Second-order Differential Equations Research Articles

Turán-type inequalities for generalized k-Bessel functions

We propose an approach to the generalized -Bessel function defined by where and p , q , r ∈ ℂ . We discuss the uniform convergence of Moreover, we prove that the analyzed function is entire and determine its growth order and type. We also find its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane. The integral representation for is found by using the representation for -beta functions. We also prove that the specified function is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions. In addition, some interesting properties, such as recurrence and differential relations, are demonstrated. Some of these properties can be used to establish some Turán-type inequalities for this function. Ultimately, we study the monotonicity and log-convexity of the normalized form of the modified \textrm{k}-Bessel function defined by as well as the quotient of the modified \textrm{k}\textit{-}Bessel function, exponential, and \textrm{k}\textit{-}hypergeometric functions. In this case, the leading concept of the proofs comes from the monotonicity of the ratio of two power series.

A Magnus-based integrator for Brownian parametric semi-linear oscillators

We introduce a numerical method for solving second-order stochastic differential equations of the form x¨=−ω2(t)x+f(t,x)+σ(t)ξ(t), describing a class of nonlinear oscillators with non-constant frequency, perturbed by white noise ξ(t). The proposed scheme takes advantages of the Magnus approach to construct an integrator for this stochastic oscillator. Its convergence properties are rigorously analyzed and selected numerical experiments on relevant stochastic oscillators are carried out, confirming the effectiveness and the competitive behavior of the proposed method, in comparison with standard integrators in the literature.

Momentum transfer across an open-channel, turbulent flow

The distribution of stress generated by a turbulent flow matters for many natural phenomena, of which rivers are a prime example. Here, we use dimensional analysis to derive a linear, second-order ordinary differential equation for the distribution of stress across a straight, open channel, with an arbitrary cross-sectional shape. We show that this equation is a generic first-order correction to the shallow-water theory in a channel of large aspect ratio. It has two adjustable parameters – the dimensionless diffusion parameter, $\chi$ , and a local-shape parameter, $\alpha$ . By assuming that the momentum is carried across the stream primarily by eddies and recirculation cells with a size comparable to the flow depth, we estimate $\chi$ to be of the order of the inverse square root of the friction coefficient, $\chi \sim C_f^{-1/2}$ , and predict that $\alpha$ vanishes when the flow is highly turbulent. We examine the properties of this equation in detail and confirm its applicability by comparing it with flume experiments and field measurements from the literature. This theory can be a basis for finding the equilibrium shape of turbulent rivers that carry sediment.

Scalar quasi-normal modes of accelerating Kerr-Newman-AdS black holes

We study linear scalar perturbations of slowly accelerating Kerr-Newman-anti-de Sitter black holes using the method of isomonodromic deformations. The conformally coupled Klein-Gordon equation separates into two second-order ordinary differential equations with five singularities. Nevertheless, the angular equation can be transformed into a Heun equation, for which we provide an asymptotic expansion for the angular eigenvalues in the small acceleration and rotation limit. In the radial case, we recast the boundary value problem in terms of a set of initial conditions for the isomonodromic tau function of Fuchsian systems with five regular singular points. For the sake of illustration, we compute the quasi-normal modes frequencies.

Research on financial risk early warning system model based on second-order blockchain differential equation

A financial risk early warning system (FREWS) is a disclosure and tracking mechanism that provides advance notice of potential issues, hazards, and potentials that might affect the business’s finances. Some elderly individuals living alone may experience financial difficulties, which may hinder their ability to pay for appropriate medical care, property maintenance, and other essential expenses. Financial difficulties can add tension and diminish their quality of life. Financial results, investment risk, and possible insolvencies may all be detected by implementing early warning systems. Management might use the window of opportunity provided by early warning systems to avert or lessen the impact of possible issues. Almost all FREWS rely on some financial statement analysis. Financial measures are combined with the EWS, accounting information, to determine the firm’s success in its field. Organizational success depends on effective financial oversight, which is at the heart of each business. Studying the enhancement of early warning capacities is relevant because there are no adequate risk evaluation methods to generate realistic estimates. To minimize the FREWS, this research provides a systemic model based on a second-order block chain differential equation (SBDE). China’s systemic financial liabilities have also been quantified using the expected investment returns of 64 selected financial enterprises in China between February 2006 and September 2020 as the datasets. The financial risk warning approach is compared and analyzed primarily using analytical and comparative techniques. The suggested method is 96% accurate in experiments. Consequently, the proposed algorithm compares favorably to others regarding both computing efficacy and precision and has strong predictability.

