Lagrange Identity Research Articles

A Study on Non-Homogeneous Multiplicative Boundary Value Problems

The paper deals with the non-homogeneous boundary value problems established in the multiplicative calculus. For this problem, the multiplicative meanings of concepts such as adjoint operator, self-adjoint operator, Lagrange identity, Green's formula are obtained. Then, a criterion is given for the existence of solutions to non-homogeneous multiplicative boundary value problems.

The spectral form of the functional model for maximally dissipative operators: A Lagrange identity approach

This paper is a contribution to the theory of functional models. In particular, it develops the so-called spectral form of the functional model where the selfadjoint dilation of the operator is represented as the operator of multiplication by an independent variable in some auxiliary vector-valued function space. With the help of a Lagrange identity, in the present version the relationship between this auxiliary space and the original Hilbert space will be explicit. A simple example is provided.

Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative

The concept of a conformable derivative on time scales is a relatively new development in the field of fractional calculus. Traditional fractional calculus deals with derivatives and integrals of non-integer order on continuous time domains. However, time scale calculus extends these concepts to more general time domains that include both continuous and discrete points. The conformable derivative on time scales has several properties that make it advantageous in certain applications. For example, it satisfies a chain rule and has a simple relationship with the conformable integral, which facilitates the development of differential equations involving fractional order dynamics. It also allows for the analysis of systems with both continuous and discrete data points, making it suitable for modeling and control applications in various fields, including physics, engineering, and finance. In this study, the Sturm-Liouville problem and its properties are examined on an arbitrary time scale using the proportional derivative, a more general form of the fractional derivative. Important spectral properties such as self-adjointness, Green formula, Lagrange identity, Abel formula, and orthogonality of eigenfunctions for this problem are expressed in proportional derivatives on an arbitrary time scale.

Duality for Stieltjes differential and integral equations

We propose new definitions of adjoint equations to first-order linear Stieltjes differential and integral equations. We investigate the existence and uniqueness of their solutions, provide explicit solution formulas, and obtain corresponding versions of Lagrange's identity. We show that our results are compatible with known results for dynamic equations on time scales, which are included as a special case.

Mathematical analysis of high-order three-phase-lagging models

The aim of this article is to provide an in-depth discussion about thermoelastic models able to take into account the effect of ultrafast strain of a deformable conductor coupled with a very refined behavior in terms of heat exchange, depicted through three distinct relaxation times and their related high-order effects. In particular, the well-posedness question is investigated dealing with a linear anisotropic and inhomogeneous medium, being able to prove the uniqueness as well as the continuous dependence of the solutions for suitable initial-boundary value problems. From a technical point of view, we underline that the main tools used are identifiable: i. in the introduction of an apposite integral operator that enters into the handling of the model, appropriately modifying the original initial-boundary value problem; ii. in the application of the Lagrange identity method in combination with the time-weighted function method and with an exponentially time-weighted Poincaré inequality. It is worth emphasizing that the results achieved are valid under very weak assumptions made on the thermoelastic features of the model.

Generalized entanglement measure for continuous-variable systems

Concurrence, introduced by Hill and Wootters [Hill and Wootters, Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is strictly positive for entangled states and vanishes for all separable states. We present an extension of the entanglement measure to general pure continuous variable states of multiple degrees of freedom by generalizing the Lagrange's identity and wedge product framework proposed by Bhaskara and Panigrahi [Bhaskara and Panigrahi, Quantum Inf. Process. 16, 118 (2017)] for pure discrete-variable systems in arbitrary dimensions and extending the concept to mixed continuous-variable states. A family of faithful entanglement measures is constructed that admits necessary and sufficient conditions for separability across arbitrary bipartitions presented by Vedral et al. [Vedral, Plenio, Rippin, and Knight, Phys. Rev. Lett. 78, 2275 (1997)]. The computed entanglement measure in the present approach for general Gaussian states, pair-coherent states, and non-Gaussian continuous-variable Bell states matches with known results. We also quantify entanglement of phase-randomized squeezed states and superposition of squeezed states. Our results also simplify several results in quantum entanglement theory.

The killed Brox diffusion

We are proposing to study a diffusion in random environment confined in bounded interval. As for many standard diffusions, we build in natural way a stochastic process with bounded domain, but with addition of considering a random environment. We carry out this construction using the Brox diffusion and applying well-known diffusion theory in a quenched fashion, which is a natural way to deal with random environment. The outcome of this procedure is an object that we may call the killed Brox diffusion. Since the generator of this process is initially an ill-posed expression we develop a Sturm–Liouville theory for one-dimensional second-order differential operators with white-noise coefficients. Our first main result is to give a close form of the Green operator associated to the generator, i.e., the inverse of the generator. We do so by setting the Lagrange identity in this context. Then, we give explicit expressions in quenched form of the probability density function of the process; such an object is given theoretically in terms of the spectral decomposition using the eigenvalues and eigenfuntions of the infinitesimal generator of the diffusion. Moreover, we characterize the eigenvalues and eigenfunctions using some integro-differential equations.

