Differential Operators Research Articles

On Certain Classes of Bi-Bazilevic Functions Defined by q-Ruscheweyh Differential Operator

In this paper, we make use of the concept of fractional q-calculus to introduce two new classes of biBazilevic functions involving q-Ruscheweyh differential operator that are subordinate to Gegenbauer polynomials and q-analogue of hyperbolic tangent functions. This study explores the characteristics and behaviors of these functions, offering estimates for the modulus of the initial Taylor series coefficients a2 and a3 within this specific class and their various subclasses. Additionally, this study delves into the classical Fekete-Szeg¨o functional problem concerning functions f that are part of our newly defined class and several of their subclasses.

Harnessing dynamic graph differential operators for efficient data-driven wind prediction

Harnessing dynamic graph differential operators for efficient data-driven wind prediction

Optimal Modeling of Oscillatory Systems for Engineering Technology and Quality Stem Education

Introduction: The article considers qualitative issues of modern education in the field of engineering technologies (SDGs 4,9,11). The problems of optimal modeling of oscillatory systems, the dynamics of which are described by differential equations with partial derivatives of the second and sixth orders with different boundary conditions corresponding to the positions of the ends of a circular arch, rods and beams, are solved. In the problems of optimal synthesis of controls, the minimization of the energy quadratic functionals of the physical system under consideration, associated with the problems of damping unwanted oscillations (flutter phenomenon) or with the excitation of oscillations of a given frequency [1,2,3], are taken as the optimality criterion. The Riesz basis of eigen-and-adjoint elements (e.g.e.) is established and applied to a nonlocal boundary value problem [4]. Using the method of spectral decomposition of non-self-adjoint differential operators, an explicit solution to the nonlinear Riccati operator equation is found, which is of great theoretical and practical importance in mathematical modeling of optimal controllers. To study engineering and technological processes, an optimal system orthogonalization scheme, i.e. the Riesz basis, based on the Schmidt orthogonalization process, is proposed. Objective: The purpose of this work is to familiarize readers of the educational system and specialists with some methods of optimal control and modeling, and study theoretical issues of new problems of optimal design of oscillatory systems; and to propose methods for calculating optimal parameters for engineering and technological calculations of a stable structure and implementation in the educational process of training specialists with high-quality education (for example, STEM education) [5] . Theoretical Framework: The theoretical basis of the research is the theory of optimal control of systems with distributed parameters. Method: The methods for studying the problem of optimal modeling of oscillatory systems are optimal control methods, dynamic programming and spectral decomposition of differential operators. Results and Discussion: Abandoned in this paper, the problem of optimal modeling of oscillations of rods (circular arch), the state of which is described by a sixth-order differential equation, is solved. The problem of optimal modeling of oscillations of systems whose states are described by a second-order equation with non-self-adjoint boundary conditions of the Bitsadze-Samarskii type has been solved. The results of the study can reveal such qualitative key moments in the design of an oscillatory system, both in the problem of suppressing undesirable phenomena and in the problem of synthesizing optimal control. The results obtained can be applied in creating stable engineering structures. Research Implications: The general problem of synthesizing optimal control for systems with distributed parameters has been solved, a special case of which is the problem of optimal modeling of oscillations of a circular arch with various fixings of its ends (including non-self-adjoint problems), encountered in modern engineering structures. Originality/Value: The relevance and value of the research are confirmed by the novelty of the results in the field of optimal control theory for systems with distributed parameters and non-self-adjoint differential operators, and the results obtained can be used by engineers and technologists, researchers, specialists interested in the theory of optimal modeling and its applications in engineering , medical biology, economics, ecology, natural science, etc. , as well as in the training of specialists with high-quality STEM education. according to the calculation of the optimal parameters of oscillatory systems.

Compressive Sensing-based Differential Channel Feedback Scheme Using Subspace Matching Pursuit Algorithm for B5G Wireless Systems

Millimeter wave (mmWave) massive multi-input multi-output (MIMO) systems are the promising technology for next-generation 5G wireless systems and beyond. Sparse signal recovery and channel feedback are challenging and fundamental problems affecting downlink transmission due to the substantial increase in channel matrix size in mmWave systems. To overcome the overhead of the channel and improve CS recovery effectiveness, this article proposes the joint use of the subspace matching search algorithm and differential operation for channel impulse response (CIR). Here, the current CIR is converted to a differential CIR using operations between the current and previous CIRs. The differential CIR is then compressed and fed back to the base station. Subsequently, this differential CIR is recovered using the subspace matching search algorithm. Such a scheme leverages effective structural sparsity through a combination of subspace and differential operations. The adaptive algorithm adaptively selects relevant subspaces based on coefficients. The simulation results show that the proposed scheme reduces channel overhead by 36% and 24% at compression ratios of 25% and 45%, respectively, over different time slots in mmWave massive MIMO systems.

