Approximate Solution of Sub diffusion Bio heat Transfer Equation

Sonawane, Jagdish; Sontakke, Bahusaheb; Takale, Kalyanrao

doi:10.21123/bsj.2023.8410

Cited by 6 publications

(7 citation statements)

References 9 publications

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“…The proposed study utilizes a time fractional drug diffusion problem described by Equations ( 3), ( 4) and (5) to evaluate the diffusion of drug within the human dermal layer. To accomplish this, we employ a fractional-order explicit finite difference method, as developed in Equations ( 8), ( 9), (10), (11) and (12), to predict drug diffusion at specific grid points (x i ,t k ) within the discretized dermal region. The simulation employs dimensionless quantities.…”

Section: Resultsmentioning

confidence: 99%

“…In this context, we examine the stability of the solution obtained through the application of the explicit method Equations ( 8), ( 9), ( 10), (11) and (12) to the TFDDE Equations ( 3), ( 4) and (5).…”

Section: Stability Of Numerical Solutionmentioning

confidence: 99%

“…In recent time, non-integer order differential equations have gained prominence in various fields due to their ability to account for memory effects and long-range dependencies. Fractional order modeling has found successful applications in diverse fields, including physics, engineering (8) , economics (9) , viscoelasticity (10) , medicine (11) , heat transfer (12) , pharmacokinetics (13) and more, to describe various physical phenomena. The application of fractional differential equations in biology has been widespread, as they demonstrate a capacity to capture memory effects, long-range dependencies, and complex dynamics within biological systems (14) .…”

Section: Introductionmentioning

confidence: 99%

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Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

Takale,

Kharde,

Takale

2023

IJST

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Objectives:The primary objective of this research paper is to gain a comprehensive understanding of drug diffusion within the human dermal region. Methods: A temporal fractional-order reaction-diffusion equation with Caputo sense is employed to get mathematical insights on the diffusion of drugs in the human dermal region. The explicit finite difference method is employed to numerically solve the modelled problem. A Python-based algorithm is employed to obtain a numerical solution through the finite difference method. Finding: In our research, we focused on examining how fractional-order parameters affect the distribution and concentration profiles of drugs in the dermal region. To convey our findings effectively, we conducted a comprehensive analysis, primarily using graphical representations. These visualizations offer a clear and insightful view of the drug's diffusion rate within the dermal region, taking into account the memory effect associated with the Caputo derivative. In addition to our exploration of fractional-order parameters and drug diffusion profiles, we conducted a comprehensive investigation into the stability and convergence of the explicit finite difference method. Novelty: The fractional order explicit finite difference method can be used to estimate drug concentration in the human skin. An algorithm based on Python provides powerful tool for obtaining numerical solution of fractional order differential equations.

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Section: Resultsmentioning

confidence: 99%

Section: Stability Of Numerical Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

Takale,

Kharde,

Takale

2023

IJST

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“…In recent years, considerable attention has been directed towards the burgeoning field of "Fractional Calculus" by numerous researchers, driven by its diverse applications across various scientific and engineering domains (1)(2)(3)(4)(5) . Introducing fractional operators into classical differential equations results in complex challenges when solving resulting fractional-order partial differential equations (6) .…”

Section: Introductionmentioning

confidence: 99%

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Dhumal,

Sontakke

2024

IJST

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Objectives: This study aims to explore solutions to the time-fractional Fitzhugh-Nagumo equation, a nonlinear reaction-diffusion equation. Method: We utilize the Homotopy Perturbation Method (HPM) as a proficient analytical approach for addressing the time-fractional Fitzhugh-Nagumo equation. The HPM offers a structured method for deriving approximate solutions in the shape of convergent series, enabling accurate solutions even for intricate nonlinear fractional equations. Finding: The series solution obtained is validated by comparing it with numerical methods, showcasing its precision and effectiveness. Additionally, we assessed the error across various time and space values. Our analysis and computations reveal that the Homotopy Perturbation Method (HPM) stands out for providing precise approximations while maintaining computational efficiency. It's clear that this method presents a robust alternative to conventional numerical techniques, particularly in situations where analytical solutions are difficult to obtain. Novelty: The application of the Homotopy Perturbation Method to the Time-fractional Fitzhugh-Nagumo Equation has been effectively explored, with specific examples showing a strong agreement between the exact solution and the obtained solution. Keywords: Time-Fractional Fitzhugh–Nagumo Equation, Homotopy Perturbation Method, Riemann-Liouville fractional integral, Caputo fractional derivative, Fractional Homotopy Perturbation Method

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“…It is observed that, most of the models are developed by using ordinary and partial derivatives, which means that they are free from the memory effect. In the last two decades, fractional differential equations (FDEs) have been widely used in the modeling of real life problems [5][6][7][8][9] .…”

Section: Introductionmentioning

confidence: 99%

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

Takale,

Kharde,

Gaikwad

2024

Baghdad Sci.J

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The fractional differential equation provides a powerful mathematical framework for modeling and understanding a wide range of complex, non-local and memory-dependent phenomena in various scientific, engineering and real-world applications. Liver metastasis is a secondary cancerous tumor that develops in the liver due to the spread of cancer cells from a primary cancer originating in another part of the human body. The primary focus of this study is to understand tumor growth in the human liver, both in the presence and absence of medication therapy. To achieve this, a temporal fractional-order parabolic partial differential equation is utilized, and its analysis is carried out using numerical methods. The Caputo derivative is employed to explore the impact of medication therapy on tumor growth. To numerically solve the mathematical model, the Crank-Nicolson finite difference method has been developed. This method is selected for its remarkable attributes, including unconditional stability and second-order accuracy in both spatial and temporal dimensions. The outcomes and insights derived from this study are effectively communicated through various graphical representations. These visual aids serve as invaluable tools for comprehending the profound impact of medication therapy on the growth of liver tumors. Through the medium of these graphical depictions, one can glean a clearer and more intuitive understanding of the complex dynamics at play. The numeric solution to this intricate problem is achieved through the implementation of algorithms meticulously crafted using versatile and powerful Python programming language. Python’s flexibility, extensive libraries, and robust numerical computing capabilities make it a good choice for handling the complexities of this study.

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Approximate Solution of Sub diffusion Bio heat Transfer Equation

Cited by 6 publications

References 9 publications

Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

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