Differential Operators Research Articles

Double honeycomb symmetry: Effective-field theory study

The double hexagonal lattice model is a new statistical model constructed by connecting two-dimensional hexagonal structured materials such as Graphene and Lie symmetries. The double hexagonal structure of this model appears in the G2 symmetry of Lie algebras. To investigate the magnetic properties of this new lattice model, an Ising-1/2 model was constructed with spin values σ = ±1. An effective-field theory (EFT) method was used with the differential operator technique to formulate the identity of the spin system, based on Callen's work. Different phase diagrams were established by varying the double-exchange interaction couplings and the super-exchange interaction coupling present in the model. The accurate results obtained by the EFT method were compared with those found by the mean-field approximation (MFA) method. The symmetry of the MFA diagrams was broken in the EFT diagrams. The critical temperature behavior was found to depend on the type of super-exchange ferromagnetism.

Criterion for ellipticity on Heisenberg group

We provide a semi-constructive criterion for ellipticity of the differential operator on the Heisenberg group H1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^1.$$\\end{document}

Resolvent expansions of 3D magnetic Schrödinger operators and Pauli operators

We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in L2(R3) and L2(R3;C2), respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g. finite rank perturbations, are discussed as well.

Discrete inverse problem for a hyperbolic equation, properties of the solution to a discrete direct and auxiliary discrete problems

This paper considers the formulation of a discrete inverse problem for a hyperbolic equation. First, the continuous inverse problem is reduced to a convenient form for research. In the inverse problem, the required function is considered even. Since the Dirac delta function is present in the problem data, the structure of a generalized solution to the Cauchy problem for a hyperbolic equation is determined. The solution to the Cauchy problem for a hyperbolic equation is determined only for positive values in time, therefore the solution to the Cauchy problem for negative values in time is determined using odd continuation. After some transformations, the formulation of the continuous inverse problem is reduced to a form convenient for research. A grid domain is introduced, and for all functions in the problem statement the corresponding grid functions and a discrete analogue of the Dirac delta function are determined. Differential operators, initial conditions and additional data of the inverse problem are approximated by finite differences. Assuming that a solution to the discrete inverse problem exists, we prove the data lemma of the discrete inverse problem. In order to study the discrete inverse problem for a hyperbolic equation, a theorem on the existence and uniqueness of the discrete direct problem is proved, as well as on the properties of the solution to this discrete problem. In the course of proving the theorem, a discrete analogue of d'Alembert's formula for solving the Cauchy problem for a hyperbolic equation was obtained. The theorem on the existence of a unique solution to the auxiliary discrete problem and its properties is proved.

Study the properties of spectral characteristics and eigenfunctions for Sturm-Liouville boundary value problems

ABSTRACT: In this study, we provide an overview of the Sturm-Liouville operator’s spectral theory on a finite interval. Also, we study the main spectral characteristics for the second-order differential operator, and we show that the eigenvalues and eigenfunctions are real and the characteristic functions are simple. The Sturm-Liouville eigenvalues of that problem are non-degenerate; for every eigenvalue, there exists only one linearly independent solution.

Research on Viscoelastic Impact Damage Constitutive Model and Its Behavioral Characteristics

To investigate the damage characteristics of viscoelastic materials under various impact scenarios, this study presents a viscoelastic impact damage constitutive model that incorporates a material coefficient, a variable fractional-order differential operator, and a damage factor. Split Hopkinson Pressure Bar experiments are conducted under single-impact and repeated-impact conditions. The results indicate that nonlinear features in stress-strain curves at different stages, exhibiting prominent strain rate dependency near the stress peak point in the single-impact scenario. In the repeated-impact scenario, the stress peak points decrease as the impact times increase, resulting in a corresponding decline in the material's load-bearing capacity. Furthermore, strain rate escalates the strain rate increases with an increase in impact times, making the material more susceptible to deformation. Parameter fitting and Sobol sensitivity analysis utilizing experimental data from both scenarios have been employed to explore the effects and patterns of single-parameter and multi-parameter coupling on the model, accurately delineating their influence. The findings reveal a strong coupling relationship among various parameters of the viscoelastic impact damage constitutive model in multiple impact scenarios, showing higher sensitivity to fluctuations in the viscoelastic coefficient and damage factor. These analyses have demonstrated the accuracy of the viscoelastic impact damage constitutive model in capturing dynamic impact damage characteristics of viscoelastic materials across various scenarios.

