Discrete inverse problem for a hyperbolic equation, properties of the solution to a discrete direct and auxiliary discrete problems
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Keywords:
hyperbolic equation, discrete direct and inverse problem, Cauchy problem.Abstract
This paper considers the formulation of a discrete inverse problem for a
hyperbolic equation. First, the continuous inverse problem is reduced to a convenient form for
research. In the inverse problem, the required function is considered even. Since the Dirac delta
function is present in the problem data, the structure of a generalized solution to the Cauchy
problem for a hyperbolic equation is determined. The solution to the Cauchy problem for a
hyperbolic equation is determined only for positive values in time, therefore the solution to the
Cauchy problem for negative values in time is determined using odd continuation. After some
transformations, the formulation of the continuous inverse problem is reduced to a form convenient
for research. A grid domain is introduced, and for all functions in the problem statement the
corresponding grid functions and a discrete analogue of the Dirac delta function are determined.
Differential operators, initial conditions and additional data of the inverse problem are
approximated by finite differences. Assuming that a solution to the discrete inverse problem exists,
we prove the data lemma of the discrete inverse problem. In order to study the discrete inverse
problem for a hyperbolic equation, a theorem on the existence and uniqueness of the discrete direct
problem is proved, as well as on the properties of the solution to this discrete problem. In the course
of proving the theorem, a discrete analogue of d'Alembert's formula for solving the Cauchy problem
for a hyperbolic equation was obtained. The theorem on the existence of a unique solution to the
auxiliary discrete problem and its properties is proved.
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Kabanikhin S. I., Novikov N. S., Shishlenin M. A. Gelfand-Levitan-Krein method in onedimensional elasticity inverse problem //Journal of Physics: Conference Series. – IOP Publishing,
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Gelfand–Levitan–Marchenko–Krein Equations //Mathematics. – 2023. – Т. 11. – №. 21. – p. 4458.
S.Kabanikhin, M.Shishlenin, G.Bakanov. Multidimensional analogue of Krein equation for the
inverse acoustic problem // Abstracts of the VII World Congress of Turkic World Mathematicians
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regularization algorithms // Journal of Physics: Conference Series. – IOP Publishing, 2021. – Т.
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