Ob ustoychivom priblizhennom reshenii odnoy nekorrektno postavlennoy kraevoy zadachi dlya metagarmonicheskogo uravneniya

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Abstract

In this paper, we consider a mixed problem for a metaharmonic equation in a region in a rectangular cylinder. On the side faces cylinder region is set to homogeneous conditions of the first kind. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i. e. the function and its normal derivative are set. The other boundary of the cylindrical region, which is flat, is free. This problem is illposed, and to construct its approximate solution in the case of Cauchy data known with some error, it is necessary to use regularizing algorithms. In this paper, the problem is reduced to the Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained. A stable solution of the integral equation is obtained by the method of Tikhonov regularization. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem as a whole is constructed. The convergence theorem of the approximate solution of the problem to the exact one is given when the error in the Cauchy data tends to zero and when the regularization parameter is agreed with the error in the data. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

About the authors

Evgeniy B. Laneev

RUDN University

Email: elaneev@yandex.ru
Doctor of Physics and Mathematics, Professor of the Mathematical Department 6 Miklukho-Maklay St., Moscow 117198, Russian Federation

Polina A. Lesik

RUDN University

Email: polinalesik@yandex.ru
Student, Mathematical Department 6 Miklukho-Maklay St., Moscow 117198, Russian Federation

Aleksandr V. Klimishin

RUDN University

Email: sa-sha-02@yandex.ru
Student, Mathematical Department 6 Miklukho-Maklay St., Moscow 117198, Russian Federation

Aleksandr M. Kotyukov

RUDN University

Email: amkotyukov@mail.ru
Post-Graduate Student, Mathematical Department 6 Miklukho-Maklay St., Moscow 117198, Russian Federation

Andrey A. Romanov

RUDN University

Email: an1romanov@gmail.com
Student, Mathematical Department 6 Miklukho-Maklay St., Moscow 117198, Russian Federation

Anna G. Khegai

RUDN University

Email: annh98@icloud.com
Student, Mathematical Department 6 Miklukho-Maklay St., Moscow 117198, Russian Federation

References

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  2. Е.Б. Ланеев, Б. Васудеван, “Об устойчивом решении одной смешанной задачи для уравнения Лапласа”, Вестник РУДН. Серия Прикладная математика и информатика, 1999, №1, 128-133.
  3. Е.Б. Ланеев, “О построении функции Карлемана на основе метода регуляризации Тихонова в некорректно поставленной задаче для уравнения Лапласа”, Дифференциальные уравнения, 54:4 (2018), 483-491.
  4. А.Н. Тихонов, В.Я. Арсенин, Методы решения некорректных задач, Наука, М., 1979.
  5. А.Н. Тихонов, В.Б. Гласко, О.К. Литвиненко, В.Р. Мелихов, “О продолжении потенциала в сторону возмущающих масс на основе метода регуляризации”, Изв. АН СССР. Физика Земли, 1968, №1, 30-48.
  6. Е.Б. Ланеев, М.Н. Муратов, “Об одной обратной задаче к краевой задаче для уравнения Лапласа с условием третьего рода на неточно заданной границе.”, Вестник РУДН. Серия Математика, 10:1 (2003), 100-110.
  7. Г.Р. Иваницкий, “Тепловидение в медицине”, Вестник РАН, 76:1 (2006), 48-58.

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