Exact Solution of Linear Volterra integro-differential Equation of First Kind Using Abaoub-Shkheam Transform

International Journal of Research and Innovation in Social Science (IJRISS) | Volume V, Issue XI, November 2021 | ISSN 2454–6186

Exact Solution of Linear Volterra integro-differential Equation of First Kind Using Abaoub-Shkheam Transform

Abejela S. Shkheam¹, Ali E. Abaoub^2*, and Yousuf A. Huwaydi^3
1,2Mathematical Dept., Faculty of Science, Sabratha University, Sabratha, LIBYA
³Mathematical Dept., School of Basic Sciences, the Libyan Academy, Tripoli, LIBYA
*Corresponding Author

Abstract: We employ Abaoub – Shkheam transformation to solve linear Volterra integro-differential equation of the first kind, we considered the kernel of that equation is a deference type kernel. Moreover, we prove the existence and uniqueness of solutions of the equation under some conditions in the Banach space and fixed-point theory. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique.

Keywords: Volterra integro-differential equation, Abaoub-Shkheam transform, Fixed-point method, Contraction mapping, convolution theorem.

I. INTRODUCTION

In recent years, many researchers have used integro-differential equations which is the combination of differential and integral equations as model of many problems of science and theoretical physics such as engineering, biological models, electrostatics, control theory of industrial [5]. This type of equations was termed as Volterra integro-differential equations, given in the form

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