Geodesic
Fundamental solution of an operator and its application for the approximate solution of initial-boundary-value problems
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 4, Tome 193 (2021), pp. 110-121.

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In this paper, we construct an approximation of the fundamental solution of a problem for a hyperbolic system of first-order linear differential equations with constant coefficients. We propose an algorithm for the approximate solution of the generalized Riemann problem on the discontinuity of a decay under additional conditions on the boundaries. This algorithm reduces the problem of finding values of variables on both sides of the discontinuity surface of the initial data to solving a system of algebraic equations whose right-hand sides depend on the values of the variables at the initial moment of time at a finite number of points. Based on these solutions, we develop a computational algorithm for the approximate solution of the initial-boundary-value problem for a hyperbolic system of first-order linear differential equations. The algorithm is implemented for a system of equations of elastic dynamics; moreover, we use it to solve some applied problems related to oil production.
Keywords: decay of a discontinuity, hyperbolic system, generalized function, Cauchy problem, matrix Green function, characteristic, equations of elastic dynamics.
Mots-clés : conjugation conditions, Riemann invariant 
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Yu. I. Skalko; S. Yu. Gridnev. Fundamental solution of an operator and its application for the approximate solution of initial-boundary-value problems. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 4, Tome 193 (2021), pp. 110-121. http://geodesic.mathdoc.fr/item/INTO_2021_193_a11/

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