Volume 33, Issue 3

In this paper, we investigate an inverse problem for non-selfadjoint Dirac operator with eigenparameter dependent boundary conditions and finite general transmission conditions. We obtain the uniqueness theorem by the generalized norming constants. Moreover, we give two reconstruction algorithms for the coefficient functions and the coefficients of the conditions by the method of spectral mappings.

In this paper, the inverse Sturm–Liouville problem with distribution potential and with polynomials of the spectral parameter in one of the boundary conditions is considered. We for the first time prove local solvability and stability of this inverse problem in the general non-self-adjoint case, taking possible splitting of multiple eigenvalues into account. The proof is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of continuous functions on some circular contour. Moreover, we introduce the generalized Cauchy data, which will be useful for investigation of partial inverse Sturm–Liouville problems with polynomials in the boundary conditions. Local solvability and stability of recovering the potential and the polynomials from the generalized Cauchy data are obtained. Thus, the results of this paper include the first existence theorems for solution of the inverse Sturm–Liouville problems with polynomial dependence on the spectral parameter in the boundary conditions in the case of multiple eigenvalues. In addition, our stability results can be used for justification of numerical methods.

Let ℋ {\mathcal{H}} and 𝒦 {\mathcal{K}} be real Hilbert spaces, let T : ℋ → 𝒦 {T:\mathcal{H}\to\mathcal{K}} be a continuous operator with a non-closed range, and let y ∈ ran ⁡ T ¯ ∖ ran ⁡ T {y\in\overline{\operatorname{ran}T}\setminus\operatorname{ran}T} . In the case where ℋ {\mathcal{H}} is finite-dimensional, or T is linear, or ℋ = 𝒦 {\mathcal{H}=\mathcal{K}} and Id - T {\operatorname{Id}-T} is nonexpansive, we show that the approximate solutions to T ⁢ x = y {Tx=y} blow up. We also show that the blow-up of approximate solutions is not necessary by constructing a continuous mapping T : L 2 ⁢ ( [ 0 , 1 ] ) → L 2 ⁢ ( [ 0 , 1 ] ) {T:L^{2}([0,1])\to L^{2}([0,1])} , a vector y ∈ ran ⁡ T ¯ ∖ ran ⁡ T {y\in\overline{\operatorname{ran}T}\setminus\operatorname{ran}T} , and a bounded sequence of approximate solutions to T ⁢ x = y {Tx=y} . Finally, if T : ℋ → 𝒦 {T:\mathcal{H}\to\mathcal{K}} is compact and self-adjoint, and y ∈ ran ⁡ T ¯ ∖ ran ⁡ T {y\in\overline{\operatorname{ran}T}\setminus\operatorname{ran}T} , we obtain a growth quantification result via an inequality of the form φ y ⁢ ( ∥ T ⁢ x - y ∥ ) ⋅ ∥ x ∥ ≥ C {\varphi_{y}(\|Tx-y\|)\cdot\|{x}\|\geq C} where φ y {\varphi_{y}} is a nondecreasing function and C is a positive constant.

We study the problem of identifying the constant parameters involved in the right-hand sides of a linear non-autonomous system of differential equations with first-order ordinary derivatives. The specificity of the problem lies in the fact that additional conditions for identifying parameters, firstly, are non-local, and secondly, they include derivatives of an unknown function. We examine the conditions for the existence and uniqueness of a solution to the problem and propose two different approaches to the numerical solution of the problem. The results of computer experiments are presented.

In this paper, the inverse problem of source term identification for Caputo–Hadamard type time-fractional diffusion-wave equation is studied. Firstly, we prove that the problem is ill-posed, and give the optimal error bound and the conditional stability results. Secondly, we apply fractional Tikhonov regularization method and fractional Landweber iterative regularization method to solve the problem. Based on the conditional stability results, we give error estimates under the a priori regularization parameter selection rule and the a posteriori regularization parameter selection rule respectively. In addition, we give three numerical examples to prove the validity and feasibility of the selected regularization method. What is novel is that we apply L2 1 formula, Crank–Nicolson format and the finite difference method to discrete equation.

The main purpose of this paper is to develop further the integrated theory of the probe and singular sources methods (IPS) which may work for a group of inverse obstacle problems. Here as a representative and typical member of the group, an inverse obstacle problem governed by the Helmholtz equation with a fixed wave number in a bounded domain is considered. It is assumed that the solutions of the Helmholtz equation outside the set of unknown obstacles satisfy the homogeneous Dirichlet or Neumann boundary conditions on each surface of obstacles. This is the case when two extreme types of obstacles are embedded in a medium. By considering this case, not only a concise technique for IPS is introduced but also a general correspondence principle from IPS to the probe method is suggested. Besides, as a corollary it is shown that the probe method together with the singular sources method reformulated in terms of the probe method has the Side B under a smallness conditions on the wave number k , which is the blowing up property of a sequence computed from the associated Dirichlet-to-Neumann map.

In this paper, we study the Kuramoto–Sivashinsky (KS) equation backward in time. First, we prove a stability estimate of Hölder type. Then the ill-posed problem is regularized by the Tikhonov regularization method, and an error estimate of Hölder type is obtained. Finally, we apply a physics-informed neural network method to solve the problem numerically.

We obtain new estimates of stability for the continuation of solutions of elliptic inequalities | ∂ ¯ ⁢ u | 2 ⩽ a ⁢ | u | 2 |\bar{\partial}u|^{2}\leqslant a|u|^{2} from finite discrete sets. For this, we use Hörmander’s type of Carleman estimates for ∂ ¯ {\bar{\partial}} with boundary terms and a special weight function equal to the linear combination of Green’s functions with singularities at points where our function is given. To estimate functions in a given domain we use not only the classical Blaschke partial sum S n {S_{n}} but we also introduce a new sequence of numbers M n {M_{n}} which gives us better estimates in the case of a finite number of observations.