About one method of approximate solution of the first boundary value problem for the fractional diffusion equation, used in gas dynamics
Аuthors
*, **Moscow State University of Civil Engineering, MSUCE, 26, Yaroslavskoe shosse, Moscow, 129337, Russia
*e-mail: kroshvanya@yandex.ru
**e-mail: aleroev@mail.ru
Abstract
This paper is devoted to the development of fundamentally new analytical and approximate methods for studying mathematical models of advection-diffusion using fractional calculus. Obtained results can have applications in various fields of applied mathematics and engineering. In particular, in the design of modern rocket engines (supersonic jet engines). The fractional operators for this problem are considered in the Caputo sense. The problem is solved by the method of separation of variables (Fourier method). In the first section of the paper the theoretical aspects of the problem are presented. An important part of this section is the construction of the basis of the systems of eigenfunctions and adjoint functions of the problem. In this question we strongly rely on the work of Dzhrbashyan and Nersesyan. Taking into account the biorthogonality of such systems and the fact that the adjoint functions are finite in number, for further solution we can consider only the problem in which the adjoint functions are not generated. Passing to the solution of the problem, we consider the system of eigenfunctions of the biorthogonal problem, since the basis of the eigenfunctions of the problem is not orthogonal in .png){kind=small}
. In order to determine the unknown coefficients of the Fourier series, due to the biorthogonality of the systems of functions, we use the scalar product of the corresponding functions. The eigenvalues of the problem are found as zeros of the Mittag-Leffler function. Thus, to solve the problem, we first find several eigenvalues. We construct eigenfunctions and functions of the conjugate problem. Since for an approximate solution of the problem it is sufficient to take only the first few terms of the series, we can construct the solution surfaces by considering only the partial sum of the obtained solution.
Keywords:
approximate calculations, fractional calculus, fractional advection-diffusion equation, fractional Caputo derivative, eigenvalue, eigenfunction, Mittag-Leffler functionReferences
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