Robin boundary value problem for the Cauchy-Riemann operator

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties.

A new algorithm for the solution of eigenvalue problems for linear operators of the form A = A + B (with a special application to high-order ordinary differential equations) is proposed and justified. The algorithm is based on the approximation of A by an operator $$ \overline{A}=A+\overline{B} $$ such that the eigenvalue problem for Ā is supposed to be simpler than for A: The algorithm for this eigenvalue problem is based on the homotopy idea and, for a given eigenpair number, recursively computes a sequence of approximate eigenpairs that converges to the exact eigenpair with a superexponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The case of multiple eigenvalues of the operator Ā is emphasized. Examples of eigenvalue problems for the high-order ordinary differential operators are presented to support the theory.

Positive solutions of four-point boundary value problems for higher-order p-Laplacian operator

The method of layer potentials has been applied with tremendous success in the treatment of boundary value problems for second-order differential operators for a very long time; the literature on this topic is enormous. By way of contrast this method is disproportionally underdeveloped in the case of higher-order operators; the difference between the higher-order and the second-order settings is striking in term of the scientific output. This paper presents new results which establish mapping properties of multiple layer potentials associated with higher-order elliptic operators in Lipschitz domains in ℝ n .

OF THE DISSERTATION Finite-difference and finite-element solution of boundary value and obstacle problems for the Heston operator by Eduardo Osorio Dissertation Director: Paul M. N. Feehan We develop finite-element and finite-difference methods for boundary value and obstacle problems for the elliptic Heston operator. For the finite-element method we first review existence and uniqueness results for these problems on weighted Sobolev spaces, where their variational formulations are formulated, and finite-dimensional subspaces are chosen to find approximating solutions, and obtain error estimates and numerical results. Similarly, for the finite-difference method, we start by reviewing the existence, uniqueness and regularity results on boundary value and obstacle problems on weighted Holder spaces, then consider finite-difference operators, establish discrete maximum principles for them, and obtain error estimates and numerical results.

This paper is concerned with inverse spectral problems for higher-order (n>2) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our approach is based on the reduction of an inverse problem to a linear equation in the Banach space of bounded infinite sequences. This equation is derived in a general form that can be applied to various classes of differential operators. The unique solvability of the linear main equation is also proved. By using the solution of the main equation, we derive reconstruction formulas for the differential expression coefficients in the form of series and prove the convergence of these series for several classes of operators. The results of this paper can be used for the constructive solution of inverse spectral problems and for the investigation of their solvability and stability.

In this article, we offer the M-Sturm Liouville problem for the diffusion operator depending on initial condition. We obtain the representation of the solution of the M-Sturm Liouville problem for diffusion operator through the M-Laplace transform. The purpose of this article is to advantage demonstrate a more generalized version of the representation of the solution of the Sturm Liouville problem for diffusion operator in classical analysis.

In this paper, we propose an approach to inverse spectral problems for the n-th order (n≥2) ordinary differential operators with distribution coefficients. The inverse problems which consist in the reconstruction of the differential expression coefficients by the Weyl matrix and by several spectra are studied. We prove the uniqueness of solution for these inverse problems, by developing the method of spectral mappings. The results of this paper generalize the previously known results for the second-order differential operators with singular potentials and for the higher-order differential operators with regular coefficients. In the future, the approach of this paper can be used for constructive solution and for investigation of solvability of the considered inverse problems.

The accuracy of the solution of a difference boundary value problem for a fourth-order elliptic operator with mixed boundary conditions

Lagrangian coordinates in free boundary problems for parabolic equations

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author’s knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance.

In this paper, we, for the first time, prove the local solvability and stability of an inverse spectral problem for higher-order (n>3) differential operators with distribution coefficients. The inverse problem consists of the recovery of differential equation coefficients from (n−1) spectra and the corresponding weight numbers. The proof method is constructive. It is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation remains uniquely solvable. Furthermore, we estimate the differences of the coefficients in the corresponding functional spaces.

The author studies the inverse spectral problem for higher-order differential operators with a singular point.

The multiscale finite volume (MSFV) method is introduced for the efficient solution of elliptic problems with rough coefficients in the absence of scale separation. The coarse operator of the MSFV method is presented as a multipoint flux approximation (MPFA) with numerical evaluation of the transmissibilities. The monotonicity region of the original MSFV coarse operator has been determined for the homogeneous anisotropic case. For grid-aligned anisotropy the monotonicity of the coarse operator is very limited. A compact coarse operator for the MSFV method is presented that reduces to a 7-point stencil with optimal monotonicity properties in the homogeneous case. For heterogeneous cases the compact coarse operator improves the monotonicity of the MSFV method, especially for anisotropic problems. The compact operator also leads to a coarse linear system much closer to an M-matrix. Gradients in the direction of strong coupling vanish in highly anisotropic elliptic problems with homogeneous Neumann boundary d...

We study the inverse spectral problem for Sturm-Liouville operators with nonseparated boundary conditions on a finite interval. The inverse problem for such operators was investigated in [1-5] and other papers. In particular, the characterization of the spectrum for the periodic case was obtained in [1, 2]. Later, Plaksina [3, 4] obtained similar results for other types of boundary conditions. Another approach to this problem is applied here. We present the solution of the inverse problem for Sturm-Liouville operators with nonseparated boundary conditions, give a characterization of spectrum for such operators, and investigate stability. The main results are stated in Theorems 1 and 3. 1. Consider the self-adjoint boundary value problem L = L(q(x) , a, d, b):