the help of (30), we obtainand so on. Using (28), the series solution of (24) is given by