Fractional Differential Analog of Biparabolic Evolution Equation and Some Its Applications
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FRACTIONAL DIFFERENTIAL ANALOG OF BIPARABOLIC EVOLUTION EQUATION AND SOME ITS APPLICATIONS
UDC 517.9:519.6
V. M. Bulavatsky
Abstract. The author analyzes fractional the differential analog of the well-known biparabolic evolution equation intended to describe the dynamics of heat and mass transfer processes that are non-equilibrium in time. Closed solution of some problems, in particular, the problem of Cauchy type and boundary-value problem for a finite interval, are obtained. A new (fractional differential) mathematical model is proposed to describe the non-equilibrium dynamics of filtration processes in fissured porous media. Keywords: biparabolic evolution equation, fractional–differential analog, fundamental solution, one-dimensional Cauchy-type problem, boundary-value problem on a finite interval, mathematical modeling of the fractional–differential dynamics of filtration processes, nonclassical models. INTRODUCTION The classical mathematical theory of heat conduction is known [1–3] to be based on the linear parabolic equation
~ æ¶ ö L1 u º ç - k 2 D ÷ u( x, t ) = 0 , è ¶t ø
(1)
where u is temperature, x Î E n , D is the Laplace operator, and k > 0 is a physical constant. This theory postulates tight constraints for processes such as infinite velocity of perturbations propagation and linear dependence of flow on field gradient as well as of energy on temperature. Violation of these conditions does not allow correct description, within the limits of this model, of the dynamics of heat and mass transfer processes and leads to some well-known paradoxes [2–4]. To describe processes of finite velocity, some authors, in particular [3, 4], propose hyperbolic equation, which takes into account relaxation of heat flow. As is mentioned in [5], replacement of Eq. (1) with a hyperbolic equation is crucial, but can hardly be explained from the group-theoretic point of view since all the nonstationary equations that include second derivatives with respect to time are not invariant with respect to Galilei transforms. Hyperbolic heat conduction equation has no respective symmetric properties. This means that it does not map the main physical conservation laws [5]. In this connection, the studies [5, 6] specify one natural generalization of Eq. (1)
~ ~ ~ ~ ~~ L u º a 1 L1 u + a 2 L2 u = 0, L2 = L1 L1 ,
(2)
where a 1 and a 2 are real parameters. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2016, pp. 89–100. Original article submitted February 16, 2016. 1060-0396/16/5205-0737 ©2016 Springer Science+Business Media New York
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The fourth-order partial differential equation (2) is invariant [6] with respect to the Galilei group G(1, 3); therefore, we assume that it can be used to describe diffusion processes independent from the inertial systems where they are observed [5, 6]. Equation (2) is called biparabolic [5]; for a 1 = 1 and a 2 = 0 , it coincides with the classical Fouri
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