On Solution of Viscoelastic Fractional Differential Equation

Abstract

In this paper we study the fractional viscoelastic differential equation and solved it in general. We notice that our method is a generalization of Coimbra’s approach and easier than Ayala method. 1- Introduction: Fractional calculus is an old and new branch of mathematics with a long history. It’s early beginning was in 1695 when G. W. Leibniz wrote a letter from Hanover, Germany, September 30, 1695 to G. A. L’Hopital said that x dx d 2 x = x 1 which is an apparent paradox and this was found in volume 2, pp.301-302, Olms Verlag, Hildesheim, Germany 1962 and first published in 1849. After two years Leibinz wrote a letter to Wallis to discuss infinite product of p and in this letter Leibinz mentioned to differential calculus and used d 2 y 1 to derivative of order 2 1 . For most details of historical background see [1,5]. The first application of fractional calculus is due to Abel in 1823 in solving an integral equation which arises in the tautochrone problem which is sometimes called isochrone problem and it is of finding the shape of a fractionless wire lying in a vertical plane such that the time of slide of a bead placed on the wire slides to the lowest point of the wire in the same time regardless of where the bead is placed.