Abstract

In mathematical physics various direct and inverse problems are studied, where the first, second and third kind Volterra equations degenerate. Problems of this nature are encountered in moisture transfer problems, in the waves theory, in inverse problems of geophysics, etc. The paper researches the coefficient inverse problem in an unbounded domain in the waves theory, where the first kind Volterra equation degenerates, whose solution is related to the Dirac function. Research in this area has not yet been completed, which is the relevance of this work

Keywords

Inverse problem, ill-posedness of the first kind Volterra integral equations, unique solvability of the problem, auxiliary functions method, regularizability of the problem, regularization method in a generalized sense

References

  1. Hadamard, J. (1978). The Cauchy problem for linear partial differential equations of hyperbolic type. Moscow: Nauka.

  2. Aller, M. (1966). Effective potential of water during soil drying. In Thermodynamics of soil moisture (pp. 325–360). Leningrad: Gidrometeoizdat.

  3. Bukhgeim, A. L. (1983). Volterra equations and inverse problems. Novosibirsk: Nauka.

  4. Dmitriev, V. I. (1987). Inverse problems of electromagnetic methods in geophysics. In Ill-posed problems of natural sciences (pp. 54–76). Moscow: Moscow State University.

  5. Imanaliev, M. I., & Ved, Yu. A. (1989). On a first-order partial differential equation with an integral coefficient. Differential Equations, 23(3), 465–477.

  6. Lavrentiev, M. M., Romanov, V. G., & Shishatsky, S. P. (1980). Ill-posed problems of mathematical physics and analysis. Moscow: Nauka.

  7. Lavrentiev, M. M. (1977). Regularization of operator equations of Volterra type. In Problems of mathematical physics and computational mathematics (pp. 199–205). Moscow: Nauka.

  8. Newell, A. (1989). Solitons in mathematics and physics (Trans. from English). Moscow: Mir.

  9. Naumkin, P. I., & Shishmarev, I. A. (1992). Generalized solutions to the Whitham equation. Differential Equations, 28(1), 121–126.

  10. Omurov, T. D. (2003). Regularization methods for Volterra integral equations of the first and third kinds. Bishkek: Ilim.

  11. Omurov, T. D., Ryspaev, A. O., & Omurov, M. T. (2014). Inverse problems in applications of mathematical physics. Bishkek: KNU.

  12. Romanov, V. G. (1973). Inverse problems for differential equations. Novosibirsk: Novosibirsk State University.

  13. Tikhonov, A. N., & Arsenin, V. Y. (1986). Methods for solving ill-posed problems. Moscow: Nauka.

  14. Trenogin, V. A. (1980). Functional analysis. Moscow: Nauka.

  15. Tobias, T. (1984). On the inverse problem of determining the kernel of a hereditary medium. Proceedings of the Academy of Sciences of the Estonian SSR. Physics and Mathematics, (2), 182–187.

  16. Whitham, G. B. (1977). Linear and nonlinear waves. Moscow: Mir.