Second-order Differential Equations Research Articles

Solution of linear and nonlinear singular value problems using operational matrix of integration of Taylor wavelets

ABSTRACT In this paper, we present an effective method under Taylor wavelets and collocation technique to find an approximate solution of linear and non-linear second-order singular value differential equations. Application of this method on various problems confirms the efficiency, easy applicability, and rapid computation. The obtained solution is compared with some other existing numerical solutions. For many problems, this method gives a solution same as the exact solution of the problem which confirms the effectiveness and accuracy of this method. Additionally, we include graphs and figures to demonstrate that the Taylor wavelet method offers better accuracy for a number of problems. MATLAB software is used to process relative data and execute calculations.

Boundedness and asymptotic Behaviour of Solutions of some second-order nonlinear stochastic differential equations with delay

This paper considers a certain second-order nonlinear stochastic differential equation with delay. Novel conditions for the existence of solutions that are uniformly bounded and ultimately bounded are obtained. Moreover, we also study the asymptotic behaviour of solutions for the considered equation. We employ Lyapunov’s second method via an appropriate complete Lyapunov functional to achieve these. Obtained results are new, and they improve and complement some existing relatively recent results in the literature. Finally, an example is provided to illustrate the obtained results.

Hankel determinant and orthogonal polynomials for a perturbed Gaussian weight: From finite n to large n asymptotics

We study the monic polynomials Pn(x; t), orthogonal with respect to a symmetric perturbed Gaussian weight function w(x)=w(x;t)≔e−x21+tx2λ,x∈R, with t>0,λ∈R. This problem is related to single-user multiple-input multiple-output systems in information theory. It is shown that the recurrence coefficient βn(t) is related to a particular Painlevé V transcendent, and the sub-leading coefficient p(n, t) of Pn(x; t) (Pn(x; t) = xn + p(n, t)xn−2 + ⋯) satisfies the Jimbo–Miwa–Okamoto σ-form of the Painlevé V equation. Furthermore, we derive the second-order difference equations satisfied by βn(t) and p(n, t), respectively. This enables us to obtain the large n full asymptotic expansions for βn(t) and p(n, t) with the aid of Dyson’s Coulomb fluid approach in the one-cut case [i.e., λt ≤ 1 (t > 0)]. We also consider the Hankel determinant Dn(t), generated by the perturbed Gaussian weight. It is found that Φn(t), a quantity allied to the logarithmic derivative of Dn(t) via Φn(t)=2t2ddtlnDn(t)−2nλt, can be expressed in terms of βn(t) and p(n, t). Based on this result, we obtain the large n asymptotic expansion of Φn(t) and then that of the Hankel determinant Dn(t) in the one-cut case.

Stability Analysis and RBF Neural Network Control of Second-Order Nonlinear Satellite System

This paper studies the open-loop stability analysis and radial basis function based neural-network(RBFNN) control of the triangular tethered satellite formation(TTSF) systems which comprise the in-plane and out-plane dynamics. The dynamical equations of TTSF are nonlinear coupled second-order differential equations(DE), whose explicit solutions are hard to work out. Therefore, the dynamics of systems are analyzed by imposing some constraints on initial states, and the norms of proper initialization for such TTSF systems are obtained by solving the DE. As a result, the nonlinear controllers achieving stable tracking to reference states of TTSF systems are designed based on open-loop analysis. Furthermore, when the structure information of the system is uncertain, delay sampled NN(DSNN) as well as offline training NN(OTNN) control algorithms are proposed to train RBFNN, and the optimal weights of NN are computed by least square(LS) method. Then RBFNN control is imposed to the uncertain TTSF system to ensure the stable tracking, and the stability proof of the control is given. Finally, simulations are taken on the specific motions of TTSF system to test the theoretical results.

