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In the present paper, we construct a first order of accuracy difference scheme for the approximate solution of the inverse problem for telegraph-parabolic equations with an unknown spacewise dependent source term. The unique solvability of constructed difference scheme and the stability estimates for its solution were obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions.

In the present paper, we study the first and second order of accuracy difference schemes for the approximate solution of the inverse problem for hyperbolic–parabolic equations with unknown time-independent source term. The unique solvability of constructed difference schemes and the stability estimates for their solutions are obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

In the present paper, a boundary value problem for a one-dimensional hyperbolic-parabolic equation with involution and the Dirichlet condition is studied. The stability estimates for the solution of the hyperbolic-parabolic problem are established. The first order of accuracy stable difference scheme for the approximate solution of the problem under consideration is constructed. Numerical algorithm for implementation of this scheme is presented. Numerical results are provided for a simple test problem.

In the present paper, we study a source identification problem for hyperbolic-parabolic equation with nonlocal conditions. The stability estimates for the solution of this source identification problem are established. Furthermore, we construct the second order of accuracy difference scheme for the approximate solution of the problem under consideration. The stability estimates for the solution of this difference scheme are presented.

In the present paper, a space-dependent source identification problem for the hyperbolic-parabolic equation with unknown parameter p $$ \left\{ \begin{array}{l} \displaystyle u''(t) + Au(t) = p + f(t), ~ 0<t<1, \\ \displaystyle u'(t) + Au(t) = p + g(t), ~ -1<t<0, \\ \displaystyle u(0^{+})=u(0^{-}), ~ u'(0^{+})=u'(0^{-}), \\ \displaystyle u(-1)=\varphi, ~ \int \limits _{0}^{1} u(z)dz=\psi \end{array} \right. $${u′′(t)+Au(t)=p+f(t),0<t<1,u′(t)+Au(t)=p+g(t),-1<t<0,u(0+)=u(0-),u′(0+)=u′(0-),u(-1)=φ,∫01u(z)dz=ψ in a Hilbert space H with self-adjoint positive definite operator A is investigated. The stability estimates for the solution of this identification problem are established. In applications, the stability estimates for the solutions of four space-dependent source identification hyperbolic-parabolic problems are obtained.

We study a nonlocal boundary value problem and a space-wise dependent source identification problem for one-dimensional hyperbolic-parabolic equation with involution and Neumann boundary condition. The stability estimates for the solutions of these two problems are established. The first order of accuracy stable difference schemes are constructed for the approximate solutions of the problems under consideration. Numerical results for two test problems are provided.

In the present paper, we establish the well-posedness of an identification problem for determining the unknown space-dependent source term in the hyperbolic-parabolic equation with nonlocal conditions. The difference scheme is constructed for the approximate solution of this source identification problem. The stability estimates for the solution of the difference scheme are presented.

In the present paper, a source identification problem for hyperbolic-parabolic equation with involution and Dirichlet condition is studied. The stability estimates for the solution of the source identification hyperbolic-parabolic problem are established. The first order of accuracy stable difference scheme is constructed for the approximate solution of the problem under consideration. Numerical results are given for a simple test problem.

An identification problem for an equation of mixed telegraph-parabolic type with an unknown parameter depending on spatial variables is considered. The unique solvability of this problem is proved, and stability inequalities for its solution are established. As applications, stability estimates are obtained for the solutions of four identification problems for telegraph-parabolic equations with an unknown source depending on spatial variables.