Partial Differential Operators Research Articles

An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators

For a constant coefficient partial differential operator P(D) with a single characteristic direction, such as the time-dependent free Schrödinger operator as well as non-degenerate parabolic differential operators like the heat operator, we characterize when open subsets X1⊆X2 of Rd form a P-Runge pair. The presented condition does not require any kind of regularity of the boundaries of X1 nor X2. As part of our result we prove that for a large class of non-elliptic operators P(D) there are smooth solutions u of the equation P(D)u=0 on Rd with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for P(D).

A sage package for the symbolic-numeric factorization of linear differential operators

We present a SageMath implementation of the symbolic-numeric algorithm introduced by van der Hoeven in 2007 for factoring linear differential operators whose coefficients are rational functions.

Squared eigenfunction symmetry of the DmKP hierarchy and its constraint

AbstractIn this paper, squared eigenfunction symmetry of the differential‐difference modified Kadomtsev–Petviashvili (DmKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the DmKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R‐Toda) hierarchy. An invertible transformation is given to connect the positive and negative R‐Toda hierarchies. The positive R‐Toda hierarchy is reduced to the differential‐difference Burgers hierarchy. We also consider another DmKP hierarchy and show that its squared eigenfunction symmetry constraint gives rise to the Volterra hierarchy. In addition, we revisit the Ragnisco–Tu hierarchy which is a squared eigenfunction symmetry constraint of the differential‐difference Kadomtsev–Petviashvili (DKP) system. It was thought the Ragnisco–Tu hierarchy did not exist one‐field reduction, but here we find a one‐field reduction to reduce the hierarchy to the Volterra hierarchy. Besides, the differential‐difference Burgers hierarchy is also investigated in the Appendix. A multidimensionally consistent three‐point discrete Burgers equation is given.

Infinite-dimensional linear algebra and solvability of partial differential equations

We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective.

Existence and Regularity of Periodic Solutions for a Class of Partial Differential Operators

Existence and Regularity of Periodic Solutions for a Class of Partial Differential Operators

Stabilization of advection dominated problems through a generalized finite element method

Traditional finite element approaches are well-known to introduce spurious oscillations when applied to advection dominated problems. We explore alleviation of this issue from the perspective of a generalized finite element formulation, which enables stabilization of a linear differential operator through enrichments based on fundamental solutions. This is demonstrated through application to steady/unsteady one- and two-dimensional advection-diffusion problems. Here, boundary layer development is efficiently captured using a set of exponential functions derived from fundamental solutions to the problems considered. Results demonstrate smooth, numerical solutions with convergence observed to be in relative agreement with expected convergence rates for elliptic problems. Furthermore, significantly improved error levels are observed compared to traditional finite element formulations. Additional insights in improvements offered by the generalized finite element method are illuminated using a consistent decomposition of the variational multiscale method, enabling comparison with classical stabilized methods.

PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNs

Spherical signals exist in many applications, e.g., planetary data, LiDAR scans and digitalization of 3D objects, calling for models that can process spherical data effectively. It does not perform well when simply projecting spherical data into the 2D plane and then using planar convolution neural networks (CNNs), because of the distortion from projection and ineffective translation equivariance. Actually, good principles of designing spherical CNNs are avoiding distortions and converting the shift equivariance property in planar CNNs to rotation equivariance in the spherical domain. In this work, we use partial differential operators (PDOs) to design a spherical equivariant CNN, PDO-eS2CNN, which is exactly rotation equivariant in the continuous domain. We then discretize PDO-eS2CNNs, and analyze the equivariance error resulted from discretization. This is the first time that the equivariance error is theoretically analyzed in the spherical domain. In experiments, PDO-eS2CNNs show greater parameter efficiency and outperform other spherical CNNs significantly on several tasks.

Floquet problem and center manifold reduction for ordinary differential operators with periodic coefficients in Hilbert spaces

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system of ordinary differential equations with constant coefficients and an infinite-dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders and for elliptic problems in quasicylinders obtained by P. Kuchment and S. A. Nazarov, respectively. As an application, a center manifold reduction is presented for a class of nonlinear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients explored by A. Mielke.

Lagrange multipliers for evolution problems with constraints on the derivatives

The existence of generalized Lagrange multipliers is proved for a class of evolution problems for linear differential operators of various types subject to constraints on the derivatives. Those Lagrange multipliers and the respective solutions are stable for the vanishing of the coercive parameter and are naturally associated with evolution variational inequalities with time-dependent convex sets of gradient type. These results are applied to the sandpile problem, to superconductivity problems, to flows of thick fluids, to problems with the biharmonic operator, and to first order vector fields of subelliptic type.

