scholarly journals Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat transfer problems

2017 ◽  
Vol 21 (3) ◽  
pp. 1161-1171 ◽  
Author(s):  
Xiao-Jun Yang
Keyword(s):  
Heat Transfer ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Complex Phenomenon ◽  
Anomalous Relaxation ◽  
Comparative Results ◽  
Transfer Problems ◽  
First Time ◽  
Derivatives Of ◽  
Relaxation Models

In this paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from mathematical view of point. The comparative results of the anomalous relaxation among the various fractional derivatives are also given. They are very efficient in description of the complex phenomenon arising in heat transfer.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

Full-Scale Bounds Estimation for the Nonlinear Transient Heat Transfer Problems with Interval Uncertainties

2021 ◽  
pp. 2150026
Author(s):  
Ruifei Peng ◽  
Haitian Yang ◽  
Yanni Xue
Keyword(s):  
Heat Transfer ◽  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Transient Heat Transfer ◽  
High Fidelity ◽  
Interval Scale ◽  
Nonlinear Transient ◽  
Transfer Problems ◽  
Derivatives Of

A package solution is presented for the full-scale bounds estimation of temperature in the nonlinear transient heat transfer problems with small or large uncertainties. When the interval scale is relatively small, an efficient Taylor series expansion-based bounds estimation of temperature is stressed on the acquirement of first and second-order derivatives of temperature with high fidelity. When the interval scale is relatively large, an optimization-based approach in conjunction with a dimension-adaptive sparse grid (DSG) surrogate is developed for the bounds estimation of temperature, and the heavy computational burden of repeated deterministic solutions of nonlinear transient heat transfer problems can be efficiently alleviated by the DSG surrogate. A temporally piecewise adaptive algorithm with high fidelity is employed to gain the deterministic solution of temperature, and is further developed for recursive adaptive computing of the first and second-order derivatives of temperature. Therefore, the implementation of Taylor series expansion and the construction of DSG surrogate are underpinned by a reliable numerical platform. The parallelization is utilized for the construction of DSG surrogate for further acceleration. The accuracy and efficiency of the proposed approaches are demonstrated by two numerical examples.

scholarly journals A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Aydin Secer
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Advantages And Disadvantages ◽  
Derivatives Of

The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

scholarly journals A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer

2019 ◽  
Vol 23 (3 Part A) ◽  
pp. 1677-1681 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Mahmoud Abdel-Aty ◽  
Carlo Cattani
Keyword(s):  
Heat Transfer ◽  
Fractional Order ◽  
Exponential Function ◽  
Power Law ◽  
Transfer Model ◽  
Mathematical Tool ◽  
Fractional Exponential Function ◽  
Anomalous Heat Transfer ◽  
First Time ◽  
Anomalous Heat

In this paper, we consider a general fractional-order derivataive of the Liouville-Caputo type with the non-singular kernel of the Rabotnov fractional-exponential function for the first time. A new general fractional-order derivataive heat transfer model is discussed in detail. The general fractional-order derivataive formula is a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

scholarly journals A new general fractional derivative Goldstein-Kac-type telegraph equation

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 3893-3898
Author(s):  
Ping Cui ◽  
Yi-Ying Feng ◽  
Jian-Gen Liu ◽  
Lu-Lu Geng
Keyword(s):  
Fractional Order ◽  
Telegraph Equation ◽  
Order Derivative ◽  
Mathematical Tool ◽  
Fractional Order Derivative ◽  
Derivative Formula ◽  
The Laplace Transform ◽  
The One ◽  
First Time ◽  
Derivatives Of

In this paper, we consider the Riemann-Liouville-type general fractional derivatives of the non-singular kernel of the one-parametric Lorenzo-Hartley function. A new general fractional-order-derivative Goldstein-Kac-type telegraph equation is proposed for the first time. The analytical solution of the considered model with the graphs is obtained with the aid of the Laplace transform. The general fractional-order-derivative formula is as a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

scholarly journals The integrating factor method for solving the steady heat transfer problems in fractal media

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 729-733
Author(s):  
Shan-Xiong Chen ◽  
Zhi-Hao Tang ◽  
Hai-Ning Wang
Keyword(s):  
Heat Transfer ◽  
Fractional Derivative ◽  
Integrating Factor ◽  
Factor Method ◽  
Fractal Media ◽  
Transfer Equations ◽  
Constant Coefficients ◽  
Transfer Problems ◽  
First Time ◽  
Steady Heat

In this paper, we propose the integrating factor method via local fractional derivative for the first time. We use the proposed method to handle the steady heat-transfer equations in fractal media with the constant coefficients. Finally, we discuss the non-differentiable behaviors of fractal heat-transfer problems.

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

Numerical solution of time-fractional three-species food chain model arising in the realm of mathematical ecology

2020 ◽  
Vol 13 (02) ◽  
pp. 2050011 ◽  
Author(s):  
Ved Prakash Dubey ◽  
Rajnesh Kumar ◽  
Devendra Kumar
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Derivative ◽  
Chain Model ◽  
Homotopy Analysis ◽  
Food Chain Model ◽  
Derivatives Of

This research paper implements the fractional homotopy analysis transform technique to compute the approximate analytical solution of the nonlinear three-species food chain model with time-fractional derivatives. The offered technique is a fantastic blend of homotopy analysis method (HAM) and Laplace transform (LT) operator and has been used fruitfully in the numerical computation of various fractional differential equations (FDEs). This paper involves the fractional derivatives of Caputo style. The numerical solutions of this selected fractional-order food chain model are evaluated by making use of the associated initial conditions. It is revealed by the adopting procedure that the more desirable estimation of the solution can be easily acquired through the calculation of some number of iteration terms only — a fact which authenticates the easiness and soundness of the suggested hybrid scheme. The variations of fractional order of time derivative on the solutions for different specific cases have been depicted through graphical presentations. The outcomes demonstrated through the graphs expound that the adopted scheme is very fantastic and accurate.

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