Partial Differential Operators Research Articles

Normal forms of a class of partial functional differential equations

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals.

Lighter and faster simulations on domains with symmetries

A strategy to improve the performance and reduce the memory footprint of simulations on meshes with spatial reflection symmetries is presented in this work. By using an appropriate mirrored ordering of the unknowns, discrete partial differential operators are represented by matrices with a regular block structure that allows replacing the standard sparse matrix–vector product with a specialised version of the sparse matrix-matrix product, which has a significantly higher arithmetic intensity. Consequently, matrix multiplications are accelerated, whereas their memory footprint is reduced, making massive simulations more affordable. As an example of practical application, we consider the numerical simulation of turbulent incompressible flows using a low-dissipation discretisation on unstructured collocated grids. All the required matrices are classified into three sparsity patterns that correspond to the discrete Laplacian, gradient, and divergence operators. Therefore, the above-mentioned benefits of exploiting spatial reflection symmetries are tested for these three matrices on both CPU and GPU, showing up to 5.0x speed-ups and 8.0x memory savings. Finally, a roofline performance analysis of the symmetry-aware sparse matrix–vector product is presented.

Non-degeneracy and Uniqueness of Periodic Solutions for a Liénard Equation with a Linear Difference Operator

Non-degeneracy and Uniqueness of Periodic Solutions for a Liénard Equation with a Linear Difference Operator

Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins et al (2010 J. Phys. A: Math. Theor. 43 265205). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (Kalnins et al 2011 SIGMA 7 031). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay et al (2009 J. Phys. A: Math. Theor. 42 242001) as well as a deformation discovered by Post et al (2011 J. Phys. A: Math. Theor. 44 505201) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realizations with finite-dimensional irreps which fill a gap in the literature.

Solvability of Vekua-type periodic operators and applications to classical equations

In this note, we investigate Vekua-type periodic operators of the form Pu=Lu−Au−Bū, where L is a constant coefficient partial differential operator. We provide a complete characterization of the necessary and sufficient conditions for the solvability and global hypoellipticity of P. As an application, we provide a comprehensive characterization of Vekua-type operators associated with classical wave, heat, and Laplace equations.

Quasi-local and frequency-robust preconditioners for the Helmholtz first-kind integral equations on the disk

We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in ℝ3. Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolu- tions and can thus be evaluated cheaply in the context of iterative methods. For the Laplace equation (i.e. for the wavenumber k = 0) with Dirichlet condition on the disk and on regular meshes, we prove that the preconditioned linear system has a bounded condition number uniformly in the mesh size. We further provide numerical evidence indicating that the preconditioners also perform well for large values of k and on locally refined meshes.

The Metivier inequality and ultradifferentiable hypoellipticity

AbstractIn 1980, Métivier characterized the analytic (and Gevrey) hypoellipticity of ‐solvable partial linear differential operators by a priori estimates. In this note, we extend this characterization to ultradifferentiable hypoellipticity with respect to Denjoy–Carleman classes given by suitable weight sequences. We also discuss the case when the solutions can be taken as hyperfunctions and present some applications.

Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory

We construct and compare two alternative quantizations, as a time-orderable prefactorization algebra and as an algebraic quantum field theory valued in cochain complexes, of a natural collection of free BV theories on the category of m-dimensional globally hyperbolic Lorentzian manifolds. Our comparison is realized as an explicit isomorphism of time-orderable prefactorization algebras. The key ingredients of our approach are the retarded and advanced Green’s homotopies associated with free BV theories, which generalize retarded and advanced Green’s operators to cochain complexes of linear differential operators.

A new kind of double phase elliptic inclusions with logarithmic perturbation terms II: Applications

This paper studies several special cases of a double phase elliptic inclusion problem (DPEI) that involves a nonlinear and nonhomogeneous partial differential operator with unbalanced growth and logarithmic perturbation terms, and two multivalued functions defined in the domain and its boundary. We establish existence and extremality results, focusing on the following two assumptions:• the multivalued terms are formulated by the Clarke’s generalized subdifferential operators of locally Lipschitz functions;• the multivalued terms are generated by discontinuous multifunctions.When the appropriate multivalued functions are formulated by two interval functions, we develop a unifying method to construct the nontrivial sub- and supersolutions for the inequalities and inclusions under considerations, hence we obtain suitable existence and extremality results.