Bernstein Operational Matrix for Solving Variable Order Fractional Differential Equations with Generalized Caputo-Type

In this research, we present novel generalized derivative founded on the newly constructed (NGCFVODs) novel generalized Caputo Fractional variable order derivatives. Utilizing these operators, a numerical methodology has been formulated to resolve the Fractional Variable Order Differential Equations (FVODEs). We estimate the solution for (FVODEs) by employing bernstein polynomials as foundational vectors. We have further expanded the derivative operational matrix of bernstein Polynomials (bPs) to generalized derivative an operational matrix in sense of (NGCFVODs). The efficiency of developed numerical methodology is tested by a taking a various test examples. We also a compare results of our suggested approach with the methodologies existing in academic papers. In this study the Fractional variable order differential operator of new generalized Caputo was described by three categories: (i) various value in ρ and Fractional variable order a parameter, (ii) various value in a fractional parameters while Fractional variable order and ρ parameter are fixed, and (iii) various value in Fractional variable order parameter controlling fractional and ρ parameter.

Torricelli’s Law in Fractal Space–Time Continuum

A new formulation of Torricelli’s law in a fractal space–time continuum is developed to compute the water discharge in fractal reservoirs. Fractal Torricelli’s law is obtained by applying fractal continuum calculus concepts using local fractional differential operators. The model obtained can be used to describe the behavior of real flows, considering the losses in non-conventional reservoirs, taking into account two additional fractal parameters α and β in the spatial and temporal fractal continuum derivatives, respectively. This model is applied to the flows in reservoirs with structures of three-dimensional deterministic fractals, such as inverse Menger sponge, Sierpinski cube, and Cantor dust. The results of the level water discharge H(t) are presented as a curve series, showing the impact and influence of fluid flow in naturally fractured reservoirs that posses self-similar properties.

Applications of Differential Inequalities Employing A New ConvolutedOperator Constructed by The Supertrigonometric Function

In order to examine various geometric features, we expanded the supertrigonometric function (STF) and superhyperbolic function (SHF) into the open unit disk. A convolution differential operator of the STF provides the formulas. The suggested operator works with both integral and double differential inequalities. As a result, we present a collection of findings that includes recent works. The idea ofsubordination and superordination serves as a guide for our method, and we then developed the primary conclusion as a double side’s theorem.

Fractional Differential Operator Based on Quantum Calculus and Bi-Close-to-Convex Functions

In this article, we first consider the fractional q-differential operator and the λ,q-fractional differintegral operator Dqλ:A→A. Using the λ,q-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass S*q,β,λ of starlike functions of order β and the class CΣλ,qα of bi-close-to-convex functions of order β. We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class S*q,β,λ. Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order β. We also investigate the erratic behavior of the initial coefficients in the class CΣλ,qα of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research.

Exploring the dynamics of nonlocal coupled systems of fractional [formula omitted]-integro-differential equations with infinite delay

In this study, we explore a coupled system of fractional integro-differential equations with infinite delay and nonlocal conditions. This system encompasses classical derivatives of different orders and the fractional derivative of Caputo-Fabrizio type, as well as the fractional integral of the q-Riemann-Liouville operator. We introduce a novel definition of the Caputo and Fabrizio differential operators, enhancing the mathematical formulation. Our main focus is to investigate the system's fundamental properties, including existence, uniqueness, and continuous dependence. Through rigorous mathematical analysis, we establish the existence and uniqueness of solutions and examine how small perturbations in initial conditions or parameters impact the solutions. For the numerical aspect, we use the finite-trapezoidal approach, a reliable method for solving fractional integro-differential equations. We provide a concise explanation of the approach and demonstrate its effectiveness through two numerical examples. Overall, this comprehensive study contributes to the understanding of coupled systems with fractional derivatives and infinite delays, with implications for various scientific and engineering fields.

Z2 classification of FTR symmetric differential operators and obstruction to Anderson localization

This paper concerns the Z2 classification of Fermionic Time-Reversal (FTR) symmetric partial differential Hamiltonians on the Euclidean plane. We consider the setting of two insulators separated by an interface. Hamiltonians that are invariant with respect to spatial translations along the interface are classified into two categories depending on whether they may or may not be gapped by continuous deformations. Introducing a related odd-symmetric Fredholm operator, we show that the classification is stable against FTR-symmetric perturbations. The property that non-trivial Hamiltonians cannot be gapped may be interpreted as a topological obstruction to Anderson localization: no matter how much (spatially compactly supported) perturbations are present in the system, a certain amount of transmission in both directions is guaranteed in the nontrivial phase. We present a scattering theory for such systems and show numerically that transmission is indeed guaranteed in the presence of FTR-symmetric perturbations while it no longer is for non-symmetric fluctuations.

