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

Urvashi Purohit Sharma, Ritu Agarwal

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.

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

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.
Paper Structure (8 sections, 18 theorems, 95 equations)

This paper contains 8 sections, 18 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Bicomplex Numbers
  3. Fractional calculus of bicomplex Prabhakar function
  4. Bicomplex Prabhakar fractional operators
  5. Bicomplex Prabhakar kernel
  6. Some properties of bicomplex Prabhakar fractional operators
  7. Laplace transform of fractional derivatives and application in solving Cauchy Problem
  8. Conclusion

Key Result

Theorem 1.4

Let $\mathcal{L}((f(t);\xi)=\tilde{f}(\xi)$ and $\mathcal{L}((g(t);\xi)= \tilde{g}(\xi)$ with $\operatorname{Re}(\xi )>M +|\operatorname{Im_j}(\xi)|$ where $M=\max(M_1,M_2)$ and $f(t)$ and $g(t)$ are of exponential order $M_1$ and $M_2$ respectively. Then

Theorems & Definitions (40)

  • Definition 1.1: Bicomplex Number
  • Definition 1.2: Bicomplex Partial Order
  • Definition 1.3: Bicomplex Laplace transform
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2: Convolution Identity
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 30 more