Some Consolidation Dynamics Problems within the Framework of the Biparabolic Mathematical Model and its Fractional-Diffe
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SOME CONSOLIDATION DYNAMICS PROBLEMS WITHIN THE FRAMEWORK OF THE BIPARABOLIC MATHEMATICAL MODEL AND ITS FRACTIONALDIFFERENTIAL ANALOG V. M. Bulavatsky1† and V. O. Bohaienko1‡
UDC 517.9:519.6
Abstract. The paper deals with mathematical modeling of dynamic processes of filtration consolidation in saturated geoporous media within the framework of non-classical mathematical models based on biparabolic evolution equation and its fractional-differential analog. We state and obtain regularized solutions of inverse retrospective problems of consolidation theory according to the above-mentioned models; obtain the convergence estimates for the found regularized solutions; and present the results of numerical experiments. Keywords: mathematical modeling, non-classical models, filtration-consolidation processes, dynamics, inverse problems, biparabolic evolution equation, fractional-differential analog. INTRODUCTION It is well-known [1] that the classical mathematical model of process dynamics of filtration consolidation of hyper-porous media is constructed based on the equation of parabolic type
æ¶ ¶2 Lu º ç - C u ç ¶t ¶x 2 è
ö ÷ u( x, t ) = 0 , ÷ ø
(1)
where u = p / g is excess pressure, p is pore pressure, g is specific weight of the liquid, and C u is the consolidation coefficient. While Eq. (1) is the main model equation of the filtration consolidation theory, it imposes rather rigid constrains such as infinite velocity of the propagation of perturbations and velocity linear dependence of the velocity on the pressure gradient. In case of breaching these conditions, the above-mentioned equation describes transfer processes inadequately and leads to a number of known paradoxes [2, 3]. A more generalized equation applicable to the consolidation processes of water-saturated hyper-porous massif taking into consideration skeleton creep is obtained in [4] and takes the form
a
¶ 2u ¶t 2
+b
¶ ö ¶ 2u ¶u æ + f ( x, t ) , = Cuç g + ÷ ¶t ø ¶x 2 ¶t è
(2)
where a, b, g , and C u are the known [4, 5] constants and f is the specified source function. Note that model equations of the form (1), (2), which are used to desñribe the corresponding consolidation processes, are hard to explain from the group-theoretical point, as they have no necessary symmetry properties [6]. In this connection, a mathematical model based on the biparabolic evolution equation 1
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2020, pp. 100–114. Original article submitted October 21, 2019.
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770
1060-0396/20/5605-0770 ©2020 Springer Science+Business Media, LLC
Lu( x, t ) + t r L2 u( x, t ) = f ( x, t ) ,
(3)
where t r is a real parameter and L2 = LL , has been proposed in [7] for more adequate description of the transfer processes. As noted in [7], this equation is invariant to the Galilean group G(1, 3) , and can therefore be used to describe heat and diffusion processes, which do not depend
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