On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters.

On the problem of solvability of nonlinear boundary value problems for shallow isotropic shells of Timoshenko type in isometric coordinates

The solvability of a boundary value problem for a system, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with loose edges referred to isometric coordinates in the Timoshenko shear model and consists of five non-linear second-order partial differential equations under given non-linear boundary conditions, is studied. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of this equation is established with the help of the contraction mapping principle.

Математичні моделі локального нагрівання елементів електронних пристроїв

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media sub-jected to internal and external local heat load, have been developed. In the case of nonlinear boundary-value problems, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equations and nonlinear boundary conditions were linearized, and as a result, linearized second-order differential equations with partial derivatives and a discontinu-ous right-hand side and partially linearized boundary conditions were obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the ap-proximation of the temperature according to one of the spatial coordinates on the boundary sur-faces of the inclusion was performed by piecewise constant functions. To solve linear bounda-ry-value problems, as well as obtained linearized boundary-value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in this envi-ronment were obtained. Numerical analysis of temperature behavior as a function of spatial co-ordinates for given values of geometric and thermophysical parameters was performed. The in-fluence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal and external heat load, software tools were developed, using which a ge-ometric image of the temperature distribution depending on the spatial coordinates was made.

New Conditions for Testing the Oscillation of Solutions of Second-Order Nonlinear Differential Equations with Damped Term

This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the standard integral averaging technique to obtain simpler oscillation theorems for new classes of nonlinear differential equations. Two examples are presented to illustrate the importance of our findings.

A second-order phase field-lattice Boltzmann model with equation of state inputting for two-phase flow containing soluble surfactants

In this paper, we propose a phase field-lattice Boltzmann (LB) model with an equation of state (EOS) inputting for two-phase flow containing soluble surfactants. In this model, both the order parameter for the phase field and the surfactant concentration are described by second-order partial differential equations, along with Navier–Stokes equations for the flow field. Changes in surfactant concentration do not affect the order parameter distribution; hence, an unwanted sharpening effect cannot arise. Most importantly, in the existing models, the EOS of surface tension is determined by posterior simulation tests instead of being directly set as an input parameter before the simulations. Hence, it is difficult to determine the model parameters in practical applications. To address this issue, we systematically develop a fully analytical EOS for surface tension based on the Gibbs–Duhem equation. Subsequently, an approximate explicit form for EOS is provided by utilizing the Jacobi–Gauss quadrature rule. Furthermore, a multiple-relaxation-time LB scheme is utilized to numerically solve the governing equations of three physical fields. Two benchmark examples are simulated to validate the accuracy of the present model. The consistency between the numerical results and the analytical EOS is verified. Moreover, the dynamics of droplets with surfactant in simple shear flow is investigated, unveiling the profound impact of various factors, such as surfactant bulk concentration, capillary number, and viscosity ratio, on single droplet deformation and two equal-sized droplets interaction. A detailed exploration of the fluid mechanism involved in two-phase flow with soluble surfactants is presented.

A high order method for oscillatory delay differential equations

This paper addresses the oscillatory properties of a class of second-order neutral delay differential equations (NDDEs) and introduces a numerical scheme based on a fourth-order predictor-corrector multistep method for solving NDDEs of this type. An example in the family of oscillatory equations has been integrated by using the proposed numerical scheme. Results confirm the theoretical assumptions.

Stability analysis of nonlinear convective boundary layer flow on a shrinking surface in the presence of viscous dissipation

This study investigates analytically the stability analysis of a nonlinear convective boundary layer (BL) flow of the nanofluids GO-EG and GO-W on a shrinking surface in the presence of a magnetic field and viscous dissipation. We provide a second-order nonlinear ordinary differential equation (NODE) for temperature distribution (TD) and a third-order NODE for velocity profile (VP) coupling based on the thermophysical characteristics of the base fluid and nanofluid as well as similarity transformations (STs) in the fundamental governing equations of momentum and energy. The analytical method HAM is used to answer the equations that have been gathered. In many different industries, such as manufacturing, automotive, microelectronics, and defense, cooling of various types of equipment and devices has very important challenges. To fulfill the demands of the engineering and industrial industries, it is hoped that this issue would significantly improve the heat transformation ratio.

