THE M TH LEVEL FRACTIONAL DERIVATIVES WITH RESPECT TO ANOTHER FUNCTION

Bany-Ahmad R., Ibrahim A., Noorani M. S. M., ABDELJAWAD T.

Journal of Applied Analysis and Computation, cilt.16, sa.3, ss.1401-1445, 2026 (SCI-Expanded, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 16 Sayı: 3
  • Basım Tarihi: 2026
  • Doi Numarası: 10.11948/20250218
  • Dergi Adı: Journal of Applied Analysis and Computation
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.1401-1445
  • Anahtar Kelimeler: Fractional calculus, generalized fractional derivatives, Riemann-Liouville fractional derivative, ψ − mth level fractional derivatives, ψ−Hilfer fractional derivative, ψ−Second level fractional derivative
  • İstanbul Gelişim Üniversitesi Adresli: Evet

Özet

We establish conditions ensuring the existence of a solution ϕ(x) in the space L1 (a, b) and derive an explicit formula for this solution. We introduce a novel parametrization of the Hilfer fractional derivative with respect to another function ψ, which unifies and extends several classical operators. In this framework, we develop an extensive variant of Luchko’s second level fractional derivative, termed the ψ−second level fractional derivative, encompassing the ψ−Riemann-Liouville, ψ−Caputo, and ψ−Hilfer derivatives as special cases. We analyze the relationships among these derivatives for various parameter values and within different function spaces, discussing their properties and significant results in fractional calculus. Finally, we propose a comprehensive generalization, the ψ − mth level fractional derivative, which facilitates the construction of fractional derivatives of any desired level. The main results focus on the ψ−second level derivative, as it unifies a wide range of established operators, simplifies the interpretation of results, and naturally supports further generalizations.