scholarly journals The Effect of Fractional Time Derivative of Bioheat Model in Skin Tissue Induced to Laser Irradiation

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 602
Author(s):  
Aatef Hobiny ◽  
Faris Alzahrani ◽  
Ibrahim Abbas ◽  
Marin Marin
Keyword(s):  
Laser Irradiation ◽  
Fractional Order ◽  
Time Derivative ◽  
Thermal Relaxation ◽  
Skin Tissue ◽  
Laplace Domain ◽  
Denatured Protein ◽  
Design Variables ◽  
New Interpretation ◽  
Fractional Time Derivative

This work uses the “fractional order bio-heat model” (Fob) model of heat conduction to offer a new interpretation to study the thermal damages in a skin tissue caused by laser irradiation. The influences of fractional order and the thermal relaxation time parameters on the temperature of skin tissue and the resulting thermal damage are studied. In the Laplace domain, the analytical solutions of temperature are obtained. Using the equation of Arrhenius, the resulting thermal injury to the tissues is assessed by the denatured protein ranges. The numerical results of the thermal damages and temperature are presented graphically. A parametric analysis is dedicated to the identifications of suitable procedures for the selection of significant design variables to achieve an effective thermal in the therapy of hyperthermia.

scholarly journals Fractional-Order Thermoelastic Wave Assessment in a Two-Dimensional Fiber-Reinforced Anisotropic Material

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1609
Author(s):  
Samah Horrigue ◽  
Ibrahim A. Abbas
Keyword(s):  
Fractional Order ◽  
Time Derivative ◽  
Relaxation Times ◽  
Thermal Relaxation ◽  
Generalized Thermoelasticity ◽  
Fiber Reinforced ◽  
Thermoelastic Wave ◽  
Practical Applications ◽  
Fractional Time Derivative ◽  
Fiber Reinforced Material

The present work is aimed at studying the effect of fractional order and thermal relaxation time on an unbounded fiber-reinforced medium. In the context of generalized thermoelasticity theory, the fractional time derivative and the thermal relaxation times are employed to study the thermophysical quantities. The techniques of Fourier and Laplace transformations are used to present the problem exact solutions in the transformed domain by the eigenvalue approach. The inversions of the Fourier-Laplace transforms hold analytical and numerically. The numerical outcomes for the fiber-reinforced material are presented and graphically depicted. A comparison of the results for different theories under the fractional time derivative is presented. The properties of the fiber-reinforced material with the fractional derivative act to reduce the magnitudes of the variables considered, which can be significant in some practical applications and can be easily considered and accurately evaluated.

scholarly journals Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Majid Bagheri ◽  
Ali Khani
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Kdv Equation ◽  
Time Derivative ◽  
Hyperbolic Function ◽  
Nonlinear Phenomena ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Fractional Time Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

GENERALIZED FRACTIONAL ORDER BLOCH EQUATION WITH EXTENDED DELAY

2012 ◽  
Vol 22 (04) ◽  
pp. 1250071 ◽  
Author(s):  
SACHIN BHALEKAR ◽  
VARSHA DAFTARDAR-GEJJI ◽  
DUMITRU BALEANU ◽  
RICHARD MAGIN
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Time Derivative ◽  
Bloch Equation ◽  
Extended Model ◽  
Order Ordinary Differential Equation ◽  
Stability Behavior ◽  
System Memory ◽  
The Stability ◽  
Fractional Time Derivative

The fundamental description of relaxation (T1 and T2) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time- and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T1 and T2 relaxation. The T1 decay is stable for the range of delays tested (1 μsec to 200 μsec), while the T2 relaxation in this extended model exhibits a critical delay (typically 100 μsec to 200 μsec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation.

scholarly journals The Effects of Fractional Time Derivatives in Porothermoelastic Materials Using Finite Element Method

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1606
Author(s):  
Marin Marin ◽  
Aatef Hobiny ◽  
Ibrahim Abbas
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Time Derivative ◽  
Thermal Relaxation ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Future Studies ◽  
Fractional Time Derivative ◽  
Insight Into ◽  
Element Method

In this work, a new model for porothermoelastic waves under a fractional time derivative and two time delays is utilized to study temperature increments, stress and the displacement components of the solid and fluid phases in porothermoelastic media. The governing equations are presented under Lord–Shulman theory with thermal relaxation time. The finite element method has been adopted to solve these equations due to the complex formulations of this problem. The effects of fractional parameter and porosity in porothermoelastic media have been studied. The numerical outcomes for the temperatures, the stresses and the displacement of the fluid and the solid are presented graphically. These results will allow future studies to gain a detailed insight into non-simple porothermoelasticity with various phases.

Fractional abstract Cauchy problem on complex interpolation scales

2020 ◽  
Vol 23 (4) ◽  
pp. 1125-1140
Author(s):  
Andriy Lopushansky ◽  
Oleh Lopushansky ◽  
Anna Szpila
Keyword(s):  
Cauchy Problem ◽  
Time Derivative ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Sectorial Operator ◽  
Complex Interpolation ◽  
Bounded Domains ◽  
Initial Boundary Value Problems ◽  
Fractional Time Derivative ◽  
Initial Boundary Value

AbstractAn fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

Quantitative measurement of skin tissue response during laser irradiation in photorejuvenation

2009 ◽  
Vol 7 (6) ◽  
pp. 512-514 ◽  
Author(s):  
黄义梅 Wei Gong ◽  
龚玮 Yimei Huang ◽  
谢树森 Shusen Xie
Keyword(s):  
Laser Irradiation ◽  
Quantitative Measurement ◽  
Tissue Response ◽  
Skin Tissue

Two-dimensional thermo-mechanical fractional responses to biological tissue with rheological properties

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Magdy A. Ezzat ◽  
Roland W. Lewis
Keyword(s):  
Rheological Properties ◽  
Human Skin ◽  
Fractional Order ◽  
Order Parameters ◽  
Bioheat Transfer ◽  
Skin Tissue ◽  
Two Dimensional ◽  
Volume Relaxation ◽  
Content Type ◽  
Human Skin Tissue

Purpose The system of equations for fractional thermo-viscoelasticity is used to investigate two-dimensional bioheat transfer and heat-induced mechanical response in human skin tissue with rheological properties. Design/methodology/approach Laplace and Fourier’s transformations are used. The resulting formulation is applied to human skin tissue subjected to regional hyperthermia therapy for cancer treatment. The inversion process for Fourier and Laplace transforms is carried out using a numerical method based on Fourier series expansions. Findings Comparisons are made with the results anticipated through the coupled and generalized theories. The influences of volume materials properties and fractional order parameters for all the regarded fields are examined. The results indicate that volume relaxation parameters, as well as fractional order parameters, play a major role in all considered distributions. Originality/value Bio-thermo-mechanics includes bioheat transfer, biomechanics, burn injury and physiology. In clinical applications, knowledge of bio-thermo-mechanics in living tissues is very important. One can infer from the numerical results that, with a finite distance, the thermo-mechanical waves spread to skin tissue, removing the unrealistic predictions of the Pennes’ model.

scholarly journals New Iterative Method for Fractional Gas Dynamics and Coupled Burger’s Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mohamed S. Al-luhaibi
Keyword(s):  
Iterative Method ◽  
Analytical Solutions ◽  
Gas Dynamics ◽  
Time Derivative ◽  
Explicit Solutions ◽  
Fractional Partial Differential Equations ◽  
Approximate Analytical Solutions ◽  
Fractional Time Derivative ◽  
Burger's Equations ◽  
New Iterative Method

This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger’s equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterative method is easy to implement and accurate when applied to time-fractional partial differential equations.

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