Regularized derivatives — Revisited

The methods used for differentiation of potential field data play an important role in interpretation, as various derivatives are included in many of the transformations often used when interpreting such data (e.g., analytical signal, tilt angles, total gradient, and others). However, the evaluation of horizontal and vertical derivatives is an unstable process. The possible (partial) answer to this instability is the Tikhonov regularization based on incorporating an additional property representing maximum smoothness, which is typically obtained by cascading low-pass filtering — managed by means of a regularization parameter (RP). Here, we develop and compare several ways to evaluate regularized differentiation operators and compare their properties. Among them, a newly introduced form (called general form) plays a very important role. As is typical for regularization algorithms, the crucial step is the setting of a quasioptimal value of the RP, which gives the best possible solution. Several methods of its estimation (focused on the use of [Formula: see text] norms) are presented and the resulting utility becomes the keystone in the valuation of the presented regularized operators.

Some Minkowski’s inequalities involving linear differential operator with associated green function

The fundamental concept of this research article is to establish some new Minkowski’s and associated inequalities through the utilization of a linear differential operator associated to the Green function. Additionally, we demonstrate some useful interconnected inequalities for the linear differential operator, which holds significant relevance in the field of applied mathematics.

Nonhomogeneous singular problems with convection

We consider a Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which has the competing effects of a singular term and of a sign-changing perturbation which is gradient dependent (convective). Using the theory of nonlinear operators of monotone type, we show that the problem has a positive smooth solution.

Extrinsic GJMS operators for submanifolds

We derive extrinsic GJMS operators and Q -curvatures associated to a submanifold of a conformal manifold. The operators are conformally covariant scalar differential operators on the submanifold with leading part a power of the Laplacian in the induced metric. Upon realizing the conformal manifold as the conformal infinity of an asymptotically Poincaré–Einstein space and the submanifold as the boundary of an asymptotically minimal submanifold thereof, these operators arise as obstructions to smooth extension as eigenfunctions of the Laplacian of the induced metric on the minimal submanifold. We derive explicit formulas for the operators of orders 2 and 4. We prove factorization formulas when the original submanifold is a minimal submanifold of an Einstein manifold. We also show how to reformulate the construction in terms of the ambient metric for the conformal manifold, and use this to prove that the operators defined by the factorization formulas are conformally invariant for all orders in all dimensions.

New Class of Complex Models of Materials withPiezoelectric Properties with Differential Constitutive Relations of Fractional Order: An Overview

Rheological complex models of various elastoviscous and viscoelastic fractional-type substances with polarized piezoelectric properties are of interest due to the widespread use of viscoelastic–plastic bodies under loading. The word “overview” used in the title means and corresponds to the content of the manuscript and aims to emphasize that it presents an overview of a new class of complex rheological models of the fractional type of ideal elastoviscous, as well as viscoelastic, materials with piezoelectric properties. Two new elementary rheological elements were introduced: a rheological basic Newton’s element of ideal fluid fractional type and a basic Faraday element of ideal elastic material with the property of polarization under mechanical loading and piezoelectric properties. By incorporating these newly introduced rheological elements into classical complex rheological models, a new class of complex rheological models of materials with piezoelectric properties described by differential fractional-order constitutive relations was obtained. A set of seven new complex rheological models of materials are presented with appropriate structural formulas. Differential constitutive relations of the fractional order, which contain differential operators of the fractional order, are composed. The seven new complex models describe the properties of ideal new materials, which can be elastoviscous solids or viscoelastic fluids. The purpose of the work is to make a theoretical contribution by introducing, designing, and presenting a new class of rheological complex models with appropriate differential constitutive relations of the fractional order. These theoretical results can be the basis for further scientific and applied research. It is especially important to point out the possibility that these models containing a Faradаy element can be used to collect electrical energy for various purposes.

