Matrix Differential Operators Research Articles

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

This article introduces two numerical methods to address boundary value problems associated with secondorder and fractional differential equations. These methods employ two parameters related to shifted Gegenbauer polynomials as their basis functions. The process involves establishing a differentiation operational matrix for the shifted Gegenbauer polynomials. Subsequently, the initial/boundary value problems for ordinary and fractional differential equations are transformed into a system of equations through the Galerkin, collocation, and tau methods. The convergence analysis is ensured by leveraging theorems pertaining to the shifted Gegenbauer polynomials. To validate the accuracy of the approach, numerous numerical examples are presented.

Liouville-Green formulas for the fourth order matrix differential operators

In this paper, Liouville-Green analytical approximations for the solutions of the fourth order matrix differential equation are constructed. Using the Gronwall’s inequality, we have also determined the error bounds for the Liouville-Green approximations.

Maxwell operator in a cylinder

The Maxwell operator in a 3D cylinder is considered. The coefficients are assumed to be scalar functions depending on the longitudinal variable only. Such an operator is represented as a sum of a countable set of matrix differential operators of first order acting in . Based on this representation we give a detailed description of the structure of the spectrum of the Maxwell operator in two particular cases: (1) in the case of coefficients stabilizing at infinity; and (2) in the case of periodic coefficients.

Schur complement dominant operator matrices

We propose a method for the spectral analysis of unbounded operator matrices in a general setting which fully abstains from standard perturbative arguments. Rather than requiring the matrix to act in a Hilbert space H, we extend its action to a suitable distributional triple D⊂H⊂D− and restrict it to its maximal domain in H. The crucial point in our approach is the choice of the spaces D and D− which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in H and its Schur complement, which allows to pass from a suitable representation of the Schur complement (e.g. by generalised form methods) to a representation of the operator matrix. We thereby generalise classical spectral equivalence results imposing standard dominance patterns.The abstract results are applied to damped wave equations with possibly unbounded and/or singular damping, to Dirac operators with Coulomb-type potentials, as well as to generic second order matrix differential operators. By means of our methods, previous regularity assumptions can be weakened substantially.

RECIPROCITY LAWS FOR OSCILLATIONS OF DISSIPATIVE SYSTEMS

) is presented. The method is based on the use of the algebraic theorem of P.L. Pasternak and on the new properties of the Duhamel integral, which are obtained for a dissipative system with internal friction of the material, which is taken into account on the basis of the non-proportional damping model. For displacements, velocities and accelerations, the dynamic reaction equations are written in the form of systems of linear equations and their symmetrical structure is shown. The functional dependence of the force parameters of the calculation model and the corresponding kinematic parameters of the reaction is determined by an arbitrary scalar function of time. An extended interpretation of the reciprocity theorems is given and sufficient conditions for their fulfillment are formulated, which consist in the requirement that the matrix differential operator of the equation of motion be symmetrical. New laws of reciprocity in dissipative systems are formulated and proved. The reciprocity of the product between the velocities / accelerations of masses and nodal forces is established. In contrast to the well-known theorem on the reciprocity of possible work, these laws are theorems on the 1st / 2nd derivative of possible work with respect to time and therefore go beyond the Betti principle. For particular cases of these theorems, the reciprocity of velocities and reciprocity of accelerations is shown. Expressions of general and particular theorems have a fairly simple mathematical form that does not require recourse to integral transformations, and are presented in an analytical form.

A matrix differential operator for passage from scalar to vector wave optics

One possible way to generalize scalar to wave optics, thus including polarization in the treatment consistent with the Maxwell equations was shown by Mukunda, Simon and Sudarshan (MSS) for paraxial systems, based on a group theoretical analysis. Later, the MSS theory for the passage from the Helmholtz scalar wave optics to the Maxwell vector wave optics was derived by casting the basic formalism in a framework very similar to the Dirac electron theory. Here, we show that the elegant MSS substitution rule is equivalent to a second-order 6 × 6 matrix differential operator for passage from scalar to vector wave optics. We explicitly obtained this matrix differential operator in a series and exponential forms respectively. Expressions for the electromagnetic field components are derived without making any assumptions on the form of the transverse beam profile. It is observed that all types of beams have certain characteristic relations among their field components. The power of the formalism is demonstrated by considering five specific beams.

Asymptotic distribution of the eigenvalues of the bending‐torsion vibration model with fully nondissipative boundary feedback

AbstractAsymptotic and spectral results on the initial boundary‐value problem for the coupled bending‐torsion vibration model (which is important in such areas of engineering sciences as bridge and tall building designs, aerospace and oil pipes modeling, etc.) are presented. The model is given by a system of two hyperbolic partial differential equations equipped with a three‐parameter family of non‐self‐adjoint (linear feedback type) boundary conditions modeling the actions of self‐straining actuators. The system is rewritten in the form of the first‐order evolution equation in a Hilbert space of a four‐component Cauchy data. It is shown that the dynamics generator is a matrix differential operator with compact resolvent, whose discrete spectrum splits asymptotically into two disjoint subsets called the α‐branch and the β‐branch, respectively. Precise spectral asymptotics for the eigenvalues from each branch as the number of an eigenvalue tends to ∞ have been derived. It is also shown that the leading asymptotical term of the α‐branch eigenvalue depends only on the torsion control parameter, while of the β‐branch eigenvalue depends on two bending control parameters.

