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

This paper introduces the bicomplex Prabhakar derivative by extending fractional calculus to four-dimensional bicomplex spaces, establishing its fundamental operational properties and integral representations to provide a versatile framework for modeling complex phenomena with memory effects and multi-dimensional coupling.

The Gist

Imagine you are trying to understand how a complex system changes over time. Maybe it's how heat spreads through a strange metal, how a population of animals grows with a "memory" of past events, or how electricity flows through a circuit with unusual components.

In the world of math, we usually use tools called calculus to measure change. But sometimes, things don't change in a simple, straight line. They have "memory." The way a material stretches today might depend on how it was stretched yesterday, last week, or even last year. To handle this, mathematicians invented Fractional Calculus. Think of this as a "super-calculus" that can measure change in steps that aren't just whole numbers (like 1st derivative or 2nd derivative) but can be fractions (like 0.5 or 1.7).

Now, imagine taking that "super-calculus" and making it even more powerful by adding a new layer of complexity: Bicomplex Numbers.

The Big Idea: A 4D Math Universe

Most of us are used to Real Numbers (1, 2, 3) and Complex Numbers (which add an imaginary "i" to handle things like square roots of negative numbers).

This paper introduces Bicomplex Numbers.

• The Analogy: Imagine a standard complex number is a flat sheet of paper (2D). A Bicomplex number is like a 4D hologram. It has two layers of "imaginary" dimensions working together.
• Why do we need it? It allows us to model systems where two different complex processes are tangled together, interacting in ways that standard math can't easily describe. It's like trying to describe a dance where two couples are dancing on the same floor, but their movements are perfectly synchronized in a way that requires a 4D map to track.

The Star of the Show: The Prabhakar Function

The authors of this paper are building a new tool using something called the Prabhakar Function.

• The Analogy: Think of the Prabhakar function as a "Smart Lens" or a "Memory Filter."
• In standard math, we have lenses that blur things out (integrals) or sharpen edges (derivatives). The Prabhakar lens is special because it has three knobs (parameters) that you can twist.
  • Knob 1: Controls how much "memory" the system has.
  • Knob 2: Controls how fast the memory fades.
  • Knob 3: Controls the shape of the memory curve.
• By twisting these knobs, you can fit the math perfectly to real-world data, whether it's the slow creep of a glacier or the rapid spike of a stock market crash.

What Did This Paper Actually Do?

The authors took this "Smart Lens" (Prabhakar) and put it inside the "4D Hologram" (Bicomplex numbers). They created a new mathematical machine called the Bicomplex Prabhakar Derivative.

Here is what they achieved, broken down simply:

1. Built the Engine: They defined exactly how to calculate this new derivative. It's like writing the instruction manual for a new type of engine that runs on 4D fuel and uses a 3-knob memory filter.
2. Tested the Rules: They proved that this new engine follows the basic rules of math (like linearity: if you double the input, you get double the output). They also showed how it connects to older, simpler math tools (like the Riemann-Liouville and Caputo derivatives), proving it's a natural upgrade, not a broken invention.
3. The "Magic" Formula (Laplace Transform): One of the hardest parts of solving these complex equations is the algebra. The authors found a "translation key" (the Laplace Transform).
   • The Analogy: Solving a differential equation is like trying to untangle a giant knot of headphones. The Laplace Transform is like a machine that instantly un-knots them, lets you fix the problem in a straight line, and then re-knots them perfectly. The authors showed how to use this machine for their new 4D, 3-knob system.
4. Solved a Puzzle (Cauchy Problem): They used their new tools to solve a specific type of puzzle: "If we know the starting state of a system, can we predict its future?" They showed that their method works efficiently, even for systems with heavy "memory" effects.

Why Should You Care?