Some Results for Ćirić–Prešić Type Contractions in F-Metric Spaces

In this study, first, we introduce Ćirić–Prešić type contraction in F-metric spaces and prove a fixed point theorem for self mappings. We apply the fixed point results for a second-order differential equation. Therefore, we define Prešić type almost contraction and F-contraction, and we prove some fixed point theorems. In the last section, we prove the best proximity point theorems for Ćirić–Prešić type proximal contraction in F-metric spaces. Our results generalize the existing results in the literature.

A singular linear statistic for a perturbed LUE and the Hankel matrices

In this paper, we investigate the Hankel determinant generated by a singular Laguerre weight with two parameters. Using ladder operators adapted to monic orthogonal polynomials associated with the weight, we show that one of the auxiliary quantities is a solution to the Painlevé III′ equation and derive the discrete σ-forms of two logarithmic partial derivatives of the Hankel determinant. We approximate the second-order differential equation satisfied by the monic orthogonal polynomials with respect to the singular Laguerre weight with two parameters to the double confluent Heun equation, leveraging the scaling limit for two parameters and the dimension of the Hankel determinant. In addition, we establish the asymptotic behavior of the smallest eigenvalue of large Hankel matrices associated with the weight with two parameters, using the Coulomb fluid method and the Rayleigh quotient.

On Solutions of a Boundary Value Problem for a Second-Order Differential Equation with a Parameter and Discontinuous Right-Hand Side

On Solutions of a Boundary Value Problem for a Second-Order Differential Equation with a Parameter and Discontinuous Right-Hand Side

LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY

In this work, we study the existence and regularity of solutions for some second order differential equations with infinite delay in Banach spaces. We suppose that the undelayed part admits a cosine operator in the sense given by Da Prato and Giusi, [ G. Da Prato and E. Giusi, \emph{Una caratterizzazione dei generatori di funzioni coseno astratte}, Bollettino dell'Unione Matematica Italiana, 22, 357-362, (1967)]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of strict solutions.

Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations

Abstract This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.

Homoclinic chaos in a perennial grass model

The second-order difference equation , where 0<a<1, 0<b and 0<c<1, has been used to model the generations of the perennial grass Agrostis scabra. The permanence of the solutions was proved. For certain parameter values in this model, and in the data from field studies, chaotic dynamics was conjectured. In this study, with the aid of a homoclinic shadowing theorem, we prove the existence of a transversal homoclinic orbit, hence the presence of chaotic dynamics, for certain parameter values of this second-order difference equation.

Relativistic bulk viscous fluids of Burgers type and their presence in neutron stars

It is well known that a mixture of two chemical components undergoing one chemical reaction is a bulk viscous fluid, where the bulk stress evolves according to the Israel-Stewart theory. Here, we show that a mixture of three independent chemical components undergoing two distinct chemical reactions can also be viewed as a bulk viscous fluid, whose bulk stress now is governed by a second-order differential equation which reproduces the Burgers model for viscoelasticity. This is a rigorous and physically motivated example of a fluid model where the viscous stress does not undergo simple Maxwell-Cattaneo relaxation, and can actually overshoot the Navier–Stokes stress. We show that, if one accounts for muons, neutron star matter is indeed a bulk viscous fluid of Burgers type.

Wavelets approach for the solution of nonlinear variable delay differential equations

Abstract In this study, the Laguerre wavelet-oriented numerical scheme for nonlinear first and second-order delay differential equations (DDEs) is offered. The proposed technique is dependent on the truncated series of the Laguerre wavelets approximation of an unknown function. Here, we transform the different ordered DDEs into a system of non-linear algebraic equations with the help of limit points of a sequence of collocation points. Four nonlinear illustrations are involved to prove the efficiency of the planned technique. the Obtained results are equated with the current results, indicating the proposed technique’s accuracy and efficiency.