Free vibrations of Timoshenko nanoscale beams based on a hybrid integral strain- and stress-driven nonlocal model

Several non-classical elasticity theories are used for considering the size-dependent behavior of structures at small scales. The nonlocal theory is widely used to reflect the softening behavior of material at small scales, and theories like the strain gradient theory are employed to reflect the hardening behavior. In this article, the most general form of integral strain- and stress-driven nonlocal models with two nonlocal parameters is developed which is able to consider both hardening and softening influences simultaneously. To this end, it is considered that the stress field at the entire points of the domain is a function of strain field of the entire points of the domain. The free vibration problem of first-order shear deformable beams is solved herein. The integral form of governing equations and associated boundary conditions are obtained first, and then directly solved in a numerical approach. Through developing an efficient matrix formulation and using differential and integral matrix operators, the discretized governing equations are obtained. The simultaneous effects of strain- and stress-driven nonlocal parameters on the natural frequencies of fully clamped, fully simply-supported, and clamped-free nanobeams are investigated. The results indicate that the paradox related to the behavior of clamped-free nanobeams is resolved using the presented integral nonlocal formulation. Also, it is revealed that it is possible to find some specific values of nonlocal parameters at which the prediction of hybrid nonlocal model coincides with that of classical elasticity theory.

Time-Periodic Thermal Boundary Effects on Porous Media Saturated with Nanofluids: CGLE Model for Oscillatory Mode

Abstract The stability of nonlinear nanofluid convection is examined using the complex matrix differential operator theory. With the help of finite amplitude analysis, nonlinear convection in a porous medium is investigated that has been saturated with nanofluid and subjected to thermal modulation. The complex Ginzburg-Landau equation (CGLE) is used to determine the finite amplitude convection in order to evaluate heat and mass transfer. The small amplitude of convection is considered to determine heat and mass transfer through the porous medium. Thermal modulation of the system is predicted to change sinusoidally over time, as shown at the boundary. Three distinct modulations IPM, OPM, and LBMOhave been investigated and found that OPM and LBMO cases are used to regulate heat and mass transfer. Further, it is found that modulation frequency (ω f varying from 2 to 70) reduces heat and mass transfer while modulation amplitude (δ 1 varying from 0.1 to 0.5 ) enhances both.

Singular Perturbations in First-Order Partial Differential Equations with Matrix Differential Operators

In this paper, we consider differential equations whose degeneracy is due to the presence of a matrix differential operator coefficient of the derivative perturbed by a small parameter. Properties of this operator are applied to the analysis of the presence of a boundary layer in solutions of initial-boundary-value problems for equations considered. Regularity conditions of degeneracy are determined.

A study on the effect of temperature modulation on Darcy–Bénard convection using a local thermal non-equilibrium model

The criterion for the onset of Darcy–Bénard convection is analyzed when the fluid and porous medium are out of thermal equilibrium and the temperatures of the boundaries vary sinusoidally with time in either a synchronous or an asynchronous manner. A stability analysis of the linearized governing equations is performed by using the matrix differential operator theory. The shift in the critical Darcy–Rayleigh number is evaluated in terms of system parameters, and the effect of those on the shift is depicted graphically to realize the significant effect of temperature modulation on the onset, especially when the thermal non-equilibrium effects are prominent.

On matching distance between eigenvalues of unbounded operators

Let AA and ~AA~ be linear operators on a Banach space having compact resolvents, and let λk(A)λk(A) and λk(~A)(k=1,2,…)λk(A~)(k=1,2,…) be the eigenvalues taken with their algebraic multiplicities of AA and ~AA~, respectively. Under some conditions, we derive a bound for the quantity md(A,~A):=infπsupk=1,2,…∣∣λπ(k)(~A)−λk(A)∣∣,md⁡(A,A~):=infπsupk=1,2,…|λπ(k)(A~)−λk(A)|, where ππ is taken over all permutations of the set of all positive integers. That quantity is called the matching optimal distance between the eigenvalues of AA and ~AA~. Applications of the obtained bound to matrix differential operators are also discussed.

