International Journal of Research in Advent Technology, Vol.7, No.4, April 2019 E-ISSN: 2321-9637 Available online at www.ijrat.org 438 A New Application of Shehu Transform for Handling Volterra Integral Equations of First Kind Sudhanshu Aggarwal 1* , Anjana Rani Gupta 2 , Swarg Deep Sharma 3 1* Assistant Professor, Department of Mathematics, National P.G. College Barhalganj, Gorakhpur-273402, U.P., India 2 Professor, Department of Mathematics, Noida Institute of Engineering & Technology, Greater noida-201306, U.P., India 3 Assistant Professor, Department of Mathematics, Nand Lal Singh College Jaitpur Daudpur Constituent of Jai Prakash University Chhapra-841205, Bihar, India sudhanshu30187@gmail.com, argad76@gmail.com, gdeep.sharma@gmail.com Abstract: Many problems of thermodynamics, nuclear reactor theory, chemotherapy and electrical systems have been described in the form of Volterra integral equations. In this paper, we have given a new application of Shehu transform for handling Volterra integral equations of first kind. In application section of this paper, some numerical applications are given to explain the importance of Shehu transform for handling Volterra integral equations of first kind. The results show that Shehu transform is a very useful integral transform for handling Volterra integral equations of first kind. Keywords: Volterra integral equation of first kind, Shehu transform, Convolution theorem, Inverse Shehu transform. 1. INTRODUCTION In the advance time, integral transforms methods (Laplace transform [1-2], Fourier transform [1], Kamal transform [3-9, 36], Mahgoub transform [10-16], Mohand transform [17-20, 37-40], Aboodh transform [21-26, 41-44], Elzaki transform [27-29, 45-46], Sumudu transform [30, 47-48] and Shehu transform [49-50]) are convenient mathematical methods for solving advance problems of engineering and sciences which are mathematically expressed in terms of differential equations, system of differential equations, partial differential equations, integral equations, system of integral equations, partial integro-differential equations and integro differential equations. Recently the comparative study of Mohand and other integral transforms was discussed by Aggarwal et al. [31-35]. Shehu transform of the function () is given by [49]: *()+ ∫ () ( ) where operator is called the Shehu transform operator. The Shehu transform of the function () for exist if () is piecewise continuous and of exponential order. These mention two conditions are the only sufficient conditions for the existence of Shehu transform of the function (). The Volterra integral equation of first kind is given by [7, 14, 24, 27] () ∫ ( )() () where the unknown function () , that will be determined, occurs only inside the integral sign. The kernel ( ) of integral equation (1) and the function () are known real-valued functions. The aim of this work is to find out exact solutions of Volterra integral equation of first kind using the new and advance integral transform “Shehu transform” without large computational work. 2. LINEARITY PROPERTY OF SHEHU TRANSFORMS [49-50] If *()+ ( ) and *()+ ( ) then *() ()+ *()+ *()+ *() ()+ ( ) ( ) where are arbitrary constants. 3. SHEHU TRANSFORM OF SOME ELEMENTARY FUNCTIONS [49-50] S.N. () *()+ ( ) 1. 2. . / 3. . / 4. . / 5. ( ) . / 6.