Existence of solutions for implicit obstacle problems involving nonhomogeneous partial differential operators and multivalued terms

In this article, we study an implicit obstacle problem with a nonlinear nonhomogeneous partial differential operator and a multivalued operator which is described by a generalized gradient. Under quite general assumptions on the data, and employing Kluge's fixed point principle for multivalued operators, Minty echnique and a surjectivity theorem, we prove that the set of weak solutions to the problem is nonempty, bounded and weakly closed.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/37/abstr.html

On solving nonlinear differential-functional equations of mechanics and other applications

The paper presents the results of a study aimed at creating a method for solving the Cauchy problem and the stationary problem for equations containing a linear differential operator with constant coefficients and a nonlinear analytical functional. The method of Volterra-Frechet series is used to obtain a stationary solution of a second-order differential equation with an analytical functional describing the oscillations of a nonlinear vibrator with hereditary behavior. The behavior of the solution and the dependence of energy losses on physical parameters are studied.

A trace formula for higher order ordinary differential operators

We obtain a first-order trace formula for a higher order differential operator on a closed interval in the case where the perturbation operator is the operator of multiplication by a finite complex-valued charge. For operators of even orders , the result contains a term of new type, previously unknown. Bibliography: 15 titles.

Jacobi polynomials and some connection formulas in terms of the action of linear differential operators

In this paper, we introduce the concept of the $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials, where $\mathbb{J}_{\alpha, \beta}$ is the raising operator $\mathbb{J}_{\alpha, \beta}:=(x^2-1)\frac{d}{dx}+ \big((\alpha+\beta)x-\alpha+\beta\big)\mathbb{I}$, with $\alpha$ and $\beta$ nonzero complex numbers and $\mathbb{I}$ representing the identity operator. Then, we show that the Jacobi polynomials $P^{(\alpha, \beta)}_n(x),\ n\geq0$, where $\alpha, \beta\in\mathbb{C}\backslash\{0,-1,-2,\ldots\}$, $(\alpha+\beta\neq-m, \ m\geq0)$, are the only $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials. As an application, we give some new connection formulas satisfied by the polynomials solution of our problem.

Multidimensional sampling theorems for multivariate discrete transforms

This paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green’s function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings.

Computing Spectral Measures and Spectral Types

Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular continuous parts) often characterise relevant physical properties such as the long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrödinger equation on L^2({mathbb {R}}^d). Computational spectral problems in infinite dimensions have led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. We classify the computation of measures, measure decompositions, types of spectra, functional calculus, and Radon–Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from orthogonal polynomials on the real line and the unit circle (giving, for example, computational realisations of Favard’s theorem and Verblunsky’s theorem, respectively), and are applied to evolution equations on a two-dimensional quasicrystal.

Integro-Differential Equation with a Higher-Order Two-Dimensional Whitham Operator

We examine the unique solvability of an initial-value problem for a certain higher-order quasilinear partial integro-differential equation with a degenerate kernel. Expressing the higher-order partial integro-differential operator as the superposition of first-order partial differential operators, we represent the integro-differential equation considered as an ordinary integro-differential equations that describes the change of the unknown function along characteristics. Using the method of successive approximations, we prove the unique solvability of the initial-value problem and obtain an estimate for the convergence rate of the Picard iterative process.

Initial-Value Problem for a Higher-Order Quasilinear Partial Differential Equation

We examine an initial-value problem for a certain higher-order quasilinear partial differential equation. Expressing the partial differential operator as the superposition of first-order operators, we apply methods of solution of first-order equations. We prove the unique solvability of the initial-value problem considered.

A method for calculating differential constitutive equation of linear viscoelastic composite model.

The stress-strain behaviors of viscoelastic materials are often simulated using a model composed of various combinations of springs and dampers. With the increase in the number of springs and dampers, the viscoelastic characteristics of the model will approach those of the actual material. This study discusses how to obtain the differential constitutive equation of a viscoelastic model composed of any number of springs and dampers. First, the general viscoelastic model is regarded as the combination of various Kelvin units. The viscoelastic model is then transformed into a digraph. Based on the relationships between the independent path of the digraph and the strain equation of the viscoelastic model and between the closed enclosure and the stress equation, the derivation of the constitutive equation is transformed into operations involving the incidence matrix of the digraph. Finally, the coefficients of the linear differential operator of the constitutive equation of the viscoelastic model can be obtained by block matrix operations. This method is suitable for computer programming and has a certain significance for accurately constructing viscoelastic models of engineering materials.

Global solvability for first order real linear partial differential operators III

In previous articles, the authors provided sufficient conditions for global solvability of linear differential operators defined on a n-dimensional manifold, with real principal part, under the hypothesis of the existence of one hyperbolic singular point. In this work, for n=2, we extend the result for non-degenerate singular semi-hyperbolic point.

Maximally Dissipative Differential Operators of First Order in the Weighted Hilbert Space

Maximally Dissipative Differential Operators of First Order in the Weighted Hilbert Space