A reliable mixed method for singular integro-differential equations of non-integer order

¹ Department of Mathematics, Çankaya University, Ankara, Turkey
² Institute of Space Sciences, Maturely-Bucharest, Romania
³ Department of Mathematics, Neka Branch, Islamic Azad University, Neka, Iran
⁴ Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

^* Corresponding author: dumitru@cankaya.edu.tr

Received: 9 November 2017
Accepted: 30 December 2017

Abstract

It is our goal in this article to apply a method which is based on the assumption that combines two methods of conjugating collocation and multiple shooting method. The proposed method can be used to find the numerical solution of singular fractional integro-differential boundary value problems (SFIBVPs)

        D^ϑ y(t) + η ∫₀^t  (t−s)^ς−1 y(s) ds = g(t),    1 < ϑ ≤ 2,    0 < ς< 1,    η ∈ ℝ,

where D^ϑ denotes the Caputo derivative of order ϑ. Meanwhile, in a separate section the existence and uniqueness of this method is also discussed. Two examples are presented to illustrate the application and further understanding of the methods.