Fourth-order Differential Operator Research Articles

Oscillatory and spectral properties of a class of fourth-order differential operators via a new Hardy-type inequality

Oscillatory and spectral properties of a class of fourth-order differential operators via a new Hardy-type inequality

A unified weighted inequality for fourth-order partial differential operators and applications

In this paper, we establish a fundamental inequality for fourth order partial differential operator P=△α∂s+β∂ss+Δ2 (▪) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator P with some suitable boundary conditions in the case of β<0 or α≠0,β=0. As application, we obtain a resolvent estimate for P, which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.

Inverse problems for the beam vibration equation with involution

This article considers inverse problems for a fourth-order hyperbolic equation with involution. The existence and uniqueness of a solution of the studied inverse problems is established by the method of separation of variables. To apply the method of separation of variables, we prove the Riesz basis property of the eigenfunctions for a fourth-order differential operator with involution in the space ${{L}_{2}}(-1,1)$. For proving theorems on the existence and uniqueness of a solution, we widely use the Bessel inequality for the coefficients of expansions into a Fourier series in the space ${{L}_{2}}(-1,1)$. A significant dependence of the existence of a solution on the equation coefficient $\alpha$ is shown. In each of the cases $\alpha &lt;-1$, $\alpha &gt;1$, $-1&lt;\alpha&lt;1$ representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are written out.

Inverse Problem for a Fourth-Order Hyperbolic Equation with a Complex-Valued Coefficient

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form of a Fourier series in terms of the eigenfunctions of a non-self-adjoint fourth-order ordinary differential operator. The proofs of the uniform convergence of the Fourier series are based on estimates of the norms of the derivatives of the eigenfunctions of a fourth-order ordinary differential operator and the uniform boundedness of the Riesz bases of the eigenfunctions.

Asymptotics of the eigenvalues of a fourth order differential operator with eigenvalue dependent and periodic boundary conditions

Asymptotics of the eigenvalues of a fourth order differential operator with eigenvalue dependent and periodic boundary conditions

Fourth-order differential operators with interior degeneracy and generalized Wentzell boundary conditions

In this article we consider the fourth-order operators \(A_1u:=(au'')''\) and \(A_2u:=au''''\) in divergence and non divergence form, where \(a:[0,1]\to\mathbb{R}_+\) degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on \(a\), we study the generation property of these operators associated to generalized Wentzell boundary conditions. We prove the well posedness of the corresponding parabolic problems.

Spectral asymptotics and a trace formula for a fourth-order differential operator corresponding to thin film equation

Spectral asymptotics and a trace formula for a fourth-order differential operator corresponding to thin film equation

Asymptotic Expressions of Fourth Order Sturm-Liouville Operator with Conjugate Conditions

In this paper, it is studied the asymptotic expression of fourth order differential operator with periodic boundary conditions. For this operator, it is also considered conjugate boundary conditions at x=0 which shows discontinuity. For this purpose, firstly asymptotic expression of solutions areobtained. Then by using the the asymptotic formulas of fundamental solutions, asymptotic expression of eigenvalues and eigenfunctions are presented. It is also dealt with the asymptotic expression of same operator with antiperiodic boundary conditions and conjugate conditions

Sharp eigenvalue asymptotics of fourth-order differential operators

Two-term self-adjoint fourth-order differential operator with summable potential on the unit interval is considered. High energy eigenvalue asymptotics and the trace formula for this operator are obtained.

Modified Morley method by free formula for fourth order elliptic singular perturbation problems

A new modified Morley method is proposed to solve fourth order elliptic singular perturbation problems of two or three dimension. The method has the fewest number of DOFs on each element. Different from the usual finite element methods, we slightly modify the bilinear term corresponding to the second order differential operator and keep that corresponding to the fourth order differential operator and the right-hand side term unchanged. The modification is based on the idea of free formula introduced by Bergan and Nygård in 1984. Theory and numerical results show that the solution is uniformly convergent with respect to the perturbed parameter ε . From the view of computation, our method is efficient since the fewest number of DOFs are employed and the local stiffness matrix takes less computation time than the usual finite element method due to the special construction of free formula.