Partial Torque Tensor and Its Building Block Representation for Dynamics of Branching Structures Using Computational Graph

The Euler–Lagrange and Newton–Euler methods are typically used to derive equations of motion for serial-link manipulators. We previously proposed a partial Lagrangian method, which is similar to the Lagrangian method, for handling the equations of motion analytically. Moreover, the proposed method can efficiently handle multi-link analyses, similar to the Newton–Euler method. The partial Lagrangian method organizes the Lagrangian, which is obtained from the link structure, and torque, which is obtained by differential operations, into a table that can be easily handled by both manual calculations and computer analysis. Furthermore, by representing it using a computational graph, it is possible to perform dynamic analysis while maintaining the structure of a system. By observing the intermediate nodes of this computational graph, it is possible to observe how the torque generated at a particular link affects the joint. Organizing the structure with graphs allows us to consider complex systems as a collection of subgraphs, making this method highly compatible with our proposed partial Lagrangian approach. This study shows that the partial torque tensor can be used as an analog of the partial torque table for serial-link systems by interpreting the meaning of the table, i.e., the partial torque, as the interaction between the links in order to simplify the treatment of branching link systems. Due to the use of the partial torque tensor, the dimensions of the tensor correspond one-to-one with the number of branches, allowing the description of any branching system. Furthermore, by using the proposed building block representation, even complex branching systems can be easily designed and analyzed.

COMPLETE HYPERSURFACES WITH LINEARLY RELATED HIGHER ORDER MEAN CURVATURES IN THE HYPERBOLIC SPACE

Abstract In this article, we study the behavior of complete two-sided hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$ . Initially, we introduce the concept of the linearized curvature function $\mathcal {F}_{r,s}$ of a two-sided hypersurface, its associated modified Newton transformation $\mathcal {P}_{r,s}$ and its naturally attached differential operator $\mathcal {L}_{r,s}$ . Then, we obtain two formulas for differential operator $\mathcal {L}_{r,s}$ acting on the height function of a two-sided hypersurface and, for the case where their support functions are related by a negative constant, we derive two new formulas for the Newton transformation $P_{r}$ and the modified Newton transformation $\mathcal {P}_{r,s}$ acting on a gradient of the height function. Finally, these formulas, jointly with suitable maximum principles, enable us to establish our rigidity and nonexistence results concerning complete two-sided hypersurfaces in $\mathbb H^{n+1}$ .

𝐵𝑀𝑂 solvability for divergence-measure equations

In this note, we obtain a sufficient and necessary condition for the existence of solutions of the divergence-measure equation div ⁡ f = μ \begin{equation*} \operatorname {div} f=\mu \end{equation*} in B M O BMO space, by resorting to the tools from harmonic analysis including the H 1 H^1 - B M O BMO dual theory and the boundedness of Riesz transform on B M O BMO . Here μ \mu is a positive Borel measure on R n ( n ≥ 2 ) \mathbb {R}^n\, (n\geq 2) . Moreover, the above result is also applicable to the adjoint equation G m ∗ ( D ) f = μ G_m^{*}(D)f=\mu , where G m ( D ) G_m(D) is the elliptic homogeneous linear differential operator of order m ∈ N + m\in \mathbb {N^+} . Note that for the other end-point space L ∞ L^\infty , only the sufficient condition was known for this adjoint equation.

Existence of solutions for some systems of superdiffusive integro-differential equations in population dynamics depending on the natality and mortality rates

We prove the existence of stationary solutions for some systems of reaction–diffusion-type equations with superdiffusion in the corresponding H 2 spaces. Our method is based on the fixed point theorem when the elliptic problems contain first-order differential operators with and without the Fredholm property, which may depend on the outcome of the competition between the natality and the mortality rates contained in the equations of our systems.

Langlands Duality and Invariant Differential Operators

Langlands Duality and Invariant Differential Operators

Spectral analysis for the almost periodic quadratic pencil with impulse

In this study, we delve into the spectral properties of a pencil of nonself-adjoint second-order differential operators characterized by almost periodic potentials and impulse conditions. Such operators arise in various physical models, particularly in quantum mechanics, where they describe systems with discontinuities in their potentials or boundary conditions. Understanding the spectrum of these operators is crucial for comprehending the stability and dynamics of the associated physical systems. By investigating the spectral gaps and accumulation points we aim to contribute to the broader understanding of non-self-adjoint operator theory and its applications in mathematical physics.

