Higher-order Differential Operators Research Articles

Spectral analysis of operator polynomials and higher-order difference operators

The study of the spectral properties of operator polynomials is reduced to the study of the spectral properties of the operator specified by the operator matrix. The results obtained are applied to higher-order difference operators. Conditions for their invertibility and for them to be Fredholm, as well as the asymptotic representation for bounded solutions of homogeneous difference equations are obtained.

Spectral Analysis of Operator Polynomials and Higher-Order Difference Operators

Исследование спектральных свойств операторных полиномов сводится к изучению спектральных свойств оператора, заданного операторной матрицей. Полученные результаты применяются к разностным операторам высокого порядка. Получены условия их обратимости, фредгольмовости, асимптотическое представление ограниченных решений однородного разностного уравнения. Библиография: 27 названий.

Existence results for a class of high order differential equation associated with integral boundary conditions at resonance

By using Mawhin’s continuation theorem, we provide some sufficient conditions for the existence of solution for a class of high order differential equations of the form $$x^{(n)} =f(t,x,x^{\prime },\dots ,x^{(n-1)})\,, \quad t \in [0, 1]\,,$$ associated with the integral boundary conditions at resonance. The interesting point is that we shall deal with the case of nontrivial kernel of arbitrary dimension corresponding to high order differential operator which will cause some difficulties in constructing the generalized inverse operator.

Blind motion image deblurring using nonconvex higher-order total variation model

We propose a nonconvex higher-order total variation (TV) method for blind motion image deblurring. First, we introduce a nonconvex higher-order TV differential operator to define a new model of the blind motion image deblurring, which can effectively eliminate the staircase effect of the deblurred image; meanwhile, we employ an image sparse prior to improve the edge recovery quality. Second, to improve the accuracy of the estimated motion blur kernel, we use L1 norm and H1 norm as the blur kernel regularization term, considering the sparsity and smoothing of the motion blur kernel. Third, because it is difficult to solve the numerically computational complexity problem of the proposed model owing to the intrinsic nonconvexity, we propose a binary iterative strategy, which incorporates a reweighted minimization approximating scheme in the outer iteration, and a split Bregman algorithm in the inner iteration. And we also discuss the convergence of the proposed binary iterative strategy. Last, we conduct extensive experiments on both synthetic and real-world degraded images. The results demonstrate that the proposed method outperforms the previous representative methods in both quality of visual perception and quantitative measurement.

Liquid-vapor transformations with surfactants. Phase-field model and Isogeometric Analysis

Surfactants are compounds that find energetically favorable to be located at the boundaries between fluids. They are able to modify the properties of those interfaces, for example, reducing surface tension. Here, we propose a new model for liquid–vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid. The aforementioned model is derived from a free energy functional by using a Coleman–Noll approach. The proposed theory emanates from the isothermal Navier–Stokes–Korteweg equations, which describe single-component two-phase flow and naturally allow for phase transformations. We believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes. From a numerical point of view, the proposed model poses significant challenges to existing discretization methods, including stiffness in space and time, internal and boundary layers as well as higher-order partial differential operators. To overcome these challenges we propose algorithms based on Isogeometric Analysis, which permit an accurate and efficient discretization. Finally, we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions.

On the spectrum of the pencil of high order differential operators with almost periodic coefficients

