On the problem of determining the parameter of an elliptic equation in a Banach space

The boundary value problem of determining the parameter of an elliptic equation -u''(t)+Au(t)=f(t)+p (0⩽t⩽T), u(0)=φ, u(T)=ψ, u(λ)=ξ, 0<λT, with a positive operator A in an arbitrary Banach space E is studied. The exact estimates are obtained for the solution of this problem in Hölder norms. Coercive stability estimates for the solution of boundary value problems for multi-dimensional elliptic equations are established

The Difference Problem of Obtaining the Parameter of a Parabolic Equation

The boundary value problem of determining the parameter of a parabolic equation ()+()=()+(0≤≤1),(0)=,(1)= in an arbitrary Banach space with the strongly positive operator is considered. The first order of accuracy stable difference scheme for the approximate solution of this problem is investigated. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained

Finite Difference Method for the Reverse Parabolic Problem

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example

Numerical solution to Bitsadze-Samarskii type elliptic overdetermined multipoint NBVP

Abstract This paper is devoted to a Bitsadze-Samarskii type overdetermined multipoint nonlocal boundary value problem (NBVP). This inverse problem is reduced to an auxiliary multipoint NBVP. Stability estimates for the solution of the auxiliary NBVP are established. The finite difference method is applied to get the first and second order of accuracy of approximate solutions of the abstract overdetermined problem. Stability estimates for the solution of difference problems are proved. Then the established abstract results are applied to get stability estimates for the solution of the Bitsadze-Samarskii type overdetermined elliptic multidimensional differential and difference problems with multipoint nonlocal boundary conditions (NBVC). Finally, numerical results with explanation on the realization for two dimensional and three dimensional elliptic overdetermined multipoint NBVPs in test examples are presented

Разностные схемы второго порядка точности для нелокальных по времени параболических задач интегрального типа

This is a discussion on the second-order accuracy difference schemes for approximate solution of the integral-type time-nonlocal parabolic problems. The theorems on the stability of r-modified Crank-Nicolson difference schemes and second-order accuracy implicit difference scheme for approximate solution of the integral-type time-nonlocal parabolic problems in a Hilbert space with self-adjoint positive definite operator are established. In practice, stability estimates for the solutions of the second-order accuracy in t difference schemes for the one and multidimensional time-nonlocal parabolic problems are obtained. Numerical results are given.Исследуются разностные схемы второго порядка точности для приближенного решения нелокальных по времени параболических задач интегрального типа. Установлены теоремы об устойчивости r-модифицированной разностной схемы Кранка-Николсона и неявной разностной схемы второго порядка точности для приближенного решения нелокальных по времени параболических задач интегрального типа в гильбертовом пространстве с самосопряженным положительно определенным оператором. В качестве приложения получены оценки устойчивости решений второго порядка точности по t разностных схем для одномерной и многомерной нелокальной во времени параболической задачи. Приведены численные результаты

Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

A …finite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the fi…rst and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem

Approximate solution for an inverse problem of multidimensional elliptic equation with multipoint nonlocal and Neumann boundary conditions

In this work, we consider an inverse elliptic problem with Bitsadze-Samarskii type multipoint nonlocal and Neumann boundary conditions. We construct the first and second order of accuracy difference schemes (ADSs) for problem considered. We stablish stability and coercive stability estimates for solutions of these difference schemes. Also, we give numerical results for overdetermined elliptic problem with multipoint Bitsadze-Samarskii type nonlocal and Neumann boundary conditions in two and three dimensional test examples. Numerical results are carried out by MATLAB program and brief explanation on the realization of algorithm is given

On the stability of hyperbolic difference equations with unbounded delay term

Abstract The paper studies the unconditionally stable difference scheme for the approximate solution of the hyperbolic differential equation with unbounded delay term $$\begin{aligned} \ \left\{ \begin{array}{l} v_{tt}(t)+A^{2}v(t)=a\left( v_{t}(t-w )+Av(t-w )\right) +f(t),t\in (0,\infty ), \\ v(t)=\varphi (t),t\in [-w,0] \end{array} \right. \end{aligned}$$ v tt ( t ) + A 2 v ( t ) = a v t ( t - w ) + A v ( t - w ) + f ( t ) , t ∈ ( 0 , ∞ ) , v ( t ) = φ ( t ) , t ∈ [ - w , 0 ] in a Hilbert space H with a self-adjoint positive definite operator A. The main theorem on unconditionally stability estimates for the solutions of this problem are established. Numerical results and explanatory illustrations are presented show the validation of the theoretical results