On first integrals, conservation laws and reduction of classes of Emden and Liénard equations

We present a general method to construct first integrals for some classes of the well known second-order ordinary differential equations, viz., the Emden and Liénard classes of equations. The approach does not require a knowledge of a Lagrangian but, rather, uses the ‘multiplier approach’ (Anco and Bluman in Eur J Appl Math 13:545–566, 2002; Eur J Appl Math 13:567–585, 2002). It is then shown how a study of the invariance properties and conservation laws are used to ‘twice’ reduce the equations to solutions. The equations admit five first integrals of which two are independent but the significance of the five are that they correspond to a five-dimensional algebra of Noether symmetries obtained without the need to construct a Lagrangian.

Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform.

Kernel functions embed into the autoencoder to identify the sparse models of nonlinear dynamics

Numerous researches have shown that there are three main challenges in data-driven model identification methods: high-dimensional measurements, system complexity and unknown underlying dynamical properties. For most nonlinear dynamics, the feature space defined by the coefficients of their control equations is sparse. Therefore, sparse regression methods are used to learn the sparse coefficients of the control equations of nonlinear dynamics. However, this method strongly depends on the appropriate selection of the sparse basis vectors. In this essay, the autoencoder is combined with the sparse regression method to simultaneously identify the sparse coordinate and a parsimonious, interpretable and generalizable model of the specified system. It also integrates kernel functions to map the intractable measurements in the hidden space of the autoencoder into a linearly distinguishable kernel space, which kernelizes the candidate function library of the sparse identification of nonlinear dynamics (SINDy) model as the sparse dictionaries. Therefore, the flexible representation of neural networks, the simplicity of sparse regression methods and the implicit non-linear representation of kernel functions are consolidated in this article. To inspect the reliability of the proposed model in this paper, a set of nonlinear dynamics formulated by ordinary differential equations (ODEs), second-order trigonometric functions and partial differential equations (PDEs) are utilized as test cases. And the comparisons between the proposed model and other model identification methods illustrate that the performance of the former is the best.

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Common Best Proximity Point Theorems for Generalized Dominating with Graphs and Applications in Differential Equations

In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations.

Superlinear distributed deviating arguments to study second-order neutral differential equations

Superlinear distributed deviating arguments to study second-order neutral differential equations

Dynamic Model of Lower Limb Motion in the Sagittal Plane during the Gait Cycle

Context: This work presents the development of a dynamic model for human lower limb motion in the sagittal plane during the gait cycle. The primary objective of this model is to serve as a powerful tool for the design of rehabilitation and assistive devices, such as exoskeletons, prostheses, and orthoses. It achieves this by facilitating the estimation of joint torques, the detailed analysis of kinematic variables, optimal actuator selection, and the exploration of advanced control techniques. Method: The dynamic model consists of two primary components: (1) the plant model and (2) a closed-loop controller. The plant model represents the forward dynamics of human gait and is based on a multi-mass pendulum composed of three segments of the lower limb (thigh, lower leg, and foot) and three joints (hip, knee, and ankle). It is analyzed using the Euler-Lagrange formulation and the nonlinear second-order differential equations are implemented in MATLAB’s Simulink. To reproduce reference human gait trajectories and simulate the functioning of the neuromusculoskeletal system and the central nervous system, a closed-loop PID controller is incorporated into the plant model. It is noteworthy that the scope of this dynamic model is specifically confined to the sagittal plane. Results: The dynamic model is evaluated in terms of angular displacement tracking using the relative maximum error (RME) and the root mean square error (RMSE) for reference trajectories of healthy adult male human gait as reported in the literature. The model demonstrates tracking with errors below 2.2 [°] in magnitude and 3,5% for all three considered segments (thigh, lower leg, and foot). Conclusions: The quantitative results show that the dynamic model developed in this work is reliable and allows for a precise reproduction of human gait trajectories.

On a class of four-step modified backward differentiation formula (MBDF) for solving second-order ordinary differential equations

This research was conducted to study a class of four-step modified backward differentiation formula (MBDF) for solving second-order ordinary differential equations, with one super-future point. A block method was constructed for the solution of problems of second-order ordinary differential equations. Adopting a step-number, k=4, we obtained a number of discrete methods in block. The stability properties of the block method were investigated using Maple application. In order to ascertain its suitability, the method was tested on some initial valued problems of second-order ordinary differential equations. The numerical solutions of the problems were compared with the respective exact solutions and of other existing methods, and are presented in both tables and graphs.