In this paper, the spectrum and the resolvent of the operator $L_{\lambda}$ which is generated by the differential expression $\ell_{\lambda }(y)=y^{(m)}+\sum_{\gamma=1}^{m} ( \sum_{k=0}^{\gamma }\lambda^{k}p_{\gamma k}(x) ) y^{(m-\gamma)}$ has been investigated in the space $L_{2}(\mathbb{R})$ . Here the coefficients $p_{\gamma k}(x)=\sum_{n=1}^{\infty }p_{\gamma kn}e^{i\alpha_{n}x}$ , $k=0,1,\ldots,\gamma-1$ ; $p_{\gamma\gamma }(x)=p_{\gamma\gamma}$ , $\gamma=1,2,\ldots,m$ , are constants, $p_{mm}\neq0$ and $p_{\gamma k}^{(\nu)}(x)$ , $\nu=0,1,2,\ldots,m-\gamma$ , are Bohr almost-periodic functions whose Fourier series are absolutely convergent. The sequence of Fourier exponents of coefficients (these are positive) has a unique limit point at +∞. It has been shown that if the polynomial $\phi (z)=z^{m}+p_{11}z^{m-1}+p_{22}z^{m-2}+\cdots+p_{m-1,m-1}z+p_{mm}$ has the simple roots $\omega_{1},\omega_{2},\ldots,\omega_{m}$ (or one multiple root $\omega_{0}$ ), then the spectrum of operator $L_{\lambda}$ is pure continuous and consists of lines $\operatorname{Re}(\lambda\omega _{k})=0$ , $k=1,2,\ldots,m$ (or of line $\operatorname{Re}(\lambda\omega_{0})=0$ ). Moreover, a countable set of spectral singularities on the continuous spectrum can exist which coincides with numbers of the form $\lambda=0$ , $\lambda _{sjn}=i\alpha_{n} ( \omega_{j}-\omega_{s} ) ^{-1}$ , $n\in \mathbb{N}$ , $s,j=1,2,\ldots,m$ , $j\neq s$ . If $\phi(z)=(z-\omega_{0})^{m}$ , then the spectral singularity does not exist. The resolvent $L_{\lambda}^{-1}$ is an integral operator in $L_{2}(\mathbb{R})$ with the kernel of Karleman type for any $\lambda\in\rho (L_{\lambda})$ .

Theory and application of fractional step characteristic finite difference method in numerical simulation of second order enhanced oil production

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results are quite interesting and satisfactory.

Theory and Application of Characteristic Finite Difference Fractional Step Method of Capillary Force Enhanced Oil Production

A kind of second-order implicit characteristic fractional steps finite difference method is presented in this paper for the numerical simulation coupled system of enhanced (chemical) oil production on consideration capillary force in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in $l^2$ norm is derived. This method has been applied successfully the numerical simulation of enhanced oil production in actual oilfields, and the simulation results are quite interesting and satisfactory.

Theory and application of numerical simulation method of capillary force enhanced oil production

A kind of second-order implicit upwind fractional step finite difference methods are presented for the numerical simulation of coupled systems for enhanced (chemical) oil production with capillary force in the porous media. Some techniques, e.g., the calculus of variations, the energy analysis method, the commutativity of the products of difference operators, the decomposition of high-order difference operators, and the theory of a priori estimate, are introduced. An optimal order error estimate in the l2 norm is derived. The method is successfully used in the numerical simulation of the enhanced oil production in actual oilfields. The simulation results are satisfactory and interesting.

Constructing Krall–Hahn orthogonal polynomials

Given a sequence of polynomials (pn)n, an algebra of operators A that acts in the linear space of polynomials and an operator Dp∈A with Dp(pn)=θnpn, where θn is any arbitrary eigenvalue, we construct a new sequence of polynomials (qn)n by considering a linear combination of m+1 consecutive pn: qn=pn+∑j=1mβn,jpn−j. Using the concept of a D-operator, we determine the structure of the sequences βn,j, j=1,…,m, such that the polynomials (qn)n are eigenfunctions of an operator in the algebra A. As an application, from the classical discrete family of Hahn polynomials, we construct orthogonal polynomials (qn)n that are also eigenfunctions of higher-order difference operators.

Holomorphic extension of fundamental solutions of elliptic linear partial differential operators of higher order with analytic coefficients

We prove that every fundamental solution of an elliptic linear partial differential operator with analytic coefficients in Ω open subset of Rn and simple complex characteristics can be holomorphically continued at least locally as a multi-valued analytic function x⟶E(x,y) in Cn up to the complex bicharacteristic conoid. This holomorphic extension is ramified around the bicharacteristic conoid and belongs to the Nilsson class.

Convergence analysis of an implicit upwind difference fractional steps method of three-dimensional enhanced oil recovery percolation coupled system

A type of implicit upwind difference fractional steps scheme is presented for the percolation coupled system of enhanced (chemical) oil recovery simulation in this paper. An optimal-order error estimate is discussed in l 2-norm by introducing many theories and techniques such as variation form, energy method, product exchange theory of difference operators, splitting of high-order difference operators, priori estimates method of differential equations. This numerical method has been applied successfully in enhanced oil recovery simulation in actual oilfields and gives rise to satisfying simulation results.