Optimal emission control and identification of an unknown pollution source

The advection-diffusion-reaction equation is used for describing the dispersion of a quasi-passive contaminant from industrial point sources in a limited area. The conditions established on the open boundary ensure that the problem is correct in the sense of Hadamard, that is, its solution exists, is unique, and is stable to initial perturbations. The Lagrange identity is used to construct the adjoint operator and formulate an adjoint problem. Equivalent direct and adjoint estimates are derived to assess the concentration of the pollutant at monitoring sites of the area. Formulas obtained on the basis of adjoint estimates are useful in analysing the sensitivity of the model to both variations in the intensity of pollution sources and variations in the initial distribution of the pollutant concentration in the area. New optimal emission control strategies based on using the adjoint estimates are developed in order to prevent violations of existing sanitary standards by timely reduction of emission rates of operating sources. Optimal control here lies in minimizing these reductions. In addition, this control is primarily aimed at reducing the intensity of emissions from sources that most pollute the monitoring site. Also, new methods are proposed for identifying the main parameters of an unknown point source that arose as a result of a dangerous incident (accident, explosion, etc.). These methods allow determining the location and intensity of a constant or non-stationary point source, as well as the moment of emission of a pollutant in the case of an instantaneous point source. This helps to quickly assess the scale of the incident and its consequences. Numerical results show the effectiveness of the methods.

Efficiency improvement on the ANCF cable element by using the dot product form of curvature

The adoption of the continuum mechanics concept has enabled the absolute nodal coordinate formulation (ANCF) cable element to handle large deformation and large rotation problems. Generally, the generalized elastic forces (GEFs) and tangent stiffness matrices are formulated by differentiating the curvature of the cross product form, which needs labour-intensive construction process, leads to complicated expressions and therefore impairs the computational efficiency of the element. This work aims to improve the computational efficiency of the ANCF cable element by simplifying the GEFs and tangent stiffness matrices based on the dot product form of curvature. The element is interpolated by the curvature constrained interpolation method and defined by following the decoupling elastic line approach, which ensures the second-order accuracy. The material curvature is presented with just dot product terms through two approaches: the original curvature concept approach (OCCA) and the Lagrange's identity approach (LIA). Based on the OCCA and the LIA, concise formulations of GEFs and tangent stiffness matrices are derived through matrix manipulation. In contrast, the GEFs and tangent stiffness matrices derived from the cross product form of curvature are traditionally constructed by deducing every single item individually, which is summarized as the itemized traversal approach (ITA) in this work. To verify the correctness and efficiency of the proposed approaches, some geometrically nonlinear problems of the straight beam and the curved beam are solved and thoroughly analysed. It is found that far fewer number of the second-order elements are needed than that of the first-order element to get the desired accuracy solution. Besides, the convergence rate of the OCCA, LIA and ITA is identical and the solutions of these three approaches coincide with the results obtained by ABAQUS for five-digits because the definitions of the GEFs and tangent stiffness matrices are essentially identical despite different curvature expressions. However, with the same number of second-order elements, the OCCA and LIA are both more than ten times more efficient than the ITA in terms of CPU time, in which the OCCA is the most efficient one. In future, the proposed approaches can be applied in finite element practices for saving computational time without loss of accuracy.

Algebraic and Topological Structures of Complex Numbers

Complex numbers play a fundamental role in every branch of mathematics. First, we review the definition of the complex numbers, discuss their arithmetic operation properties and give proof of the Lagrange identity. At last, we take the investigation of topological notations of the complex plane

Uniqueness, continuous dependence and reciprocity theorems in thermoelastic relaxed micromorphic continuum

A mixed initial-boundary value problem is formulated for linear thermo-elastic relaxed micromorphic continuum with asymmetric micro-distortion tensor, micro-dislocation tensor and temperature field. First, Lagrange identity involving two admissible processes at different instants for the considered mixed initial-boundary value problem is established. Using this identity, four theorems are proved: (i) uniqueness of solution without recourse either to an energy conservation law or to any boundedness assumptions on the thermoelastic coefficients, (ii) continuous dependence of solution of the problem on loads, (iii) continuous dependence of solution of the problem on the initial data, and (iv) reciprocity relation. The use of positive definiteness conditions on the thermoelastic coefficients is also avoided.

SPECTRAL PROPERTIES OF A CONFORMABLE BOUNDARY VALUE PROBLEM ON TIME SCALES

We study a self-adjoint conformable dynamic equation of second order on an arbitrary time scale $\mathbb{T}$. We state an existence and uniqueness theorem for the solutions of this equation. We prove the conformable Lagrange identity on time scales. Then, we consider a conformable eigenvalue problem generated by the above-mentioned dynamic equation of second order and we examine some of the spectral properties of this boundary value problem.