Nonlocal numerical simulation of thermoelectric coupling field by using peridynamic differential operator

Nonlocal numerical simulation of thermoelectric coupling field by using peridynamic differential operator

Структура электрического поля в конвективно-турбулентном приземном слое атмосферы

Electrodynamic models of the surface layer of the atmosphere, taking into account turbulent and convective transport, are constructed and studied. The models are based on the so-called total current equation derived from the equations of the theory of the electrode effect in the atmosphere. From the point of view of mathematics, this is a partial differential equation of a parabolic type with variable coefficients. Since in this type of equations the differential operator is not necessarily the Sturm-Liouville operator, in appropriate cases it is necessary to use different approaches to integrating the total current equation. This, in turn, makes it necessary to consider various hypotheses about the ratio of the intensities of local factors that affect the structure of the equation. The paper considers several cases of the model corresponding to the convective turbulent and convective-turbulent electrode layer. Analytical solutions of the stationary equation of the total electric current are obtained. in various approximations. Electric field strength profiles in the surface layer of the atmosphere for various physical conditions are constructed and studied.

Data-driven identification of stable sparse differential operators using constrained regression

Identifying differential operators from data is essential for the mathematical modeling of complex physical and biological systems where massive datasets are available. These operators must be stable for accurate predictions for dynamics forecasting problems. In this article, we propose a novel methodology for learning differential operators that are theoretically linearly stable and have sparsity patterns of common discretization schemes. These differential operators are obtained by solving a constrained regression problem, involving local constraints to ensure the linear stability of the global dynamical system. We further extend this approach for learning nonlinear differential operators by determining linear stability constraints for linearized equations around an equilibrium point. The applicability of the proposed method is demonstrated for both linear and nonlinear partial differential equations such as 1-D scalar advection-diffusion equation, 1-D Burgers equation and 2-D advection equation. The results indicated that solutions to constrained regression problems with linear stability constraints provide accurate and linearly stable sparse differential operators.

Constrained multiobjective optimization design for ordinary shovel attachment of hydraulic excavator based on evolutionary algorithm

Shovel attachment is one of the most commonly used working attachments in hydraulic excavators. Its mechanical design remains a challenging optimization problem. To tackle this issue, the optimizing model of ordinary shovel attachment is firstly established which contains twenty-three design variables, six objective functions, and sixty-three constraints. Then, an improved decomposition-based constrained many-objective evolution algorithm is proposed, which integrates advanced push & pull search constraint handling technique and differential evolution operator to deal with the optimization problems of shovel attachment. Finally, the proposed algorithm has demonstrated its superior optimization ability for designing shovel attachments through the case studies of a 70-ton face-shovel hydraulic excavator, outperforming seventeen state-of-the-art constrained multiobjective evolutionary algorithms. Furthermore, the first quantitative comparison between ordinary shovel attachment and TriPower shovel attachment under the same main machine is presented by the optimized results. The result demonstrates the complexity of the optimal design for shovel attachment and the effectiveness of multiobjective evolutionary algorithm.

Invariance of Iterated Global Differential Operator for Slice Monogenic Functions

Invariance of Iterated Global Differential Operator for Slice Monogenic Functions

Antiperiodic Solutions for Impulsive ω-Weighted ϱ–Hilfer Fractional Differential Inclusions in Banach Spaces

In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, D0,tσ,v,ϱ,ω, of order σ∈(1,2), in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator D0,tσ,v,ϱ,ω, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions.

Eyring–Kramers law for Fokker–Planck type differential operators

We consider Fokker–Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions ensuring suitable resolvent estimates, we prove Eyring–Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes and avoids supersymmetry or PT-symmetry assumptions.

Basis for high order divergence-free finite element spaces

A method classically used in the lower polynomial degree for the construction of a finite element basis of the space of divergence-free functions is here extended to any polynomial degree for a bounded domain without topological restrictions. The method uses graphs associated with two differential operators: the gradient and the divergence, and selects the basis using a spanning tree of the first graph. It can be applied for the two main families of degrees of freedom, weights and moments, used to express finite element differential forms.