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5Methods of computational algebra and solving mathematical physics equations 39
A new variant of the discontinuous particle method is presented. We use a new particle rearrangement criterion without analyzing particle overlaps. It is assumed that the nonlinear elastic transport preserves not only the mass of the particles, but also the mass located between the centers of these particles. This require�ment leads to the fact that the change in the distance between the particles in the process of their movement and the conservation of mass in the space between them leads to a change in the density of one of theparti�cles. The new version applies to solving the one.dimensional and two.dimensional quasi.linear transport equa�tion problems. The main feature of the new variant is minimal smearing of discontinuities.
References
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Bogomolov S.V., Esikova N.B.,Kuvshinnikov A.E. Micro.macro Fokker�Planck�Kolmogorov models for a gas of rigid spheres // Math.Models Comput. Simul. 2016.V. 8, N. 5. P. 533.547.
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Bogomolov S.V., Kuvshinnikov A.E. Discontinuous particle method on gas dynamic examples // Math. Models Comput. Simul.2019. V. 11,N. 5. P. 768.777.
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Bogomolov S.V., Filippova M.A., Kuvshinnikov A.E. A discontinuous particle method for the inviscid Burgers� equation // J. Phys.: Conf. Ser. 2021. V. 1715.012066.
Ontheoptimalapproximationoffunctionsintheboundarylayer
I. V. Boikov, V. A. Ryazantsev
Penza State University Email: i.v.boykov@gmail.com
DOI 10.24412/cl.35065.2021.1.00.07
In this study we consider the problem of approximation of functions belonging to the class of functions with high gradients in boundary layer. For such functions we build an algorithm of approximation both in one.dimensional and multidimensional cases. The idea of the algorithm is based on the results for optimal approx�imations of specific functional classes [1.3]. These classes include functions with modules of derivatives having power.type singularity that is a function of distance from the point to the boundary of the domain.
In order to develop the proposed algorithm we introduce the specific functional class and use the appa�ratus of continuous local splines providing approximation of functions from the mentioned class that is optimal with respect to accuracy. Solving model problems demonstrate the efficiency of the proposed method.
References
1. Boikov I. V., Ryazantsev V. A. On the optimal approximation of geophysical field // Siberian J. Num. Math. 2021.
V. 24, N. 1. P. 17.34.
2.
Boykov I. V., Ryazantsev V. A. On a difference method of potential fields� extension // University proceedings. Volga region. Physical and mathematicalsciences.2014. N. 2 (30).P. 20.33.
3.
Boikov I.V. Approximation of some classes of functions by local splines // Computational Mathematics and Mathematical Physics. 1998. V. 38, N. 1. P. 21�29.
Ononeiterativemethodforsolvingtheamplitude.phaseproblem
I. V. Boikov, Ya.V. Zelina, D. I. Vasyunin
Penza State University Email: zelinayana@gmail.com
DOI 10.24412/cl.35065.2021.1.00.08
Methods for solving amplitude and phase problems for one and two.dimensional discrete signals are pro�posed. Methods are based on using nonlinear singular integral equations. In the one.dimensional case ampli�
