A derivative concept with respect to an arbitrary kernel and applications to fractional calculus. (18th October 2018)

• Record Type: Journal Article
• Authors: Jleli, Mohamed; Kirane, Mokhtar; Samet, Bessem
• Is Part Of: Mathematical methods in the applied sciences. Volume 42:Number 1(2019)
• Page Start: 137
• Page End: 160
• Publication Date: 2018-10-18
• Subjects: boundary value problem -- conjugate kernels -- fractional calculus -- k‐derivative; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5329
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we propose a new concept of derivative with respect to an arbitrary kernel function. Several properties related to this new operator, like inversion rules and integration by parts, are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann‐Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.