Partial Differential Operators Research Articles

Tchebycheffian B-splines in isogeometric Galerkin methods

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.

Approaches to decomposing acoustic data

AbstractThis note compares different approaches to estimating acoustic plane‐wave components from a set of traces recorded by receivers at known locations. The parameters for these plane waves are estimated following somewhat different approaches: A heuristic approach without reference to any optimization schemes). A frequency‐domain least‐squares approach: With explicit derivation of separate real and imaginary parts of the complex‐valued wave‐field parameters. Using complex partial differential operators. All approaches give the same solution. Of the three, the first approach gives a better understanding of the physics – the last approach is far more direct and easier. That the results are identical argues for our use of Wirtinger complex partial differential operators for our least‐squares minimization problem. Using these operators allows the complex numbers z and to be considered independent variables, giving .

On multivariate orthogonal polynomials and elementary symmetric functions

We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi-xj|2γ+1,x∈(a,b)d,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_{\\gamma }(\\mathtt {x}) = \\prod \\limits _{i=1}^{d} \\omega (x_{i}) \\prod \\limits _{i<j} |x_{i}-x_{j}|^{2\\gamma +1}, \\quad \\mathtt {x}\\in (a,b)^{d}, \\end{aligned}$$\\end{document}for gamma >-1, where omega (t) is an univariate weight function in t in (a,b) and mathtt {x} = (x_{1},x_{2}, ldots , x_{d}) with x_{i} in (a,b). Applying the change of variables x_{i},i=1,2,ldots ,d, into u_{r},r=1,2,ldots ,d, where u_{r} is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having x_{i},i=1,2,ldots ,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables u_{r} for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables.

$\Gamma$-Convergence for Functionals Depending on Vector Fields. II. Convergence of Minimizers

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study integral functionals depending on $X$. Using the results in \cite{MPSC1}, we study the convergence of minima, minimizers and momenta of those functionals. Moreover, we apply these results to the periodic homogenization in Carnot groups and to prove a $H$-compactness theorem for linear differential operators of the second order depending on $X$.

Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians

We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\langle \alpha_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $\alpha_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\check{S}^{\dagger}_{W}$ of the quotient difference operator $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\widehat{S}$ is a linear difference operator with constant coefficients depending on $\alpha_{i}$ and $\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which is annihilated by $\check{S}^{\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\in\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

This paper is concerned with inverse spectral problems for higher-order (n&gt;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.

Positive weak solutions of uniformly elliptic inequalities and applications to population models

By using the uniformly ellipticity of linear partial differential operators (which contain p-Laplacian operators with 1 < p < ∞ ), weak comparison principle and Urysohn's lemma, in the paper, we introduce and study a class of new variational inequalities involving uniformly elliptic operators and demicontinuous S-contractive operators to extend some corresponding tasks and scope on population genetics. Further, as applications, we employ the main results presented in this paper to second-order uniformly elliptic inequalities and a population model of one species arising in disrupted environments.

Fast algorithms for interpolation with L-splines for differential operators [formula omitted] of order 4 with constant coefficients

In the classical theory of cubic interpolation splines there exists an algorithm which works with only On arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces their computation to interpolation cubic splines and also performs with On arithmetic operations. In this paper it is shown that many features of the polynomial cubic spline setting carry over to the larger class of L-splines where L is a linear differential operator of order 4 with constant coefficients. Criteria are given such that the associated matrix R is strictly diagonally dominant which implies the existence of a fast algorithm for interpolation. We provide an example of two L-splines interpolating S&P500 data.

Bilevel Training Schemes in Imaging for Total Variation--Type Functionals with Convex Integrands

In the context of image processing, given a $k$-th order, homogeneous and linear differential operator with constant coefficients, we study a class of variational problems whose regularizing terms depend on the operator. Precisely, the regularizers are integrals of spatially inhomogeneous integrands with convex dependence on the differential operator applied to the image function. The setting is made rigorous by means of the theory of Radon measures and of suitable function spaces modeled on $BV$. We prove the lower semicontinuity of the functionals at stake and existence of minimizers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the space-dependent regularization parameters, thus allowing for good flexibility and preservation of details in the reconstructed image. We establish existence of optima for the scheme and we finally substantiate its feasibility by numerical examples in image denoising. The cases that we treat are Huber versions of the first and second order total variation with both the Huber and the regularization parameter being spatially dependent. Notably the spatially dependent version of second order total variation produces high quality reconstructions when compared to regularizations of similar type, and the introduction of the spatially dependent Huber parameter leads to a further enhancement of the image details.

Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem

Consider the tri-harmonic differential expression $L_{V}^{\nabla}u=\left(\nabla^{+}\nabla\right)^{3}u+Vu$, on sections of a hermitian vector bundle over a complete Riemannian manifold $\left(M,g\right)$ with metric $g$, where $\nabla$ is a metric covariant derivative on bundle E over complete Riemannian manifold, $\nabla^{+}$ is the formal adjoint of $\nabla$ and $V$ is a self adjoint bundle on $E$. We will give conditions for $L_{V}^{\nabla}$ to be essential self-adjoint in $L^{2}\left(E\right).$ Additionally, we provide sufficient conditions for $L_{V}^{\nabla}$ to be separated in $L^{2}\left( E\right)$. According to Everitt and Giertz, the differential operator $L_{V}^{\nabla}$ is said to be separated in $L^{2}\left( E\right) $ if for all $u$ $\in L^{2}\left( E\right)$ such that $L_{V}^{\nabla}u\in L^{2}\left( E\right) $, we have $Vu\in L^{2}\left( E\right)$.

On mixed problems for a class of fractional order equations with involution

In this paper, we consider new classes of differential equations of fractional order related to Hadamard derivatives. These equations generalize the well-known heat conduction equation for the fractional exponent of the time derivative. For the equations under consideration, mixed problems with Dirichlet and Neumann boundary conditions are studied. The Fourier method is used to solve these problems. Two auxiliary problems are obtained for ordinary differential equations of fractional order and ordinary differential equations with involution. The spectral properties of ordinary differential operators with involution are studied. For the main problems, theorems on the existence and uniqueness of solutions are proved.

Recursion Operators for Multidimensional Integrable PDEs

We present a novel construction of recursion operators for integrable second-order multidimensional PDEs admitting isospectral scalar Lax pairs with Lax operators being first-order scalar differential operators linear in the spectral parameter. Our approach, illustrated by several examples and applicable to many other PDEs of the kind in question, employs an ansatz for the sought-for recursion operator of the equation under study based on the Lax pair for the latter.

Two-scale cut-and-projection convergence for quasiperiodic monotone operators

Averaging a certain class of quasiperiodic monotone operators can be simplified to the periodic homogenization setting by mapping the original quasiperiodic structure onto a periodic structure in a higher dimensional space using the cut-and projection method. We characterize the cut-and-projection convergence limit of the nonlinear monotone partial differential operator −divσx,Rxη,∇uη for a bounded sequence uη in W01,p(Ω), where 1<p<∞, and Ω is a bounded open subset in Rn with Lipschitz boundary. We identify the homogenized problem with a local equation defined on a hyperplane, or a lower dimensional plane in the higher-dimensional space. A new corrector result is established.

Uncertainty Propagation for General Stochastic Hybrid Systems on Compact Lie Groups

This paper deals with uncertainty propagation of general stochastic hybrid systems (GSHS) where the continuous state space is a compact Lie group. A computational framework is proposed to solve the Fokker-Planck (FP) equation that describes the time evolution of the probability density function for the state of GSHS. The FP equation is split into two parts: the partial differential operator corresponding to the continuous dynamics, and the integral operator arising from the discrete dynamics. These two parts are solved alternatively using the operator splitting technique. Specifically, the partial differential equation is solved by the spectral method where the density function is decomposed into a linear combination of a complete orthonormal function basis brought forth by the Peter-Weyl theorem, thereby resulting an ordinary differential equation. Next, the integral equation is solved by approximating the integral by a finite summation using a quadrature rule. The proposed method is then applied to a three-dimensional rigid body pendulum colliding with a wall, evolving on the product of the three-dimensional special orthogonal group and the Euclidean space. It is illustrated that the proposed method exhibits numerical results consistent with a Monte Carlo simulation, while explicitly generating the density function that carries the complete stochastic information of the hybrid state.

