Linear Fractional Diffusion-Wave Equation for Scientists and Engineers

Povstenko, Yuriy

doi:10.1007/978-3-319-17954-4

Cited by 216 publications

(155 citation statements)

References 108 publications

(184 reference statements)

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“…where , α Moreover, the conditions of the perfect contact of the layers are satisfied [3] Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab 107…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…These conditions in the theory of fractional heat transfer are also called mathematical Robin conditions [3]. In this case the condition (6) of equality of the heat fluxes at the interface assumes also a classical form.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The fractional heat equation includes a fractional derivative with respect to space and/or time variable. The problems of fractional heat conduction are the subject of numerous works, for instance references [3][4][5][6][7]. In book [3] by Povstenko the equations obtained by generalizations of the time-nonlocal Fourier's, Fick's and Darcy's laws are discussed.…”

Section: Introductionmentioning

confidence: 99%

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Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab

Kukla¹,

Siedlecka²

2015

J. Appl. Math. Comput. Mech.

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Abstract. In this paper the Laplace transformation for solving the problem of fractional heat conduction in a two-layered slab has been applied. The different orders of Caputo derivative in the time-fractional equation governed the heat transfer in the layers are assumed. The inverse Laplace transform by using a numerical method is determined. The numerical results obtained by using of the eigenfunctions method and by numerically inverting the Laplace transform are compared.

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Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab

Kukla¹,

Siedlecka²

2015

J. Appl. Math. Comput. Mech.

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“…The fractional differential equation governing the heat equation includes the fractional derivatives with respect to time and/or space variables. Properties of the fractional derivatives and methods to the solution of the fractional differential equations are presented in books [4][5][6]. A method of solution of a time-fractional heat conduction equation in a solid sphere has been discussed by Ning and Jiang in paper [7].…”

Section: Introductionmentioning

confidence: 99%

Fractional heat conduction in multilayer spherical bodies

Kukla

Siedlecka

2016

J. Appl. Math. Comput. Mech.

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Abstract. In this paper an analytical solution of the time-fractional heat conduction problem in a spherical coordinate system is presented. The considerations deal the two--dimensional problem in multilayer spherical bodies including a hollow sphere, hemisphere and spherical wedge. The mathematical Robin conditions are assumed. The solution is a sum of time-dependent function satisfied homogenous boundary conditions and of a solution of the steady-state problem. Numerical example shows the temperature distributions in the hemisphere for various order of time-derivative.

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“…Moreover, the boundary conditions may also include the fractional derivatives. The fundamentals of fractional calculus and of the theory of fractional differential equations are given in [11][12][13][14][15]. Some applications of fractional order calculus to modelling of real-world phenomena are presented in [16][17][18].…”

mentioning

confidence: 99%

An analytical solution to the problem of time-fractional heat conduction in a composite sphere

Kukla

Siedlecka

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative. analytical solution to the problem for thermal and mechanical properties of the sphere was obtained in the form of power functions of the radial direction. Thermal stresses in a sphere of a functionally graded material were also considered by Pawar et al. [9]. Transient temperature distribution was determined by assuming that the material properties of the sphere were exponential functions of the radial direction. The parabolic differential equation of heat conduction, derived under the framework of the classical theory of heat conduction is based on the local Fourier law. Non-local generalizations of the Fourier law lead to non-classical theories, in which the parabolic equation is replaced by a time-fractional and/or space-fractional heat conduction differential equation [10]. In these fractional differential equations, different kinds of derivatives of fractional order are used (the Riemann-Liouville derivative, the Caputo derivative and the Grűnwald-Letnikov derivative). Moreover, the boundary conditions may also include the fractional derivatives. The fundamentals of fractional calculus and of the theory of fractional differential equations are given in [11][12][13][14][15]. Some applications of fractional order calculus to modelling of real-world phenomena are presented in [16][17][18].Heat conduction problems formulated under the framework of the non-classical theories in the spherical coordinates with fractional Caputo or Riemann-Liouville derivatives were studied in [19,20]. An approximate analytical solution of time-fractional heat conduction in a composite medium consisting of an infinite matrix and a spherical inclusion is presented by Povstenko in [19]. The perfect thermal contact was realized by the conditions of equality of temperatures and heat fluxes at the boundary surfaces, wherein the heat fluxes are expressed by a Riemann-Liouville fractional derivative. An analytical solution to the problem of time-fractional heat conduction in a multilayered slab was presented by Siedlecka and Kukla in [20]. Ning and Jiang [21] use the Laplace transform and the method of variable separation to determine an analytical

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Linear Fractional Diffusion-Wave Equation for Scientists and Engineers

Cited by 216 publications

References 108 publications

Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab

Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab

Fractional heat conduction in multilayer spherical bodies

An analytical solution to the problem of time-fractional heat conduction in a composite sphere

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