You might think, "I don't need to calculate 4D fractional derivatives." But this math is the engine behind the scenes for many real-world technologies:

• Medical Imaging: Better models for how drugs move through the human body (which has memory and complex structures).
• Battery Technology: Understanding how energy degrades in batteries over time, which isn't a simple linear process.
• Earthquake Engineering: Modeling how soil and buildings react to vibrations that have "memory" of past stresses.
• Quantum Physics: Describing particles that exist in multiple states simultaneously.

The Bottom Line

This paper is like a blueprint for a new, super-advanced calculator.

• Old Calculator: Good for simple, straight-line changes.
• Fractional Calculator: Good for things with memory.
• Bicomplex Calculator: Good for things with multiple, tangled layers of complexity.
• This Paper's Calculator: The ultimate tool that combines memory, complexity, and flexibility into one powerful package.

The authors have laid the foundation. Now, scientists and engineers can use this new "4D memory lens" to build better models, design smarter materials, and understand the universe in a deeper, more connected way.

Technical Summary

Here is a detailed technical summary of the paper "Fractional differ-integral involving bicomplex Prabhakar function in the kernel and applications" by Urvashi Purohit Sharma and Ritu Agarwal.

1. Problem Statement

The paper addresses the need to extend fractional calculus from one-dimensional real or complex domains to four-dimensional bicomplex spaces. While fractional calculus (specifically operators involving the Prabhakar function) has been established in real and complex domains to model systems with memory effects and non-local dynamics, these tools lack the algebraic structure to handle coupled multi-dimensional systems naturally.

Specifically, the authors identify a gap in the literature regarding the bicomplex Prabhakar derivative. Although bicomplex Riemann-Liouville operators and various bicomplex Mittag-Leffler functions have been studied, a rigorous definition and operational theory for the bicomplex Prabhakar derivative (a three-parameter generalization) were missing. This limitation hinders the modeling of complex phenomena in fields like electromagnetic theory, coupled population dynamics, and quantum mechanics where variables interact in a commutative, four-dimensional algebraic structure.

2. Methodology

The authors employ a component-wise decomposition approach based on the algebraic properties of bicomplex numbers.

• Idempotent Representation: The core methodology relies on the idempotent representation of bicomplex numbers. Any bicomplex number $\xi$ is decomposed into two complex components ($\xi_1, \xi_2$) using idempotent units $e_1$ and $e_2$ (where $e_1 + e_2 = 1$ and $e_1 e_2 = 0$).
  $$\xi = \xi_1 e_1 + \xi_2 e_2$$
• Reduction to Complex Domain: By utilizing this representation, the authors reduce the problem of defining bicomplex operators to defining corresponding operators in two separate complex domains ($T_1$ and $T_2$). They leverage existing results for the complex Prabhakar function and fractional operators.
• Reconstruction: Once the properties are established for the complex components, the results are reconstructed into the bicomplex domain using the linearity of the idempotent basis.
• Transforms: The Bicomplex Laplace Transform is utilized extensively to derive operational formulas, solve integral equations, and analyze the Cauchy problem.

3. Key Contributions

A. Definition of Bicomplex Prabhakar Operators

The paper formally defines the following operators for bicomplex-valued functions:

1. Bicomplex Prabhakar Kernel: Defined as $e^l_{m,n,r}(t) = t^{n-1} E^l_{m,n}(rt^m)$, where $E^l_{m,n}$ is the bicomplex Prabhakar function.
2. Bicomplex Prabhakar Integral: Defined as a convolution of the function with the kernel:
   $$(E^l_{m,n,r,a+}f)(t) = \int_\Upsilon (t-y)^{n-1} E^l_{m,n}[r(t-y)^m] f(y) dy$$
3. Bicomplex Prabhakar Derivative (Riemann-Liouville type):
   $$(D^l_{m,n,r,a+}f)(t) = \frac{d^k}{dt^k} (E^{-l}_{m,k-n,r,a+}f)(t)$$
4. Bicomplex Regularized Prabhakar Derivative (Caputo type):
   $$(^C D^l_{m,n,r,a+}f)(t) = (E^{-l}_{m,k-n,r,a+} \frac{d^k}{dt^k} f)(t)$$
   This derivative is crucial for handling initial value problems as it incorporates standard initial conditions.