Mixtures of phase transforming fluids and gases: Phase field model and stabilized isogeometric discretization

Liquid–vapor phase change in the presence of non-condensable gases is a classical problem, which continues to challenge continuum modeling. Here, we propose a new model based on the phase field method, which describes the dynamics of the non-condensable gas, phase change and flow simultaneously. The model is built by extending van der Waals and Korteweg’s theory for phase-transforming mixtures. The model equations have fourth order partial differential operators which presents significant challenges to standard spatial discretization methods. In addition, the isentropic form of the model equations is not hyperbolic at the liquid–gas interface, which inhibits the direct application of most algorithms for systems of hyperbolic equations. We propose a novel numerical scheme based on Isogeometric Analysis and the Taylor–Galerkin method, and also implement a residual based discontinuity capturing scheme. We show numerical results, at micron length scales, to study the accuracy and stability of our algorithm. We study the flow of a shock wave past a liquid droplet to test our numerical algorithm when steep gradients are present in the solution. Finally, we use the model to study the problem of gas–vapor bubble collapse near a solid wall.

Varying coefficients in parallel shared-memory variational splitting solvers for non-stationary Maxwell equations

Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originallye mployed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variationals plitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structureo f the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally usedf or fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regularm odel parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with nonregularm aterial data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element methods imulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.

Real Gas Effects on Receptivity to Roughness in Hypersonic Swept Blunt Flat-Plate Boundary Layers

Temperatures within the boundary layers of high-enthalpy hypersonic flows can soar to thousands or even tens of thousands of degrees, leading to significant real gas phenomena. Although there has been significant research on real gas effects on hypersonic boundary layer stability, their impact on the boundary layer’s receptive stage is still poorly understood. Most aerodynamic boundary layers in flight vehicles are three-dimensional. Because of complex geometry and significant crossflow effects, the crossflow mode in three-dimensional boundary layers is crucial in hypersonic vehicle design. In this study, a linear stability analysis (LST) accounting for chemical nonequilibrium effects (CNE) and its adjoint form (ALST) is developed to investigate the real gas effects on the stability and receptivity of stationary crossflow modes. The results indicate that real gas effects significantly influence the receptivity of stationary crossflow modes. Specifically, chemical nonequilibrium effects destabilize the crossflow modes but reduce the receptivity coefficients of the stationary crossflow modes. The Mach number effect was also investigated. It was found that increasing the Mach number stabilizes the stationary crossflow modes, but the receptivity coefficients increase. As the Mach number progressively rises, these effects alternately dominate, leading to a non-monotonic shift in the transition position.

On the solutions of Fermat type quadratic trinomial equations in ℂ2 generated by first order linear C-SHIFT and partial differential operators

On the solutions of Fermat type quadratic trinomial equations in ℂ2 generated by first order linear C-SHIFT and partial differential operators

REDUCTION OF A LINEAR MATRIX OPERATOR OF DIFFERENTIATION IN MANY VARIABLES TO AN OPERATOR WITH AN ORDINARY DIFFERENTIATION OPERATOR

Abstract. In the theory of equations with a linear partial differential operator of the first order, the question of differentiation and integration of vector functions is associated with considering them alternately along the characteristics of the operator, and the representation of their integrals by characteristics in the form of functions of many variables depended on the characteristic first integrals of the operator. The difficulty in this matter was to manage the characteristics and the first integrals for each of their variables so that they were in their places. Such control was implemented by projectors having pairwise ductile properties. The main results of this study are: 1) the use of the reduction method to reduce the matrix operator in partial derivatives to an operator with ordinary differentiations for one variable and 2) the use of projectors in matters of the composition of functions from many variables and the characteristics of the operator for each of the variables. These novelties of the research have a serious perspective.