The Lagrangian structure, the Euler equation, and second Newton’s law of ultrafast nonlinear optics

With a notable exception of space–time variable swap, pulse evolution equations of ultrafast optics are mathematically similar to the equations of quantum mechanics and beam dynamics. The Lagrangian structure of these equations is, however, much less intuitive. Here, we show that such a structure can be identified via a path-integral analysis of the pulse evolution equation, offering useful insights into the Lagrangian underpinning of ultrafast nonlinear optics. The Euler–Lagrange equation applied to the optical analog of the Lagrange function leads to a second-order differential equation that parallels in its structure second Newton’s law. Stationary-phase space–time paths found by solving such an equation reveal the hidden inner workings behind a vast class of field-waveform transformations in ultrafast optics, including the buildup of nonlinear phase, modulation instabilities, and soliton breather dynamics.

Low-threshold all-optical nonlinear activation function based on injection locking in distributed feedback laser diodes.

We experimentally demonstrate an all-optical nonlinear activation unit based on the injection-locking effect of distributed feedback laser diodes (DFB-LDs). The nonlinear carrier dynamics in the unit generates a low-threshold nonlinear activation function with optimized operating conditions. The unit can operate at a low threshold of -15.86 dBm and a high speed of 1 GHz, making it competitive among existing optical nonlinear activation approaches. We apply the unit to a neural network task of solving the second-order ordinary differential equation. The fitting error is as low as 0.0034, verifying the feasibility of our optical nonlinear activation approach. Given that the large-scale fan-out of optical neural networks (ONNs) will significantly reduce the optical power in one channel, our low-threshold scheme is suitable for the development of high-throughput ONNs.

A regular interior solution of Einstein field equations

Starting from the solution of the Einstein field equations in a static and spherically symmetric space–time that contains an isotropic fluid, we construct a model to represent the interior of compact objects with compactness rate [Formula: see text]. The solution is obtained by imposing the isotropy condition for the radial and tangential pressures; this generates an ordinary differential equation of second order for the temporal gtt and radial grr metric potentials, which can be solved for a specific function of gtt. The graphic analysis of the solution shows that it is physically acceptable, which is to say that the density, pressure, and speed of sound are positive, regular and monotonically decreasing functions; also, the solution is stable due to meeting the criteria of the adiabatic index. When taking the data of mass [Formula: see text] and radius [Formula: see text] that corresponds to the estimations of the star PSR J0030+045, we obtain values of central density ρc = 7.5125 × 1017 kg/m3 for the maximum compactness u = 0.19628 and of ρc = 2.8411 × 1017 kg/m3 for the minimum compactness u = 0.13460, which are consistent with those expected for this type of star.

CONVECTIVE MHD FLOW PAST AN IMPULSIVELY STARTED VERTICAL PLATE WITH PERIODIC TEMPERATURE AND CONCENTRATION IN PRESENCE OF THERMAL RADIATION AND CHEMICAL REACTION

This study investigated a two-dimensional unsteady MHD-free convection of a viscous, incompressible fluid past through an impulsively started vertical plate in the presence of thermal radiation and chemical reactions. The Laplace transform method is used to solve the second-order partial differential equations. The effect of various related parameters on the velocity field, temperature field, and concentration field are demonstrated graphically. The effects of the Nusselt number, Sherwood number, and skin friction for the influence of different parameters are also graphically visualized. It is noticed that the increment in Soret number hikes the fluid velocity and concentration. Also, the Hartmann number reduces the velocity of the fluid. Present work is important in various industrial and engineering sectors, such as chemical and power industries, glass production, steel rolling etc.

