On the solvability fractional of a boundary value problem with new fractional integral

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On the solvability fractional of a boundary value problem with new fractional integral M. Moumen Bekkouche1,2

· H. Guebbai2

· M. Kurulay3

Received: 3 February 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020

Abstract This work presents new techniques for finding solutions of linear fractional differential equation boundary value problem when the derivation is conformable fractional of Caputo type, the first technique we will study the method that converts an initial value problem to an equivalent linear ordinary differential equation of second order. In order to find solution by an other technique we introduce a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo. Also, some examples are included to improve the validity and applicability of the techniques. Keywords Fractional Boundary value problem · Green’s function · Fractional derivative · fractional integral Mathematics Subject Classification 26A33 · 34A08 · 34K37 · 34A08

Introduction The usual integral and derivative are (to say the least) a staple for the technology professional, essential as a means of understanding and working with natural and

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M. Moumen Bekkouche [email protected] H. Guebbai [email protected] M. Kurulay [email protected]

1

Faculty of Exact Sciences - Department of Mathematics, El oued University, 39000 El oued, Algeria

2

Laboratoire des Mathématiques Appliquées et Modélisation, Université 8 Mai 1945, 24000 Guelma, Algeria

3

Department of Mathematical Engineering, Yildiz Technical University, 34210 Davutpasa, Istanbul, Turkey

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M. Moumen Bekkouche et al.

artificial systems. Recently, many authors have participated in the development of the fractional calculus (differentiation and integration of arbitrary order). The applications of fractional calculus often appeared in the fields such as generalized voltage dividers [21], the electric conductance of biological systems [6], capacitor theory [16], engineering [5], electrode-electrolyte interface models [12], feedback amplifiers, medical [6,7], fractional order models of neurons [2], analysis of special functions [10], and fitting experimental data [3], and there are some papers deal with the existence and multiplicity of solution of nonlinear initial fractional differential equation by the use of techniques of nonlinear analysis, see [1,3,7,20]. However, most papers offer the problem using the standard Riemann-Liouville differentiation. But, Our aim is to find the solution for a class of fractional boundary value problems (Fig. 1). To the best of our knowledge, this is the first work that solve problem with the conformable fractional of Caputo type [4,12]. We will use some important theorems and lemmas to get solutions of the problem (Fig. 2). In this work, we consider a type of fractional boundary value problems. The governing equation is given with the fractional differential equation as follows D (γ ) u(x) + q(x)u(x) = f (x), 0 < x < 1 u(0) = u(1) = 0

(1)

where 1 < γ < 2

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