Conservativeness and uniqueness of invariant measures related to non-symmetric divergence type operators

ABSTRACT We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by -uniqueness and imply that a given infinitesimally invariant measure becomes an invariant measure. We explore further conditions on the coefficients of the partial differential operators that ensure the uniqueness of the invariant measure beyond the case where the corresponding semigroups are recurrent. A main observation is that for conservativeness and uniqueness of invariant measures in this article, no growth conditions are required for the partial derivatives related to the anti-symmetric matrix of functions that determine a part of the drift coefficient. As stochastic counterparts, our results can be applied to show not only the existence of a pathwise unique and strong solution up to infinity to a corresponding Itô-SDE, but also the existence and uniqueness of invariant measures for the family of strong solutions.

On symmetric partial differential operators

Let \(\sigma _{1}, \dots , \sigma _{k}\) be the elementary symmetric functions of the complex variables \(x_{1}, \dots , x_{k}\). We say that \(F \in {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\) is a trace function if their exists \(f \in {\mathbb {C}}[z]\) such that \(F(\sigma _{1}, \dots , \sigma _{k}) = \sum _{j=1}^{k} f(x_{j})\) for all \(\sigma \in {\mathbb {C}}^{k}\). We give an explicit finite family of second order differential operators in the Weyl algebra \(W_{2}:= {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\langle \frac{\partial }{\partial \sigma _{1}}, \dots , \frac{\partial }{\partial \sigma _{k}}\rangle \) which generates the left ideal in \(W_{2}\) of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As an application, we obtain for each integer k a holonomic system which is a quotient of \(W_{2}\) by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: \(z^{k} + \sum _{h=1}^{k} (-1)^{h}\sigma _{h}z^{k-h} = 0\).

Global synchronization for BAM delayed reaction-diffusion neural networks with fractional partial differential operator

This paper is devoted to the exploration of the global asymptotic synchronization (GAS) and global Mittag–Leffler synchronization (GMLS) issues on bidirectional associative memory reaction-diffusion neural networks (BAMRDNNs), which include a Caputo’s fractional partial differential operator, leakage and discrete delays. By means of the Green’s theorem, Jensen’s integral inequality and Lyapunov functional approach, several synchronization criteria are established by using the hybrid controllers. The presented results are described in the algebraic inequalities form, which can immensely reduce the computational complexity. Moreover, these results reveal the impact of the network system coefficient matrices on GAS and GMLS. Finally, in the three-dimensional space, the effectiveness and practicability of these synchronization criteria are confirmed by choosing different derivative orders and diffusion coefficients.

The Completeness of Classical Electromagnetic Theory

The core of classical electromagnetic field theory is Maxwell electromagnetic field equations. Each of its formulas is supported by experimental facts, and its mathematical logic is rigorous and self-consistent, which has been tested for a long time in application.Therefore, its correctness is beyond doubt. In this paper, on the premise of affirming the great significance and important contribution of Maxwell equations(ME), the possible problems and solutions are discussed. Since ME and its derived wave equation (MWE) are vectorized partial differential equations, it is difficult to solve them strictly. In this paper, the solutions of vectorized ME and vectorized MWE are discussed in depth, and the necessity of Hansen function analysis is illustrated before the concrete calculation of engineering structures. Another method is to use vector partial differential operator theory to derive the standard basic equations expressed by state functions, and change the traditional calculation in Euclid space to the calculation in wave function space. Singularities and singularity are also avoided in the process. This paper argues that the completeness and modernity of traditional electromagnetic theory can only be improved by introducing the concept of particle, because the wave field processed by ME is actually a quantum field of light. The motion equation of electromagnetic field in classical theory is ME. In quantum theory, the field vector is an operator, governed by ME. The photon theory successfully explained the photoelectric effect in a way that ME could not. But in the mainstream theory, the photon rest mass is equal to zero, which blocks the way to add particle concept to ME. In 1936, Proca equations assumed ≠0, and with the support of quantum electrodynamics(QED), ME was improved. In this paper, the Proca equations are approved and the Proca wave equation (PWE) is further derived. In short, it reflects the adherence to the concept of wave-particle duality. The extended Maxwell equations proposed by Chinese scientist recently should be emphasized and discussed in this paper.

Sobolev regularity for linear growth functionals acting on ℂ-elliptic operators

Abstract In this paper, we prove the higher Sobolev regularity of minimizers for convex integral functionals evaluated on linear differential operators of order one. This work intends to generalize the already existing theory for the cases of full and symmetric gradients to the entire class of ${\mathbb C}$-elliptic operators therein including the trace-free symmetric gradient for dimension $n \geq 3$.

Parameter Estimation for Several Types of Linear Partial Differential Equations Based on Gaussian Processes

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously.