Fractional differ-integral involving bicomplex Prabhakar function in the kernel and applications

TL;DR

This paper extends fractional calculus to bicomplex spaces using the Prabhakar function, establishing fundamental properties and applications in modeling complex phenomena with memory effects.

Contribution

It introduces the bicomplex Prabhakar derivative, expanding fractional calculus into four-dimensional bicomplex domains with new operational properties.

Findings

Defined the bicomplex Prabhakar derivative and proved its properties.

Derived integral representations and transformation formulas.

Established connections to classical fractional operators.

Abstract

This paper introduces the bicomplex Prabhakar derivative, extending fractional calculus to four-dimensional bicomplex spaces. Using the generalized kernel involving bicomplex Prabhakar function, we construct the bicomplex Prabhakar derivative and prove fundamental operational properties including linearity, composition rules, and connections to Riemann-Liouville and Caputo operators. We further investigate how fractional operators act on the bicomplex Prabhakar function itself, developing integral representations and transformation formulas. This work provides a rigorous foundation for modeling complex phenomena with memory effects and multi-dimensional coupling in bicomplex domains. The rich algebraic structure of bicomplex numbers, combined with the flexibility of Prabhakar kernels, offers a versatile framework applicable across diverse scientific and engineering disciplines.