Inverse Problem of Finding the Coefficient of the Lowest Term in Two-Dimensional Heat Equation with Ionkin-Type Boundary Condition

Abstract

We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The well-posedness of the problem is obtained by generalized Fourier method combined by the Banach fixed poind theorem. For obtaining a numerical solution of the inverse problem, we propose the discretization method from a new combination. On the one hand, it is known the traditional method of uniform finite difference combined with numerical integration on a uniform grid (trapezoidal and Simpson's), on the other hand, we give the method of non-uniform finite difference is combined by a numerical integration on a non-uniform grid (with Gauss-Lobatto nodes). Numerical examples illustrate how to implement the method.

О Некоторых Классах Обратных Задач Об Определении Функции Источников Для Систем Тепломассопереноса

“…Далее, вопрос об определении правой части такого вида по интегральным условиям рассматривался в [4,22,28,35,37]. Вопросы определения того или иного коэффициента, зависящего от времени, в параболическом уравнении по интегральным условиям переопределения был рассмотрен, например, в [3,5,40,41,43,46] и многих других. Выделим книгу [42], где в случае n = 1, был исследован ряд коэффициентных обратных задач (см., например, [42, теорема 4.114]).…”

Numerical Solution to Recover Time-dependent Coefficient and Free Boundary from Nonlocal and Stefan Type Overdetermination Conditions in Heat Equation

“…We consider a one-sided free domain problem in one-dimensional space for the parabolic heat equation, with non-homogenous Dirichlet boundary condition when the thermal conductivity is equal to unity. This problem contains free boundary depending on time only [2,3]. Therefore, the shape of the problem varies with time step marching.…”

“…In [4], the authors suggested an iterative method for solving the inverse problem to determine the right-hand side part (heat source problem) with separating variables on spatial variables and time. Determination of time-dependent lowest order coefficient in heat equation was investigated in [5]. The authors in [6 -9] investigated the theoretical and numerical aspects of several types of parabolic heat equations, in fixed and free domains of oneand two-dimensions for various types of additional information.…”

Inverse problem for higher order ultraparabolic equation with unknown minor coefficient and right-hand side function

“…In the works [9,18] the conditions of the unique solvability were obtained for the inverse problems for second order ultraparabolic equations with a single unknown function in its right-hand side, and in [17] -with two unknown parameters in minor coefficient and in the right-hand side function. The problems of identifying of the minor coefficient in the parabolic or hyperbolic equations were studied in [1,5,6,22], of the several coefficients in right-hand side function -in [6,21]. The authors used the methods: of the integral equations, regularization and the Shauder principle [6,9], of successive approximations [16][17][18][19], of continuation in a parameter [13], the generalized Fourier method [5].…”

“…The problems of identifying of the minor coefficient in the parabolic or hyperbolic equations were studied in [1,5,6,22], of the several coefficients in right-hand side function -in [6,21]. The authors used the methods: of the integral equations, regularization and the Shauder principle [6,9], of successive approximations [16][17][18][19], of continuation in a parameter [13], the generalized Fourier method [5]. The main aim of the present paper is to find the sufficient conditions of the existence and the uniqueness of the solution for the inverse problem for higher order ultraparabolic equation with a right-hand side function of the special type.…”