Свойства одного матрично-дифференциального оператора высокого порядка

The article considers a linear matrix-differential operator of the n-th order of the form A^n. For it and for the operator (A ̃^(-1) )^n, an analytical expression is derived, for which an operator analog of the Newton binomial is obtained. A lemma on the solution of a linear equation is given. It is used in the study of the abstract Cauchy problem for an algebro-differential equation in a Banach space with the cube of the operator A at the highest derivative. The operator A has the property of having 0 as a normal eigenvalue. Conditions for the existence and uniqueness of the solution are determined; the solution is found, for which the method of cascade splitting of the equation and conditions into the corresponding equations and conditions in subspaces of lower dimensions is used. As an application, the results obtained for n=3 are used in solving a mixed problem for a fourth-order partial differential equation. These equations include the generalized shallow water wave equation and the generalized Liouville equation.

Dubrovin’s method and Ablowitz-Kaup-Newell-Segur hierarchy

Previously, the Dubrovin method was used to construct spectral curves of multiphase solutions of such integrable models of nonlinear optics as vector and derived nonlinear Schrödinger equations. The method of constructing spectral curves of multiphase solutions of the scalar nonlinear Schrödinger equation was based on replacing matrix differential operator from the Lax pair by the scalar one. The present paper shows that Dubrovin’s method gives the same results. At the same time, it has the advantage that it consists in a simpler construction of multiphase solutions expressed in terms of elementary functions.

An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation

This paper presents the fractional formulation and numerical solution of a non-linear fractional diffusion equation with advection and reaction terms. The fractional derivative employed in this equation is of non-singular type with Mittag-Leffler kernel. To investigate this fractional equation numerically, we adopted a collocation method with the Legendre operational matrix. The numerical value of Atangana-Baleanu Caputo (ABC) fractional differentiation of polynomial function f(ϑ)=ϑk is determined by using the definition of Mittag-Leffler kernel fractional differentiation. The operational matrix of fractional differentiation of Mittag-Leffler kernel fractional derivative is determined numerically employing a polynomial function. Within the above context, we solve the fractional equation in-hand by using this matrix and using the collocation method based upon Legendre polynomial. The effectiveness of these techniques is illustrated through different cases that include fractional diffusion, advection-diffusion, and reaction-diffusion equation. The feasibility of the proposed derived ABC fractional differentiation operational matrix is clearly demonstrated by validating the numerical resolutions against exact solutions. The results demonstrated that the method is capable of resolving the non-linear fractional diffusion equation with different fractional order.

Degeneracy Property of a Matrixdifferential Operator and Applications

We prove that a matrix-differential operator A in a Banach space is a Fredholm operator with zero index and construct the corresponding subspaces and projections. We find the spectrum of A. Based on properties of the operator A, we study solutions to the initial-boundary value problem for a second order partial differential equation with a small parameter e at the higher order derivatives and some parameter c at lower order terms. We clarify how the parameter c affects the behavior of the solution as e → 0.

AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

AbstractLet A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

Bending analysis of nanoscopic beams based upon the strain-driven and stress-driven integral nonlocal strain gradient theories

Strain gradient and nonlocal influences have important roles in the mechanical characteristics of structures as they are scaled down to nanoscopic dimensions. In this research, an attempt is made to capture both effects on the static bending response of nanoscopic Bernoulli–Euler beams by means of a comprehensive model. For this purpose, the strain gradient theory of Mindlin is combined with the integral (original) form of Eringen’s nonlocal theory. Also, the integral nonlocal formulation is written on the basis of both strain-driven and stress-driven versions of nonlocal theory. This mixed nonlocal strain gradient formulation is capable of reducing to the size-independent and combination of integral nonlocal strain gradient family theories by employing simple substitutions. Through constructing finite difference-based differential and integral matrix operators, numerical solution approaches are developed to obtain the deflection of nanobeams under various end conditions. In addition, for the solution of governing equation of strain-driven nonlocal strain gradient theory, an efficient technique is presented that is applied to the variational statement of the problem in a direct approach. Selected numerical results are provided to explore the simultaneous influences of strain gradient terms and nonlocality based on different theories on the static bending response of beams. It is revealed that the developed size-dependent model has the ability of considering strain gradient and nonlocal influences in the most general way.

Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

Nonlinear bending analysis of hyperelastic Mindlin plates: a numerical approach

In this paper, a numerical solution strategy is proposed for studying the large deformations of rectangular plates made of hyperelastic materials in the compressible and nearly incompressible regimes. The plate is considered to be Mindlin-type, and material nonlinearities are captured based on the Neo-Hookean model. Based on the Euler–Lagrange description, the governing equations are derived using the minimum total potential energy principle. The tensor form of equations is replaced by a novel matrix–vector format for the computational aims. In the solution strategy, based on the variational differential quadrature technique, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing differential and integral matrix operators. Fast convergence rate, computational efficiency and simple implementation are advantages of this approach. The results are first validated with available data in the literature. Selected numerical results are then presented to investigate the nonlinear bending behavior of hyperelastic plates under various types of boundary conditions in the compressible and nearly incompressible regimes. The results reveal that the developed approach has a good performance to address the large deformation problem of hyperelastic plates in both regimes.