scholarly journals Time-Fractional Heat Conduction in Two Joint Half-Planes

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 800 ◽  
Author(s):  
Yuriy Povstenko ◽  
Joanna Klekot
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Fourier Transforms ◽  
Graphical Representation ◽  
Thermal Contact ◽  
The Cauchy Problem ◽  
The Laplace Transform ◽  
Fourier And Laplace Transforms ◽  
Perfect Thermal Contact ◽  
Fractional Heat Conduction

The heat conduction equations with Caputo fractional derivative are considered in two joint half-planes under the conditions of perfect thermal contact. The fundamental solution to the Cauchy problem as well as the fundamental solution to the source problem are examined. The Fourier and Laplace transforms are employed. The Fourier transforms are inverted analytically, whereas the Laplace transform is inverted numerically using the Gaver–Stehfest method. We give a graphical representation of the numerical results.

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.

Effect of a Heat-Conducting Well Casing on Temperature Distribution in an Observation Well

1979 ◽  
Vol 101 (1) ◽  
pp. 20-27
Author(s):  
P. J. Closmann ◽  
E. R. Jones ◽  
E. A. Vogel
Keyword(s):  
Heat Conduction ◽  
Temperature Distribution ◽  
Heat Source ◽  
Constant Temperature ◽  
Thermal Contact ◽  
Heat Conducting ◽  
Formation Temperature ◽  
Maximum Difference ◽  
Observation Well ◽  
Perfect Thermal Contact

The effect of heat conduction on temperature along the wall of a well casing has been determined by solution of the equations of heat conduction. The casing was assumed to pass vertically through a planar heat source of constant temperature. The casing and formation were assumed to be in perfect thermal contact. Numerical results were obtained for two sizes of steel casing and one size of aluminum casing. At any given distance from the heat source, the casing temperature differs most at early times from the formation temperature computed in the absence of casing. This difference decreases rapidly with time. Furthermore, the maximum difference occurs at greater distances from the heat source as time increases. In general, after about three months of heating, errors in measured temperatures due to conduction along the casing wall are negligible.

scholarly journals The inversion of a transform related to the Laplace transform and to heat conduction

1964 ◽  
Vol 4 (1) ◽  
pp. 1-14 ◽  
Author(s):  
David V. Widder
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Laplace Transform ◽  
Physical Interpretation ◽  
Integral Transform ◽  
The Laplace Transform ◽  
Image Position

In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

scholarly journals Heat Conduction Through a Flat Punch

1981 ◽  
Vol 48 (4) ◽  
pp. 871-875 ◽  
Author(s):  
Maria Comninou ◽  
J. R. Barber ◽  
John Dundurs
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Half Plane ◽  
Contact Region ◽  
Thermal Contact ◽  
Total Heat Flux ◽  
Flat Punch ◽  
Imperfect Contact ◽  
Perfect Contact ◽  
Perfect Thermal Contact

An elastic half plane is indented by a perfectly conducting rigid flat punch, which is maintained at a different temperature from the half plane. It is found that, depending on the magnitude and direction of the total heat flux, one of the following states occurs: separation at the punch corners, perfect thermal contact throughout the punch face, or an imperfect contact region at the center with adjacent perfect contact regions.

scholarly journals Time-Fractional Fourier Law in a finite hollow cylinder under Gaussian-distributed heat flux

2018 ◽  
Vol 180 ◽  
pp. 02008 ◽  
Author(s):  
Slawomir Blasiak
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Hollow Cylinder ◽  
Integral Transformation ◽  
Temperature Fields ◽  
Radial Coordinate ◽  
Finite Hollow Cylinder ◽  
The Laplace Transform ◽  
Transformation Methods ◽  
Fractional Heat Conduction

This paper presents the solution of the theoretical model of heat conduction based on timefractional Fourier equation for a finite hollow cylinder treated with heat flux on one of the front surfaces. A derivative of fractional order in the Caputo sense was applied to record the temperature derivative in time. The distributions of temperature fields in the hollow cylinder were determined with the use of Fourier-Bessel series, as surface functions of two variables (r, θ) . The distributions of temperature fields were determined using analytical methods and applying integral transformation methods. The Laplace transform with reference to time, the Fourier finite cosine transform with reference to axial coordinate z and Marchi-Zgrablich transform for radial coordinate r. The fractional heat conduction equation was analysed for 0 < α ≤ 2

scholarly journals Heat conduction in a composite sphere - the effect of fractional derivative order on temperature distribution

2018 ◽  
Vol 157 ◽  
pp. 08008
Author(s):  
Urszula Siedlecka ◽  
Stanisław Kukla
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Laplace Transform ◽  
Time Derivative ◽  
Spherical Layer ◽  
Solid Sphere ◽  
Liouville Derivative ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

The aim of the contribution is an analysis of time-fractional heat conduction in a sphere with an inner heat source. The object of the consideration is a solid sphere with a spherical layer. The heat conduction in the solid sphere and spherical layer is governed by fractional heat conduction equation with a Caputo time-derivative. Mathematical (classical) or physical formulations of the Robin boundary condition and the perfect contact of the solid sphere and spherical layer is assumed. The boundary condition and the heat flux continuity condition at the interface are expressed by the Riemann-Liouville derivative. An exact solution of the problem under mathematical conditions is determined. A solution of the problem under physical boundary and continuity conditions using the Laplace transform method has been obtained. The inverse of the Laplace transform by using the Talbot method are numerically determined. Numerical results show the effect of the order of the Caputo and the Riemann-Liouville derivatives on the temperature distribution in the sphere.

Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Yuriy Povstenko
Keyword(s):  
Heat Conduction ◽  
Contact Point ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Heat Fluxes ◽  
Half Line ◽  
Cauchy Problems ◽  
The Laplace Transform ◽  
Mainardi Function ◽  
Fractional Heat Conduction

AbstractHeat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.

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