Abstract
In the paper, a fundamental solution of the fractional dual-phase-lagging heat conduction problem is obtained. The considerations concern the 1D Cauchy problem in a whole-space domain. A solution of the initial-boundary problem is determined by using the Fourier–Laplace transform technique. The final form of solution is given in a form of a series. One of the properties of the derived fundamental solution of the considered problem with the initial condition expressed be the Dirac delta function is that it is symmetrical. The effect of the time-fractional order of the Caputo derivatives and the phase-lag parameters on the temperature distribution is investigated numerically by using the method which is based on the Fourier-series quadrature-type approximation to the Bromwich contour integral.
Highlights
In recent years, we have observed a significant increase in interest in fractional calculus, which is used in engineering and in other sciences such as biology or economics
The fractional model of heat conduction has been presented. This model can be regarded as a generalization of the classical DPLM in which two additional parameters αq ∈
The time-fractional differential equation was solved by using the Fourier–Laplace transform technique
Summary
We have observed a significant increase in interest in fractional calculus, which is used in engineering and in other sciences such as biology or economics. We use fractional calculus primarily for the mathematical modelling of physical phenomena such as, for example, heat conduction [5,6]. The starting point for considerations of many authors is the classical theory of heat transfer based on Fourier’s law This model has a certain non-physical property, i.e., the speed of heat is infinite. Many examples of the application of this method to the problem of heat conduction can be found in books [12,13] The advantage of this method is that the Green’s function always exists, but in the case of a more complex domain, it cannot always be written explicitly. We presented a fundamental solution of the fractional dual-phaselag equation of heat transfer with appropriate initial-boundary conditions. In the final part of the paper, several examples concerning the analysis of temperature distribution are presented
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