On the Decay of Solutions for the Negative Fractional KdV Equation

We explore the limits of fractional dispersive effects and their incidence in the propagation of polynomial weights. More precisely, we consider the fractional KdV equation when a differential operator of negative order determines the dispersion. We investigate what magnitude of weights and conditions on the initial data that allow solutions of the equation to persist in weighted spaces. As a consequence of our results, it follows that even in the presence of negative dispersion, it is still possible to propagate weights whose maximum magnitude is related to the dispersion of the equation. We also observe that our results in weighted spaces do not follow specific properties and limits that their counterparts with positive dispersion.

Optimization of the Approximate Integration Formula Using the Quadrature Formula

The article explores Sard’s problem of constructing optimal quadrature formulas in the space W(4,0)2(0,1) by using Sobolev’s method. This problem involves two steps: first, calculating the norm of the error functional, and then finding the minimum of this norm by the coefficients of the quadrature formulas. The norm of the error functional is computed using the extremal function. Then, by using the method of Lagrange multipliers, a system of linear equations for the coefficients of the optimal quadrature formulas in the space W(4,0)2(0,1) is derived, and the existence and uniqueness of the solution to this system are discussed. The paper then proceeds to construct the optimal quadrature formula using the discrete analogue D4(hβ) of the high-order differential operator. Finally, the optimal quadrature formulas that are exact for exponential-trigonometric functions are obtained.

Extended Separation Method of Semi‐Fixed Variables Together With Analytical Method for Solving Time Fractional Equation

ABSTRACTIt is well known that investigation of exact solutions on nonlinear fractional partial differential equations (PDEs) is a very difficult work. In this paper, the separation method of semi‐fixed variables is extended. Based on the original separation method, two kinds of new structures of the solutions in a hypothetical way are proposed. With the extended separation method of semi‐fixed variables and the mapping method of Riccati equation combined, a new approach for searching exact solutions on time‐fractional PDEs is introduced. To demonstrate the effect of the extended method, the time‐fractional porous medium equation, time‐fractional Hunter‐Saxton equation, and time‐fractional Fornberg‐Whitham equation are solved under the Riemann‐Liouville fractional differential operator. Different kinds of new exact solutions of the above three equations are obtained. In order to intuitively show the dynamic property of these exact solutions, the 3D‐graphs of some solutions are illustrated as examples. Compared to the previous method, more abundant results can be obtained on solving some complex nonlinear time‐fractional PDEs by use of this extended method.

Physics-informed neural network based on control volumes for solving time-independent problems

Physics-informed neural networks (PINNs) have been employed as a new type of solver of partial differential equations (PDEs). However, PINNs suffer from two limitations that impede their further development. First, PINNs exhibit weak physical constraints that may result in unsatisfactory results for complex physical problems. Second, the differential operation using automatic differentiation (AD) in the loss function may contaminate backpropagated gradients hindering the convergence of neural networks. To address these issues and improve the ability of PINNs, this paper introduces a novel PINN, referred to as CV-PINN, based on control volumes with the collocation points as their geometric centers. In CV-PINN, the physical laws are incorporated in a reformulated loss function in the form of discretized algebraic equations derived by integrating the PDEs over the control volumes by means of the finite volume method (FVM). In this way, the physical constraints are transformed from a single local collocation point to a control volume. Furthermore, the use of algebraic discretized equations in the loss function eliminates the derivative terms and, thereby, avoids the differential operation using AD. To validate the performance of CV-PINN, several benchmark problems are solved. CV-PINN is first used to solve Poisson's equation and the Helmholtz equation in square and irregular domains, respectively. CV-PINN is then used to simulate the lid-driven cavity flow problem. The results demonstrate that CV-PINN can precisely predict the velocity distributions and the primary vortex. The numerical experiments demonstrate that enhanced physical constraints of CV-PINN improve its prediction performance in solving different PDEs.

Numerical approximation and simulation of a Volterra integro‐differential equation with a peridynamic differential operator

AbstractWe study the numerical approximation to a nonlocal Volterra integro‐differential equation, in which the integral term is the convolution product of a positive‐definite kernel and a nonlocal peridynamic differential operator (PDDO). Compared with the classical differential operators, the nonlocal PDDOs describe, for example, discontinuities and have demonstrated more widespread applications. The equation is discretized in space by the Galerkin finite element method, and we accordingly prove its error estimate. We then discretize the equation in time by the backward Euler method, and a positive quadrature rule is combined to approximate the convolution term. The convergence rate of the fully‐discrete finite element scheme is proved, and numerical experiments are carried out to substantiate the theoretical findings.