scholarly journals Time-Fractional Heat Conduction in a Plane with Two External Half-Infinite Line Slits under Heat Flux Loading

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 689 ◽  
Author(s):  
Yuriy Povstenko ◽  
Tamara Kyrylych
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Graphical Representation ◽  
Integral Transform ◽  
Conduction Equation ◽  
Infinite Plane ◽  
Conservation Of Energy ◽  
Infinite Line ◽  
Fractional Heat Conduction

The time-fractional heat conduction equation follows from the law of conservation of energy and the corresponding time-nonlocal extension of the Fourier law with the “long-tail” power kernel. The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with two external half-infinite slits with the prescribed heat flux across their surfaces. The integral transform technique is used. The solution is obtained in the form of integrals with integrand being the Mittag–Leffler function. A graphical representation of numerical results is given.

Fractional thermoelasticity problem for an infinite solid with a penny-shaped crack under prescribed heat flux across its surfaces

2020 ◽  
Vol 378 (2172) ◽  
pp. 20190289 ◽  
Author(s):  
Y. Povstenko ◽  
T. Kyrylych
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Stress Field ◽  
Fractional Calculus ◽  
Heat Conduction Equation ◽  
Thermoelasticity Problem ◽  
Conduction Equation ◽  
Penny Shaped Crack ◽  
Shaped Crack ◽  
Fractional Heat Conduction

The time-nonlocal generalization of the Fourier law with the ‘long-tail’ power kernel can be interpreted in terms of fractional calculus and leads to the time-fractional heat conduction equation with the Caputo derivative. The theory of thermal stresses based on this equation was proposed by the first author ( J. Therm. Stresses 28 , 83–102, 2005 ( doi:10.1080/014957390523741 )). In the present paper, the fractional heat conduction equation is solved for an infinite solid with a penny-shaped crack in the case of axial symmetry under the prescribed heat flux loading at its surfaces. The Laplace, Hankel and cos-Fourier integral transforms are used. The solution for temperature is obtained in the form of integral with integrands being the generalized Mittag-Leffler function in two parameters. The associated thermoelasticity problem is solved using the displacement potential and Love’s biharmonic function. To calculate the additional stress field which allows satisfying the boundary conditions at the crack surfaces, the dual integral equation is solved. The thermal stress field is calculated, and the stress intensity factor is presented for different values of the order of the Caputo time-fractional derivative. A graphical representation of numerical results is given. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Yuriy Povstenko
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Linear Combination ◽  
Heat Conduction Equation ◽  
Integral Transform ◽  
Convective Heat Exchange ◽  
Robin Boundary Condition ◽  
Conduction Equation ◽  
Robin Boundary ◽  
Fractional Heat Conduction

AbstractThe central symmetric time-fractional heat conduction equation with Caputo derivative of order 0 < α ≤ 2 is considered in a ball under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of values of temperature and values of its normal derivative at the boundary, and the physical condition with the prescribed linear combination of values of temperature and values of the heat flux at the boundary, which is a consequence of Newton’s law of convective heat exchange between a body and the environment. The integral transform technique is used. Numerical results are illustrated graphically.

Mesoscopic Heat Conduction and Onset of Periodicity

2003 ◽  
Author(s):  
Kal Renganathan Sharma
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Time Domain ◽  
Heat Conduction Equation ◽  
Higher Order ◽  
Heat Propagation ◽  
Conduction Equation ◽  
Transient Temperature ◽  
Higher Order Terms ◽  
Fourier Heat Conduction

Mesoscopic approach deals with study that considers temporal fluctuations which is often averaged out in a macroscopic approach without going into the molecular or microscopic approach. Transient heat conduction cannot be fully described by Fourier representation. The non-Fourier effects or finite speed of heat propagation effect is accounted for by some investigators using the Cattaneo and Vernotte non-Fourier heat conduction equation: q=−k∂T/∂x−τr∂q/∂t(1) A generalized expression to account for the non-Fourier or thermal inertia effects suggested by Sharma (5) as: q=−k∂T/∂x−τr∂q/∂t−τr2/2!∂2q/∂t2−τr3/3!∂3q/∂t3−…(2) This was obtained by a Taylor series expansion in time domain. Manifestation of higher order terms in the modified Fourier’w law as periodicity in the time domain is considered in this study. When a CWT is maintained at one end of a medium of length L where L is the distance from the isothermal wall beyond which there is no appreciable temperature change from the initial condition during the duration of the study the transient temperature profile is obtained by the method of Laplace transforms. The space averaged heat flux is obtained and upon inversion from Laplace domain found to be a constant for the the case obeying Fourier’s law; 1 − exp(−τ) using the Cattaneo and Vernotte non-Fourier heat conduction equation, and upon introduction of the second derivative in time of the heat flux the expression becomes, 1 − exp(−τ)(Sin(τ) + Cos(τ)). Thus the periodicity in time domain is lost when the higher order terms in the generalized Fourier expression is neglected.

Time-fractional heat conduction in a two-layer composite slab

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Yuriy Povstenko
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Composite Body ◽  
Composite Slab ◽  
Layer Composite ◽  
Fractional Heat Conduction

AbstractThe heat conduction equation is considered in a composite body consisting of two regions: 0 <

Reconstruction of the Robin boundary condition and order of derivative in time fractional heat conduction equation

2018 ◽  
Vol 13 (1) ◽  
pp. 5 ◽  
Author(s):  
Rafał Brociek ◽  
Damian Słota
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Fractional Order ◽  
Heat Conduction Equation ◽  
Robin Boundary Condition ◽  
Conduction Equation ◽  
Additional Information ◽  
Bee Colony ◽  
Robin Boundary ◽  
Fractional Heat Conduction

This paper describes an algorithm for reconstruction the boundary condition and order of derivative for the heat conduction equation of fractional order. This fractional order derivative was applied to time variable and was defined as the Caputo derivative. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution an Artificial Bee Colony (ABC) algorithm and Nelder-Mead method were used. In order to stabilize the procedure the Tikhonov regularization was applied. The paper presents examples to illustrate the accuracy and stability of the presented algorithm.

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