Vitaly Kamynin - Academia.edu (original) (raw)
Papers by Vitaly Kamynin
Matematicheskie Zametki
Доказана однозначная разрешимость обратной задачи определения функции источника в неравномерно па... more Доказана однозначная разрешимость обратной задачи определения функции источника в неравномерно параболическом уравнении со многими независимыми переменными в ограниченной области. В качестве дополнительного условия задается условие интегрального наблюдения. Неизвестная функция источника может быть найдена методом итераций из операторного уравнения со сжимающим оператором. Приведен пример обратной задачи, для которой применимы доказанные в работе результаты. Полученные результаты для обратной задачи основаны на предварительном исследовании однозначной разрешимости соответствующей прямой задачи, которое представляет также и самостоятельный интерес. Библиография: 20 названий.
Lecture Notes in Computer Science, 2017
We establish existence and uniqueness theorems as well as the theorem on stability under perturba... more We establish existence and uniqueness theorems as well as the theorem on stability under perturbations of the input data for the solution of the inverse problem for a degenerate higher-order parabolic equation on a plane with integral observation. We also obtain the estimates of the solution with constants explicitly written out in terms of the input data of the problem.
Eurasian Journal of Mathematical and Computer Applications, 2021
Finite Difference Methods. Theory and Applications, 2019
We prove the existence and uniqueness theorems for inverse problem of determination of the lower ... more We prove the existence and uniqueness theorems for inverse problem of determination of the lower coefficient in the Black-Scholes type equation with additional condition of integral observation. These results are based on the investigation of unique solvability of corresponding direct problem which is of independent interest. We give the example of the inverse problem for which the conditions of the theorems proved are fulfilled.
Journal of Inverse and Ill-posed Problems, 2020
Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading co... more Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problem...
Journal of Mathematical Sciences, 2019
We consider the inverse multifactor source problem for a uniformly parabolic equation with integr... more We consider the inverse multifactor source problem for a uniformly parabolic equation with integral overdetermination conditions and study the existence, uniqueness, and stability of a solution. We obtain two forms of sufficient conditions for the unique solvability and present examples of inverse problems satisfying the assumptions of the proved theorems. Bibliography: 22 titles.
Matematicheskie Zametki, 2019
Получены теоремы существования и единственности решения обратной задачи одновременного определени... more Получены теоремы существования и единственности решения обратной задачи одновременного определения зависящих от переменной xxx коэффициентов перед uuu и uxu_xux в недивергентном параболическом уравнении при условии интегрального наблюдения. Даны оценки максимумов модулей этих коэффициентов с константами, явно выписанными через входные данные задачи. Приведен пример обратной задачи, для которой применимы доказанные теоремы. Библиография: 15 названий.
Journal of Physics: Conference Series, 2019
We find two type conditions sufficient for unique solvability of inverse problem of source determ... more We find two type conditions sufficient for unique solvability of inverse problem of source determination in degenerate parabolic equation with two independent variables.
Journal of Inverse and Ill-posed Problems, 2018
We consider the inverse problem of source determination in nonuniformly parabolic equation under ... more We consider the inverse problem of source determination in nonuniformly parabolic equation under the additional condition of integral observation. We investigate the questions of existence and uniqueness of solution. Two types of sufficient conditions for the unique solvability of the inverse problem are obtained. Examples of inverse problems are given for which the conditions of the proved theorems are fulfilled.
Matematicheskie Zametki, 2017
Lecture Notes in Computer Science, 2015
This work deals with the inverse problem of simultaneous determination of two unknown time-depend... more This work deals with the inverse problem of simultaneous determination of two unknown time-dependent terms in a one-dimensional parabolic equation. The additional information is given by two integral observations. We prove theorems of existence and uniqueness of solution. We also give estimates of maximum modulus of unknown right-hand side and unknown coefficient of the equation with constants derived explicitly in terms of input data.
Математические заметки, 2015
Journal of Inverse and Ill-Posed Problems, 2012
We consider the inverse multidimensional problem for wave equation with unknown variable speed an... more We consider the inverse multidimensional problem for wave equation with unknown variable speed and absorption. A linear procedure based on the BC method for absorption reconstruction is proposed. The speed stays unknown.
Matematicheskie Zametki, 2019
Доказаны теоремы существования и единственности решения обратной задачи одновременного определени... more Доказаны теоремы существования и единственности решения обратной задачи одновременного определения зависящих от ttt коэффициентов перед uuu и uxu_xux в недивергентном параболическом уравнении с двумя независимыми переменными при условии интегрального наблюдения по xxx. Даны оценки максимумов модулей этих коэффициентов с константами, явно выписанными через входные данные задачи. Приведен пример обратной задачи, для которой применимы доказанные теоремы. Библиография: 19 названий.
