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On a New Analog of the Biparabolic Evolution Equation with Conformable Fractional Derivatives
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ABSTRAKT
Due to the fact that the classical Fourier mathematical model of heat and mass transfer does not allow, in some cases, obtaining a correct enough description of the dynamics of the process and leads to several known paradoxes, V.I. Fushchych and his students proposed in their works a model based on the biparabolic evolution equation, which was further repeatedly used to model various thermal and diffusion processes in natural science, in particular, when modeling the dynamics of deformable water-saturated geoporous media. Nowadays, an intensively developing theory of anomalous transfer processes, created with the use of ideas of fractional order integral-differentiation, is of increasing interest. So, for example, the work Bulavatsky V.M. Fractional differential analog of biparabolic evolution equation and some of its applications (Cybernetics and Systems Analysis, 2016, 52, No. 5, 737-747) proposes a fractional differential analog of the biparabolic evolution equation (based on the Caputo-Gerasimov fractional derivatives) designed to simulate the dynamics of nonlocal in time transfer processes, and within this approach, constructs a nonclassical mathematical model to describe anomalous dynamics of filtration processes in fractured porous formations. In this paper, a new analog of the biparabolic evolution equation based on conformable fractional derivatives is proposed and some boundary value problems are solved. In particular, a solution is found to a boundary value problem on a finite interval for the analog of the biparabolic evolution equation with conformable fractional derivatives; a problem with nonlocal boundary conditions is posed and solved; an inverse retrospective problem of the restoration of the initial field function from its given final value is considered. Some estimates of the convergence rate of the regularized solution of the inverse retrospective problem are obtained and the results of numerical experiments are presented.
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