Thermoelastic Interaction in an Infinite Long Hollow Cylinder with Fractional Heat Conduction Equation

2017 ◽  
Vol 9 (2) ◽  
pp. 378-392
Author(s):  
Ahmed. E. Abouelregal
Keyword(s):  
Heat Conduction ◽  
Laplace Transform ◽  
Hollow Cylinder ◽  
Heat Conduction Equation ◽  
Thermal Stresses ◽  
Generalized Thermoelasticity ◽  
Constant Heat Flux ◽  
Conduction Equation ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

AbstractIn this work, we introduce a mathematical model for the theory of generalized thermoelasticity with fractional heat conduction equation. The presented model will be applied to an infinitely long hollow cylinder whose inner surface is traction free and subjected to a thermal and mechanical shocks, while the external surface is traction free and subjected to a constant heat flux. Some theories of thermoelasticity can extracted as limited cases from our model. Laplace transform methods are utilized to solve the problem and the inverse of the Laplace transform is done numerically using the Fourier expansion techniques. The results for the temperature, the thermal stresses and the displacement components are illustrated graphically for various values of fractional order parameter. Moreover, some particular cases of interest have also been discussed.

Thermoelasticity of thin shells based on the time-fractional heat conduction equation

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Yuriy Povstenko
Keyword(s):  
Heat Conduction ◽  
Heat Exchange ◽  
Fractional Derivative ◽  
Heat Conduction Equation ◽  
Generalized Thermoelasticity ◽  
Convective Heat Exchange ◽  
Conduction Equation ◽  
Thin Shells ◽  
Median Surface ◽  
Fractional Heat Conduction

AbstractThe time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 < α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.

scholarly journals Time-Fractional Heat Conduction in a Half-Line Domain due to Boundary Value of Temperature Varying Harmonically in Time

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Yuriy Povstenko
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Heat Conduction ◽  
Dirichlet Problem ◽  
Heat Conduction Equation ◽  
Boundary Value ◽  
Conduction Equation ◽  
Half Line ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

The Dirichlet problem for the time-fractional heat conduction equation in a half-line domain is studied with the boundary value of temperature varying harmonically in time. The Caputo fractional derivative is employed. The Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate are used. Different formulations of the considered problem for the classical heat conduction equation and for the wave equation describing ballistic heat conduction are discussed.

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.

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