Oscillatory and spectral properties of fourth-order differential operator and weighted differential inequality with boundary conditions

In the paper, we study the oscillatory and spectral properties of a fourth-order differential operator. These properties are established based on the validity of some weighted second-order differential inequality, where the inequality’s weights are the coefficients of the operator. The inequality, in turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of its weights at infinity and at zero.

Spectral Estimates for the Fourth-Order Differential Operator with Periodic Coefficients

Spectral Estimates for the Fourth-Order Differential Operator with Periodic Coefficients

Dependence of Eigenvalues of Discontinuous Fourth-order Differential Operators with Eigenparameter Dependent Boundary Conditions

In this paper, we investigate a fourth-order differential operator with eigenparameter dependent boundary conditions and transmission conditions. To study the eigenvalues of the problem, we establish a new operator associated with the considered problem. Furthermore, we prove that the eigenvalues are differentiable depending on the parameters of the problem. Finally, the differential expressions of the eigenvalues with respect to all parameters are given.

Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions

&lt;abstract&gt;&lt;p&gt;This paper is concerned with a fourth-order differential operator with eigenparameter dependent boundary conditions. We prove that each of the eigenvalues of the problem can be embedded in a continuous eigenvalue branch. Furthermore, the differential expressions of the eigenvalues with respect to each of parameters are given.&lt;/p&gt;&lt;/abstract&gt;

Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions

&lt;abstract&gt;&lt;p&gt;In this paper, a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions is considered. A new operator associated with the problem is established, and the self-adjointness of this operator in an appropriate Hilbert space $ H $ is proved. The fundamental solutions are constructed. Sufficient and necessary conditions of the eigenvalues are investigated. Then asymptotic formulas for the fundamental solutions and the characteristic functions are given. Finally, the completeness of eigenfunctions in $ H $ is given and the Green function is also involved.&lt;/p&gt;&lt;/abstract&gt;

Weighted Second-Order Differential Inequality on Set of Compactly Supported Functions and Its Applications

In the paper, we establish the oscillatory and spectral properties of a class of fourth-order differential operators in dependence on integral behavior of its coefficients at zero and infinity. In order to obtain these results, we investigate a certain weighted second-order differential inequality of independent interest.

Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

A Second Regularized Trace Formula for a Fourth Order Differential Operator

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

Eventual domination for linear evolution equations

We consider two C_0-semigroups on function spaces or, more generally, Banach lattices and give necessary and sufficient conditions for the orbits of the first semigroup to dominate the orbits of the second semigroup for large times. As an important special case we consider an L^2-space and self-adjoint operators A and B which generate C_0-semigroups; in this situation we give criteria for the existence of a time t_1 ge 0 such that e^{tB} ge e^{tA} for all subsequent times tge t_1. As a consequence of our abstract theory, we obtain many surprising insights into the behaviour of various second and fourth order differential operators.

Method of moments with a choice of special basic functions for fourth-order partial differential equations

This paper presents an approximation method to solve the boundary value problem (BVP) for partial differential equations (PDEs) of a kind of the fourth order with the idea of discretization of a spatial variable by using the method of moments with a choice of special basic functions. A system of ordinary differential equations (ODEs) is obtained by multiplying the original equation by some auxiliary functions, followed by interpolation and integration over the spatial variable. Newton-Stirling, Hermite-Birkhoff and Hermite interpolations are flexibly applied to internal and pre-boundary nodes. In addition, boundary conditions are automatically satisfied without being approximated separately as in classical numerical methods (for example, method of grids, method of lines). Thus, the proposed schemes have a higher order of approximation. The main goal of the work is the construction of basic functions for a fourth-order differential operator and a possible increase in the order of the remainder term when passing to difference equations.