The inhomogeneous complex partial differential equations for bi-polyanalytic functions

<abstract><p>In this paper, we study a Riemann-Hilbert problem related to inhomogeneous complex partial differential operators of higher order on the unit disk. Applying the Cauchy-Pompeiu formula, we find out the solvable conditions and obtain the representation of the solutions. Then, we investigate the boundary value problems for bi-polyanalytic functions with the Dirichlet and Riemann-Hilbert boundary conditions, obtain the specific solution and the solvable conditions, and extend the conclusion to the corresponding higher-order problems. Therefore, we obtain the solution to the half-Neumann problem of higher order for bi-polyanalytic functions.</p></abstract>

Effective orientifolds from broken supersymmetry

We recently proposed a class of type IIB vacua that yield, at low energies, four–dimensional Minkowski spaces with broken supersymmetry and a constant string coupling. They are compactifications with an internal five-torus bearing a five–form flux Φ and warp factors depending on a single coordinate. The breaking of supersymmetry occurs when the internal space includes a finite interval. A probe-brane analysis revealed a gravitational repulsion and a charge attraction of equal magnitude from the left end of the interval, together with a singularity at the other end. Here we complete the analysis revealing the presence, at one end, of an effective of negative tension and positive five–form charge. We also determine the values of these quantities, showing that , and characterize the singularity present at the other end of the interval, which hosts an opposite charge. Finally, we discuss various forms of the gravity action in the presence of a boundary and identify a self–adjoint form for its fluctuations. Invited contribution to the special issue of Journal of Physics A: Mathematical and Theoretical on ‘Fields, Gravity, Strings and Beyond: In Memory of Stanley Deser’

A second order quadratic integral inequality with an application to ordinary differential operators

A second order quadratic integral inequality with an application to ordinary differential operators

Spectral asymptotics of elliptic operators on manifolds

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator [Formula: see text] directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function [Formula: see text] and the heat trace [Formula: see text]. The kernel [Formula: see text] of the heat semigroup [Formula: see text], called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as [Formula: see text], that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as [Formula: see text], that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

Investigation of individual problems of calculation of drainage pipes in complex geotechnical conditions

The article notes that changes in the loads acting on the drainage pipe (pipeline) in accordance with the relative ability of the drainage pipe and the ground filling to deform are essential. To determine the magnitude of the loads on the drainage pipeline laid in the trench, the formula proposed by A. Marston most accurately corresponds to the results of experiments and is convenient for engineering practice. The perforation of the walls of the drainage pipe affects their strength not only when the pipes are crushed, but also for the conditions of transportation and installation of pipes. The contact interaction of the cross-section of the drainage pipe with the soil base is investigated, in this case, the interaction of cylindrical (having a semicircular contact) or reloid (having a contact of the circle sector) shells lying on an elastic soil base is considered. A differential equation is considered to describe the deformation of an elastic shell, which includes the stiffness parameter, linear differential operators, components of displacement of the neutral axis of the drainage pipe, external load and dimensionless coordinates. For the elastic base models of E.Winkler and V.Z.Vlasov, a general solution of the differential equation describing the deformation of the drainage pipe is proposed. The solution of this equation and the external load is presented in the form of double or single trigonometric series, depending on the task (two-dimensional or one-dimensional). A horizontally lying cylindrical (or reloid) shell (drainage pipe) supported on a ground base is considered. The contact pressure is determined by summing the Fourier coefficients. This solution additionally takes into account the distribution capacity of the ground base of the drainage pipe.

Estimates of Integrally Bounded Solutions of Linear Differential Inequalities

We study integrally bounded solutions of the differential equation A(x)=z, where A is a linear differential operator of order l defined on functions x:R→H (R=(−∞,∞), H () and H is a finite-dimensional Euclidean space). The right-hand side z is an integrally bounded function on R ranging in H and satisfying the inequality (ψ(t),z(t))≥δ|z(t)|, t∈R, δ0. Conditions are given on the operator A and the function ψ:R→H that guarantee an inverse inequality of the following form for the solutions x under consideration: ∫τ+1τ|x(l)(t)|dt≤c∫τ+2τ−1|x(t)|dt, where the constant is independent of the choice of a real number t and function x.