Original Article |
Open Access |
Int. J. Mat. Math. Sci., 2024; 6(5), 129-133 |
doi: 10.34104/ijmms.024.01290133

A Novel Approach to Solving Fractional Diffusion Equations Using Fractional Beta Derivative

This paper introduces a novel application of fractional beta derivatives in solving fractional diffusion equations with an emphasis on systems exhibiting anomalous diffusion and memory effects. The work explores the fractional beta derivative as an extension of classical fractional derivatives by incorporating a parameter β (beta) that controls the systems memory behavior. We investigate both the analytical and numerical solutions of these equations, demonstrating the superior flexibility of fractional beta derivatives in modeling complex diffusion processes. Additionally, we provide a comparison between classical fractional derivatives and the new fractional beta approach to highlight the advantages in terms of accuracy and computational efficiency. We begin by reviewing the theoretical background of fractional derivatives and proceed to introduce the beta derivative as a modification that provides additional flexibility in modeling complex systems. Applications in fields such as control theory, signal processing, and bioengineering are highlighted. Furthermore, numerical methods for solving fractional beta differential equations are discussed, along with potential areas for future research.

The concept of Fractional calculus, developed by Podlubny, (1999) has become an essential tool in various fields such as fluid mechanics, viscoelasticity, and diffusion processes, providing new ways to model systems with memory effects and anomalous behavior. Traditional derivatives, like Caputo and Riemann-Liouville, have been extensively studied, but they offer limited flexibility in tuning the memory effect in fractional-order systems. A fractional beta derivative introduces an additional parameter β (beta) given by Diethelm, (2010) allowing more control over the long-term behavior of the solution, which can enhance modeling accuracy for systems with variable memory. Despite the promise of fractional beta derivatives, developed by Tarasov, (2016) their application in time-fractional diffusion equations remains relatively unexplored. This paper aims to bridge that gap by presenting a novel approach to solving diffusion problems in fluid mechanics and other areas, using this powerful tool. Fractional calculus has gained substantial attention in recent years due to its ability to describe complex systems with memory and hereditary properties. Among the well-known fractional operators, the Caputo and Riemann-Liouville derivatives have been instrumental in modeling various physical phenomena. However, there is growing interest in exploring generalized fractional derivatives to further refine these models investigated by (Samko *et al., *1993; Miller *et al., *1993).

Fractional calculus extends traditional calculus by allowing differentiation and integration of non-integer (fractional) order. This allows for more general models that are well-suited for capturing memory and hereditary properties in physical and engineering systems given by (Metzler *et al., *2000; Oldham *et al., *2006). The concept of a fractional derivative has been widely used in various scientific fields, including mechanics, electrical engineering, biology, and economics. Recent advances have been introduced by (Machado *et al.,* 2011; Luo *et al.,* 2021), the idea of fractional beta derivatives which add an additional parameter β(beta) to the classical definition of a fractional derivative. This beta parameter enhances flexibility, providing better control over the behavior of the system which have been modeled by Li, (2015). The aim of this paper is to present a rigorous analysis of fractional beta derivatives, their theoretical properties, and their applications. We briefly review the Caputo and Riemann-Liouville fractional derivatives and highlight their limitations when modeling systems with variable memory. The fractional beta derivative offers a more flexible approach due to its tunable parameter β (beta), making it particularly suitable for complex systems. The fractional beta derivative is a relatively new concept that introduces a flexible parameter, beta, which allows the adjustment of the memory effect in the system. This parameter can provide more control over the behavior of solutions, making it a potentially powerful tool in fractional differential equations (FDEs). Despite its potential, the use of fractional beta derivatives in fluid mechanics remains unexplored.

In this paper, we aim to investigate the application of fractional beta derivatives in solving time-fractional differential equations commonly used in fluid mechanics. We summarize the effects of the beta parameter on the solutions accuracy and convergence. Numerical experiments reveal that increasing β (beta) leads to slower diffusion and more pronounced memory effects, providing a better fit for anomalous diffusion phenomena.

**Riemann-Liouville Fractional Derivative**

The Riemann-Liouville fractional derivative is defined as the Riemann-Liouville derivative was developed by Jumarie, (2006)

D^α f(t)=1/Γ(n-α) d^n/(dt^n ) ∫_0^t▒〖(t-τ)^(n-α-1) f(τ)dτ〗, (1)

where n-1<α

**Caputo Fractional Derivative**

The Caputo derivative is widely used due to its applicability in real-world problems where initial conditions are given in terms of integer-order derivatives Grünwald-Letnikov derivative was developed by Caputo, (1967),

〖(_^C)D〗^α f(t)=1/Γ(n-α) ∫_0^t▒〖(t-τ)^(n-α-1) d^n/(dt^n ) f(τ)dτ〗. (2)

**Fractional Beta Derivatives: Definition and Properties**

**Definition of Beta Derivatives**

The fractional beta derivative is a generalization of fractional derivatives with the inclusion of an additional beta parameter:

D^(α,β) f(t)=1/Γ(β) ∫_0^t▒〖(t-τ)^(β-1) (d/dτ)^α f(τ)dτ〗. (3)

The parameter beta allows greater flexibility in defining the dynamics of the system, enabling the derivative to be more sensitive to certain features of the system under study and the definition was given by (Zhang *et al.,* 2011; Baleanu *et al., *2009).