On the quasi-static approximation in the initial boundary value problem of linearised elastodynamics

Continuous data dependence estimates are employed to rigorously derive conditions that validate the quasi-static approximation for the initial homogeneous boundary value problem in the theory of small elastic deformations superposed upon large elastic deformations. This theory imposes no sign-definite assumptions on the linearised elastic moduli and in consequence the requisite estimates are established using methods principally motivated by known Lagrange identity arguments.

The mixed problem in the theory of strain gradient thermoelasticity approached with the Lagrange identity

In our paper we address the thermoelasticity theory of the strain gradient. First, we define the mixed problem with initial and boundary data in this context. Then, with the help of an identity of Lagrange type, we prove some uniqueness theorems with regards to the solution of this problem and two theorems with regards to the continuous dependence of solutions on loads and on initial data. We want to highlight that the use of the approach proposed in this work enables obtaining results without recourse to any boundedness assumptions on the coefficients or to any laws of conservation of energy. Also, we do not impose restrictions on thermoelastic coefficients regarding their positive definition.

Sound propagation using an adjoint-based method

Abstract

On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation

It is known that in the case that several constitutive tensors fail to be positive definite the system of the thermoelasticity could become unstable and, in certain cases, ill-posed in the sense of Hadamard. In this paper, we consider the Moore–Gibson–Thompson thermoelasticity in the case that some of the constitutive tensors fail to be positive and we will prove basic results concerning uniqueness and instability of solutions. We first consider the case of the heat conduction when dissipation condition holds, but some constitutive tensors can fail to be positive. In this case, we prove the uniqueness and instability by means of the logarithmic convexity argument. Second we study the thermoelastic system only assuming that the thermal conductivity tensor and the mass density are positive and we obtain the uniqueness of solutions by means of the Lagrange identities method. By the logarithmic convexity argument we prove later the instability of solutions whenever the elasticity tensor fails to be positive, but assuming that the conductivity rate is positive and the thermal dissipation condition hold. We also sketch similar results when conductivity rate and/or the thermal conductivity fail to be positive definite, but the elasticity tensor is positive definite and the dissipation condition holds. Last sections are devoted to considering the case when a third-order equation is proposed for the displacement (which comes from the viscoelasticiy). A similar study is sketched in these cases.

On Porous Matrices with Three Delay Times: A Study in Linear Thermoelasticity

Through the present work, we want to lay the foundation of the well-posedness question for a linear model of thermoelasticity here proposed, in which the presence of voids into the elastic matrix is taken into account following the Cowin–Nunziato theory, and whose thermal response obeys a three-phase lag time-differential heat transfer law. By virtue of the linearity of the model investigated, the basic initial-boundary value problem is conveniently modified into an auxiliary one; attention is paid to the uniqueness question, which is addressed through two alternative paths, i.e., the Lagrange identity and the logarithmic convexity methods, as well as to the continuous dependence issue. The results are achieved under very weak assumptions involving constitutive coefficients and delay times, at most coincident with those able to guarantee the thermodynamic consistency of the model.

An approach with Lagrange identity of the mixed problem in theory of strain gradient thermoelasticity

In our paper we first define the mixed initial-boundary values problem in the theory of strain gradient thermoelasticity. With the help of an identity of Lagrange’s type, we then prove some theorems regarding the uniqueness of the solution of this mixed problem and also two results regarding the continuous dependence of solutions on initial data and on the charges. We must ouline that we obtain these qualitative results without recourse to any laws of conservation of energy and without recourse to any boundedness assumptions on the coefficients. It is equally important to note that we do not impose restrictions on the elastic coefficients regarding their positive definition.

Modeling triple porosity under local thermal non-equilibrium

In this paper, we present a model describing the evolutionary behavior of materials with triple porosity under local thermal non-equilibrium. The constitutive equations are presented for anisotropic and inhomogeneous materials and so they take into account the terms representing connectivity between the pressures and temperatures in the various pore scales. Further, the basic initial boundary value problems are investigated by using the Lagrange identity and the logarithmic convexity methods. We introduce a functional representing an appropriate measure of the solution to the initial boundary value problem for the model in concern and by means of the Lagrange identity method we are able to establish a uniqueness result. We also establish some estimates describing the continuous data dependence of solution with respect to the initial data and with respect to the various supply terms. Finally, we show how the above functional can be used in combination with the logarithmic convexity method in order to prove again that the solution is unique.

Quasi-Noether Systems and Quasi-Lagrangians

We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green–Lagrange) identity. We discuss quasi-Noether systems, and some of their properties, and generate classes of quasi-Noether differential equations of the second order. We next introduce a more general version of quasi-Lagrangians which allows us to extend Noether theorem. Here, variational symmetries are only sub-symmetries, not true symmetries. We finally introduce the critical point condition for evolution equations with a conserved integral, demonstrate examples of its compatibility, and compare the invariant submanifolds of quasi-Lagrangian systems with those of Hamiltonian systems.