Constructing Bispectral Orthogonal Polynomials from the Classical Discrete Families of Charlier, Meixner and Krawtchouk

Given a sequence of polynomials $$(p_n)_n$$ , an algebra of operators $${\mathcal A}$$ acting in the linear space of polynomials, and an operator $$D_p\in {\mathcal A}$$ with $$D_p(p_n)=np_n$$ , we form a new sequence of polynomials $$(q_n)_n$$ by considering a linear combination of $$m+1$$ consecutive $$p_n$$ : $$q_n=p_n+\sum _{j=1}^m\beta _{n,j}p_{n-j}$$ . Using the concept of $$\mathcal {D}$$ -operator, we determine the structure of the sequences $$\beta _{n,j}, j=1,\ldots ,m,$$ so that the polynomials $$(q_n)_n$$ are eigenfunctions of an operator in the algebra $${\mathcal A}$$ . As an application, from the classical discrete families of Charlier, Meixner, and Krawtchouk, we construct orthogonal polynomials $$(q_n)_n$$ which are also eigenfunctions of higher-order difference operators.

Strichartz estimates for parabolic equations with higher order differential operators

The present paper first obtains Strichartz estimates for parabolic equations with nonnegative elliptic operators of order 2m by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. Some conclusions can be viewed as the improvements of the previously known ones. Furthermore, an endpoint homogeneous Strichartz estimates on BMO x (ℝ n ) and a parabolic homogeneous Strichartz estimate are proved. Meanwhile, the Strichartz estimates to the Sobolev spaces and Besov spaces are generalized. Secondly, the local well-posedness and small global well-posedness of the Cauchy problem for the semilinear parabolic equations with elliptic operators of order 2m, which has a potential V (t, x) satisfying appropriate integrable conditions, are established. Finally, the local and global existence and uniqueness of regular solutions in spatial variables for the higher order elliptic Navier-Stokes system with initial data in L r (ℝ n ) is proved.

Characteristic fractional step finite difference method for nonlinear section coupled system

For the section coupled system of multilayer dynamics of fluids in porous media, a parallel scheme modified by the characteristic finite difference fractional steps is proposed for a complete point set consisting of coarse and fine partitions. Some techniques, such as calculus of variations, energy method, twofold-quadratic interpolation of product type, multiplicative commutation law of difference operators, decomposition of high order difference operators, and prior estimates, are used in theoretical analysis. Optimal order estimates in l2 norm are derived to show accuracy of the second order approximation solutions. These methods have been used to simulate the problems of migration-accumulation of oil resources.

Hamiltonian functional integrals representing regularized traces of higher-order differential operators

In this paper, we obtain a representation for the regularized traces (1) of evenorder differential opera� tors in terms of Hamiltonian functional integrals and the corresponding Feynman formulas. We also present an example showing how such a representation can be applied to obtain formulas not containing functional integrals. A Feynman formula (2) is a representation of a solution of an evolution differential equation or an object somehow related to this equation in terms of limit integrals over Cartesian powers of some space. If the space under consideration coincides with the spa� tial domain of the solution, then the corresponding formulas are said to be Lagrangian; if this is the prod� uct of this spatial domain and the dual of the linear space containing it, then the formulas are said to be Hamiltonian. A similar terminology is used for func� tional integrals being approximated by Feynman for� mulas. In this paper, we extend results of (3) to higher� order differential operators, which requires substan� tially modifying the technique; namely, we use Hamil� tonian (rather that Lagrangian, as in (3)) Feynman formulas and functional integrals. Moreover, in those Hamiltonian functional integrals which are used in the paper, integration is over the set of functions taking values in the product of the momentum space and a domain in the configuration space not coinciding with the entire space. Main attention is paid to the algebraic aspects of the problems under consideration; some analytic assumptions are omitted.

Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D) be a nonnegative homogeneous elliptic operator of order 2m with real constant coefficients on Rn and V be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e−tH generated by H=P(D)+V with Kato type perturbing potential V, which naturally generalizes the classical result for Schrödinger semigroup e−t(Δ+V) as V∈K2(Rn), the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e−tP(D) on L1(Rn). As a consequence of the Gaussian upper bound, the Lp-spectral independence of H is concluded.

Value Distribution and Linear Operators

AbstractNevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

Inverse Problems on the Least Eigenvalue

Some uniqueness theorems on the least eigenvalue are provided for wide classes of self-adjoint operators: differential operators with operator-valued potentials, higher-order partial differential operators and the p-Laplacian.

On spectral properties of certain regular boundary value problems for higher-order differential operators

On spectral properties of certain regular boundary value problems for higher-order differential operators