**Key Properties of Beta Derivatives**

The fractional beta derivative possesses several properties analogous to those of traditional fractional derivatives, including,

(i) Linearity property: The operator is linear, i.e.,

D^(α,β) (c_1 f(t)+c_2 g(t))=c_1 D^(α,β) f(t)+c_2 D^(α,β) g(t) . (4)

(ii) Beta Adjustment: The beta parameter allows for adjusting the rate at which memory effects decay, making it highly suitable for applications in systems with varying degrees of temporal memory.

(iii) Convolution Property: The fractional beta derivative exhibits a convolution-type relationship, similar to the classical fractional derivatives. A detailed comparison with classical fractional calculus methods highlights the advantages of using fractional beta derivatives, particularly in cases where the memory effect varies over time.

**Anomalous Diffusion with Variable Memory**

Anomalous diffusion, where the diffusion rate deviates from the classical Brownian motion, is common in complex media such as porous materials and biological systems. The fractional beta derivative provides a natural framework to model such systems, offering better control over the long-term behavior of diffusion. We explore real-world applications, such as diffusion in porous media and biological tissue, demonstrating how the beta parameter can be adjusted to match experimental data more closely than with traditional fractional derivatives.

**Applications of Fractional Beta Derivatives**

**Application in Fluid Mechanics**

In fluid mechanics, non-Newtonian fluids exhibit memory effects that cannot be captured by classical models. Using fractional beta derivatives, we model the flow of such fluids and analyze the impact of β (beta) on the velocity profile and shear stress. Simulations show that the beta parameter allows for a more accurate representation of the memory effects in viscoelastic fluids.

**Control Theory**

In the control theory introduced by (Duarte *et al.,* 2020; Wei, *et al.,* 2020) the fractional beta derivatives are used to model systems with complex dynamics and long-range temporal dependencies. Their inclusion allows for greater accuracy in predicting system behavior over time.

**Signal Processing**

Fractional beta derivatives have found applications in signal processing, where they help in modeling and analyzing signals that exhibit anomalous behavior or noise with memory cited by Mainardi, (2010).

**Bioengineering**

In bioengineering, the fractional beta derivative is used to model biological systems, such as neural networks and biological tissues, where processes exhibit both short- and long-term memory effects presented by Zhang, (2021).

**Numerical Methods for Solving Fractional Beta Differential Equations**

**Discretization Methods**

Numerical solutions to fractional beta differential equations often involve discretization techniques, such as the finite difference method or spectral methods, adapted to handle the non-integer nature of the derivatives.

**Convergence and Stability**

Ensuring convergence and stability in numerical methods is a critical aspect of solving fractional beta differential equations. The impact of the beta parameter on the stability of numerical methods needs further investigation.

**Future Directions and Open Problems**

There are numerous open problems and potential research directions in the field of fractional beta derivatives. These include:

- Developing more efficient numerical methods for solving fractional beta differential equations
- Investigating the role of beta derivatives in other fields such as finance, fluid dynamics, and material science
- Exploring fractional beta derivatives in non-linear systems

Fractional beta derivatives represent a significant advancement in the field of fractional calculus, offering increased flexibility in modeling complex systems. This paper has provided a comprehensive overview of the theory behind fractional beta derivatives, their properties, and their applications. Future research is expected to further explore the potential of these derivatives in both theoretical and applied contexts. We conclude that fractional beta derivatives provide a powerful extension to classical fractional calculus, offering more flexibility and control over memory effects in diffusion and fluid mechanics problems. Future research could extend this work to more complex, nonlinear systems, and explore the implications of beta in other fields such as heat transfer and wave propagation.

M.A.A. Conceptualization, Resources, Methodology, Investigation, Writing-Original Draft. M.A.I. Software, Validation, Project administration, Super-vision, Writing-Review Editing. M.S.I. Visualization, Data Curation, Funding acquisition, Writing-Review Editing. A.M.A.M. Formal Analysis, Validation. S.K. Writing-Review Editing.

No data was used for the research described in the article.

Firstly I want to thank Almighty Allah. I would like to thank my family for their support throughout my graduate studies. It is a matter of great pleasure to express my heartiest gratitude and profound respect to my brother Md. Ashik Iqbal, Depart-ment of Mathematics and Physics, Khulna Agricultural University, Khulna, for his valuable guidance, spontaneous encouragement, untiring efforts, keen interest, and wholehearted supervision throughout the progress of my work without which this work would never have been materialized. With hearty thanks, I acknow-ledge all my research partner for their encour-agement, well wishes, technical and material support in different stages of this work.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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**Academic Editor**

Dr. Wiyanti Fransisca Simanullang Assistant Professor Department of Chemical Engineering Universitas Katolik Widya Mandala Surabaya East Java, Indonesia.

Aziz MA, Iqbal MA, Ikbal MS, Al-Mamun AM, and Khatun S. (2024). A novel approach to solving fractional diffusion equations using fractional beta derivative, Int. J. Mat. Math. Sci., 6(5), 129-133. https://doi.org/10.34104/ijmms.024.01290133

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