Математические заметки, 2015
Matematicheskie Zametki
Доказана однозначная разрешимость обратной задачи определения функции источника в неравномерно па... more Доказана однозначная разрешимость обратной задачи определения функции источника в неравномерно параболическом уравнении со многими независимыми переменными в ограниченной области. В качестве дополнительного условия задается условие интегрального наблюдения. Неизвестная функция источника может быть найдена методом итераций из операторного уравнения со сжимающим оператором. Приведен пример обратной задачи, для которой применимы доказанные в работе результаты. Полученные результаты для обратной задачи основаны на предварительном исследовании однозначной разрешимости соответствующей прямой задачи, которое представляет также и самостоятельный интерес. Библиография: 20 названий.
Lecture Notes in Computer Science, 2017
We establish existence and uniqueness theorems as well as the theorem on stability under perturba... more We establish existence and uniqueness theorems as well as the theorem on stability under perturbations of the input data for the solution of the inverse problem for a degenerate higher-order parabolic equation on a plane with integral observation. We also obtain the estimates of the solution with constants explicitly written out in terms of the input data of the problem.
Eurasian Journal of Mathematical and Computer Applications, 2021
Finite Difference Methods. Theory and Applications, 2019
We prove the existence and uniqueness theorems for inverse problem of determination of the lower ... more We prove the existence and uniqueness theorems for inverse problem of determination of the lower coefficient in the Black-Scholes type equation with additional condition of integral observation. These results are based on the investigation of unique solvability of corresponding direct problem which is of independent interest. We give the example of the inverse problem for which the conditions of the theorems proved are fulfilled.
Journal of Inverse and Ill-posed Problems, 2020
Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading co... more Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problem...
Journal of Mathematical Sciences, 2019
We consider the inverse multifactor source problem for a uniformly parabolic equation with integr... more We consider the inverse multifactor source problem for a uniformly parabolic equation with integral overdetermination conditions and study the existence, uniqueness, and stability of a solution. We obtain two forms of sufficient conditions for the unique solvability and present examples of inverse problems satisfying the assumptions of the proved theorems. Bibliography: 22 titles.
Matematicheskie Zametki, 2019
Получены теоремы существования и единственности решения обратной задачи одновременного определени... more Получены теоремы существования и единственности решения обратной задачи одновременного определения зависящих от переменной xxx коэффициентов перед uuu и uxu_xux в недивергентном параболическом уравнении при условии интегрального наблюдения. Даны оценки максимумов модулей этих коэффициентов с константами, явно выписанными через входные данные задачи. Приведен пример обратной задачи, для которой применимы доказанные теоремы. Библиография: 15 названий.
Journal of Physics: Conference Series, 2019
We find two type conditions sufficient for unique solvability of inverse problem of source determ... more We find two type conditions sufficient for unique solvability of inverse problem of source determination in degenerate parabolic equation with two independent variables.
Journal of Inverse and Ill-posed Problems, 2018
We consider the inverse problem of source determination in nonuniformly parabolic equation under ... more We consider the inverse problem of source determination in nonuniformly parabolic equation under the additional condition of integral observation. We investigate the questions of existence and uniqueness of solution. Two types of sufficient conditions for the unique solvability of the inverse problem are obtained. Examples of inverse problems are given for which the conditions of the proved theorems are fulfilled.
Matematicheskie Zametki, 2017
Lecture Notes in Computer Science, 2015
This work deals with the inverse problem of simultaneous determination of two unknown time-depend... more This work deals with the inverse problem of simultaneous determination of two unknown time-dependent terms in a one-dimensional parabolic equation. The additional information is given by two integral observations. We prove theorems of existence and uniqueness of solution. We also give estimates of maximum modulus of unknown right-hand side and unknown coefficient of the equation with constants derived explicitly in terms of input data.
Математические заметки, 2015
Journal of Inverse and Ill-Posed Problems, 2012
We consider the inverse multidimensional problem for wave equation with unknown variable speed an... more We consider the inverse multidimensional problem for wave equation with unknown variable speed and absorption. A linear procedure based on the BC method for absorption reconstruction is proposed. The speed stays unknown.
Matematicheskie Zametki, 2019
Доказаны теоремы существования и единственности решения обратной задачи одновременного определени... more Доказаны теоремы существования и единственности решения обратной задачи одновременного определения зависящих от ttt коэффициентов перед uuu и uxu_xux в недивергентном параболическом уравнении с двумя независимыми переменными при условии интегрального наблюдения по xxx. Даны оценки максимумов модулей этих коэффициентов с константами, явно выписанными через входные данные задачи. Приведен пример обратной задачи, для которой применимы доказанные теоремы. Библиография: 19 названий.
Математические заметки, 2015