МОДЕЛЮВАННЯ ЗМІН АКТИВНОСТІ ВІТЧИЗНЯНОГО БІЗНЕСУ ЗА УМОВ ВІЙНИ

The article examines the effects of changes in the activity of domestic business during the war, as a result of the system of unfavorable factors. The authors analyze the possibilities of using different types of mathematical models to reproduce the dynamics of business activity at the level of a particular country, and selects the most successful one. A phenomenological model describing the dynamics of the process was proposed, which uses the business activity index as the main variable and is based on an ordinary heterogeneous second-order differential equation with a source term. Using a three-component production function, a transition to a system of differential equations describing the dynamics of various components of the business activity index was made, which, in the absence of a source term responsible for the impact of external stimulating factors, is conservative. The modeling results are compared with empirical data, which indicates a good ability of the model to reproduce real processes in the economy under the influence of powerful negative factors, including the restoration of business optimism caused by the return of territories in September-October 2022 and overcoming the consequences of attacks on energy infrastructure in November 2022-January 2023. It is shown that the involvement of other equations describing the dynamics of business activity in the system leads to a loss of conservatism. The frequency and phase-frequency analysis of the dynamics of Ukrainian business activity during the war allows us to conclude that business activity will recover fairly quickly after the cessation of negative depressing factors. Based on the descriptive data, the article clusters the negative factors that suppress domestic business during hostilities and compares them with the parameters that determine the ease of doing business, an internationally recognized indicator that allows assessing obstacles to business activity. Based on the research of recognized foreign experts, a retrospective analysis of obstacles to the development of Ukrainian business was conducted. An empirical study of a number of enterprises from the Dnipropetrovs’k region belonging to various business sectors was implemented, which proves that the main problems that emerged during the war are not much more severe than the problems that domestic business experienced before the aggression. The «classical» factors associated with hostilities are reduced to ensuring uninterrupted supply of electricity and preventing the outflow of human resources through conscription and forced migration. It is concluded that an interesting direction for further research on this problem is the use of the Granger model, which will allow for joint analysis of time series and take into account hidden relationships.

Solving nonlinear filtering problems with correlated noise based on Hermite–Galerkin spectral method

Nonlinear filtering problem has important applications in various fields. One of the core issues in nonlinear filtering is to numerically solve the Duncan–Mortensen–Zakai (DMZ) equation, which is an evolution equation satisfied by the unnormalized conditional density of state process under noisy observations, in a real-time and memoryless manner. When the noise in observations is correlated to the state process, the DMZ equation we need to deal with is a second-order stochastic partial differential equation. In this paper, we will propose an algorithm to solve the DMZ equation in this case, based on Hermite–Galerkin spectral method. According to this method, the DMZ equation is converted into a system of linear stochastic differential equations generated by the observation process. The effects of different discretization schemes on this stochastic differential system will also be discussed. Moreover, rigorous convergence analysis of the algorithm is given under mild conditions. Numerical results show that the method proposed in this paper can provide an instantaneous and accurate estimation to the state process of the system.

On the existence of a positive solution to a boundary value problem for one nonlinear functional-differential equation of the second order

This article considers a boundary value problem for one non-linear second-order functional differential equation on the segment [0, 1] with an integral boundary condition at one of the ends of the segment. Using the well-known Go-Krasnoselsky theorem, sufficient conditions for the existence of at least one positive solution to the problem under consideration are established. A non-trivial example is given, illustrating the fulfillment of the conditions for the unique solvability of the problem posed.

Semi-Analytical Closed-Form Solutions for the Rikitake-Type System through the Optimal Homotopy Perturbation Method

The goal of this work is to build semi-analytical solutions of the Rikitake-type system by means of the optimal homotopy perturbation method (OHPM) using only two iterations. The chaotic behaviors are excepted. By taking into consideration the geometrical properties of the Rikitake-type system, the closed-form solutions can be established. The obtained solutions have a periodical behavior. These geometrical properties allow reducing the initial system to a second-order nonlinear differential equation. The latter equation is solved analytically using the OHPM procedure. The validation of the OHPM method is presented for three cases of the physical parameters. The advantages of the OHPM technique, such as the small number of iterations (the efficiency), the convergence control (in the sense that the semi-analytical solutions are approaching the exact solution), and the writing of the solutions in an effective form, are shown graphically and with tables. The accuracy of the results provides good agreement between the analytical and corresponding numerical results. Other dynamic systems with similar geometrical properties could be successfully solved using the same procedure.