B. Fundamental Operational Properties

The authors prove that these operators satisfy essential properties analogous to their complex counterparts:

• Linearity: The operators are linear over the bicomplex field.
• Semigroup Property: The composition of two Prabhakar integrals results in a single integral with modified parameters:
  $$E^l_{m,n,r,a+} E^\sigma_{m,\nu,r,a+} \equiv E^{l+\sigma}_{m,n+\nu,r,a+}$$
• Composition with Riemann-Liouville Operators: The paper establishes commutative and non-commutative composition rules between standard fractional integrals/derivatives and the Prabhakar operators.
• Inversion Theorem: A left inverse operator is constructed, proving that the Prabhakar integral is invertible under specific conditions.

C. Laplace Transform Analysis

A significant contribution is the derivation of the Laplace transform for the bicomplex Prabhakar derivative and the regularized derivative.

• For the regularized derivative, the transform is given by:
  $$\mathcal{L}\{^C D^l_{m,n,r,0+}f(t); \xi\} = \xi^{n-ml}(\xi^m - r)^l \left[ \tilde{f}(\xi) - \sum_{p=0}^{k-1} \xi^{-p-1} f^{(p)}(0+) \right]$$
  This formula is pivotal for solving initial value problems.

4. Results and Applications

Solving the Cauchy Problem

The paper applies the derived Laplace transforms to solve a Cauchy problem involving the regularized Prabhakar derivative:
$$^C D^l_{m,n,r,0+} f(t) = A f(t), \quad f^{(k)}(0+) = \tau_k$$
Using the Laplace transform and Neumann series expansion, the authors derive an analytic solution expressed as a series involving the bicomplex Prabhakar function:
$$f(t) = \sum_{k=0}^{m-1} \sum_{j=0}^{\infty} A^j t^{n_j+k} E^{jl}_{m, n_j+k+1}(r t^m) \tau_k$$
This demonstrates the practical utility of the theory in solving fractional differential equations with memory kernels.

Convolution Identity

The paper proves a convolution identity for the Prabhakar function, which is fundamental for establishing the semigroup property and solving integral equations:
$$\int_0^x (x-t)^{n-1} E^l_{m,n}(r(x-t)^m) t^{\nu-1} E^\sigma_{m,\nu}(rt^m) dt = x^{n+\nu-1} E^{l+\sigma}_{m,n+\nu}(rx^m)$$

5. Significance and Impact

• Theoretical Advancement: This work fills a critical theoretical gap by establishing the foundations of fractional calculus in bicomplex spaces using the most flexible kernel available (Prabhakar). It unifies the algebraic richness of bicomplex numbers with the modeling power of fractional calculus.
• Modeling Capability: The bicomplex Prabhakar derivative offers a versatile framework for modeling coupled multi-dimensional systems with memory effects and non-local interactions. This is particularly relevant for:
  • Electromagnetic Theory: Fractional Maxwell equations in anisotropic media.
  • Physics: Time-fractional Schrödinger equations and anomalous diffusion.
  • Engineering: Analysis of electric circuits with coupled fractional elements and viscoelastic materials with hierarchical relaxation.
• Generalization: The results generalize previous work on bicomplex Mittag-Leffler functions and Riemann-Liouville operators. The inclusion of the parameter $l$ (from the Prabhakar function) allows for better fitting of experimental data compared to standard fractional operators.
• Future Directions: The authors suggest that this framework can be extended to multicomplex spaces (dimension $2^n$) and k-parameter Prabhakar functions, opening new avenues for high-dimensional coupled systems.

In conclusion, the paper provides a rigorous mathematical toolkit for analyzing complex, memory-dependent systems in four-dimensional spaces, bridging the gap between abstract algebraic structures and applied fractional differential equations.