Second-order Differential Equations Research Articles

Development of stability charts for double salience reluctance machine modeled using hill’s equation

The paper presents a novel algorithm for the development of stability charts. The second-order differential homogeneous equation describing a double salient reluctance machine with a capacitance connected to its stator winding is transformed into hill’s equation. The circuit components are the stator coil time-varying inductance of a double salient reluctance machine, capacitance and resistance. All these are modeled by hill’s equation. The double salient reluctance machine acts as an energy conversion system. The maximum and minimum inductance of the energy conversion system is measured in laboratory by inductance, capacitance, and resistance (LCR) meter. These values help to determine the inductance modulation index. The inductance modulation indetx, the characteristic constant and the characteristic parameter obtained from modeling equations are used in the MATLAB/Simulink model. The MATLAB/Simulink simulations generate stable and unstable oscillations to form stability charts. The proposed stability charts are in good agreement with the Ince-Stritt stability chart, which is widely applied in physics, mechanics and in electrical engineering, especially where the state of stability of a system or an electric oscillatory circuit is to be determined.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

Structural and Organizational Strategies of Locomotor Modules during Landing in Patients with Chronic Ankle Instability.

Human locomotion involves the coordinated activation of a finite set of modules, known as muscle synergy, which represent the motor control strategy of the central nervous system. However, most prior studies have focused on isolated muscle activation, overlooking the modular organization of motor behavior. Therefore, to enhance comprehension of muscle coordination dynamics during multi-joint movements in chronic ankle instability (CAI), exploring muscle synergies during landing in CAI patients is imperative. A total of 22 patients with unilateral CAI and 22 healthy participants were recruited for this research. We employed a recursive model for second-order differential equations to process electromyographic (EMG) data after filtering preprocessing, generating the muscle activation matrix, which was subsequently inputted into the non-negative matrix factorization model for extraction of the muscle synergy. Muscle synergies were classified utilizing the K-means clustering algorithm and Pearson correlation coefficients. Statistical parameter mapping (SPM) was employed for temporal modular parameter analyses. Four muscle synergies were identified in both the CAI and healthy groups. In Synergy 1, only the gluteus maximus showed significantly higher relative weight in CAI compared to healthy controls (p = 0.0035). Synergy 2 showed significantly higher relative weights for the vastus lateralis in the healthy group compared to CAI (p = 0.018), while in Synergy 4, CAI demonstrated significantly higher relative weights of the vastus lateralis compared to healthy controls (p = 0.030). Furthermore, in Synergy 2, the CAI group exhibited higher weights of the tibialis anterior compared to the healthy group (p = 0.042). The study suggested that patients with CAI exhibit a comparable modular organizational framework to the healthy group. Investigation of amplitude adjustments within the synergy spatial module shed light on the adaptive strategies employed by the tibialis anterior and gluteus maximus muscles to optimize control strategies during landing in patients with CAI. Variances in the muscle-specific weights of the vastus lateralis across movement modules reveal novel biomechanical adaptations in CAI, offering valuable insights for refining rehabilitation protocols.

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis.

Comparative analysis of thermal behaviour in functionally graded material fin under convective-radiative heat transfer: Homotopy perturbation method

In this study, the thermal characteristics of a fully wet, moving longitudinal fin made from linear Functionally Graded Material (FGM) is explored. The fin response to convective-radiative heat transfer is examined through three distinct FGM scenarios namely, a Homogeneous material (HM), Functionally Graded Material I (FGM I) having greater thermal grading near the base, and Functionally Graded Material II (FGM II) having increased thermal grading towards the tip. The derived energy equation is a second-order nonlinear ordinary differential equation is solved using the Homotopy Perturbation Method (HPM) compared with results found in the existing literature. The fin thermal profile has been graphically analysed for the thermal conductivity grading parameter (β), Peclet number (Pe) and other pertinent parameters. It is found that the temperature distribution along the fin is highest for the FGM II fin and lowest for the HM fin, with the FGM I fin displaying intermediate values. Efficiency of the fin (η) is discovered to be lowest in the case of HM fin structures followed by FGM I and FGM II fin structures at the base and vice versa at the tip. The results show significant potential of HPM which benefits the manufacture and design of FGM fin structures.

Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients

We study the question of the existence of a unique bounded in the mean solution for the second-order difference equation with piecewise constant operator coefficients and of the stationary solution of the corresponding difference equation with constant operator coefficients. The case is considered when the corresponding “algebraic” operator equations have separated roots.

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.

Mathematical model of a lightweight three-axle off-road vehicle construction for Arctic zone of Russia

Introduction. The model and results of calculating the smoothness of a light three-axle off-road vehicle for the Arctic zone of Russia are considered. The model on standard approaches and uses a system of assumptions that limits the number of degrees of freedom for the vehicle body to three, as well as one degree of freedom for the unsprung masses is based. The mathematical model is a system of ordinary differential equations and is supplemented with the necessary algebraic equations, as well as initial conditions. The system is integrated using the 4th order RungeKutta method, for which a program was written in C++. The calculations presented in the article demonstrate the possibility of conducting research on the smoothness of a vehicle under conditions of arbitrary terrain, typical for off-road conditions in the winter conditions of the Arctic zone. The dimensions and other parameters of the vehicle were taken from a full-scale model that was used in real expeditions in 2003-2019. Based on the model, suspension characteristics will be developed for a new all-terrain vehicle.Theory. When operating a wheeled vehicle in a wide range of conditions, even in northern regions, transverseangular vibrations are very often insignificant, so only vertical linear and longitudinal-angular vibrations of the frame can be considered. This problem enables to construct a system of equations of vehicle motion using selected degrees of freedom. From a mathematical point of view, these equations are classified as second-order ordinary differential equations with a variable structure of the right-hand sides, which reflects the non-linear nature of the behavior of the suspension in terms of its geometric constraints.Methods. The work uses numerical methods to solve the equations of the constructed model, which enables to gradually weaken the accepted assumptions and build more general calculation algorithms. The main integration method for ensuring the stability of solutions is the multi-step Adams method, which, with the correct choice of step, ensures the necessary stability of the solution over sufficiently long model times. However, in this work, the 4th order Runge-Kutta method was adopted, which turned out to be quite sufficient.Results and conclusions. The paper presents the results of a numerical study of the oscillatory processes of an off-road vehicle during uniform translational motion of the vehicle on a horizontal surface with a given profile of irregularities. The graphs show a transition process of oscillations, which ends with reaching a steady state. The shape of oscillations in a steady state can be irregular and significantly depends on the given speed of the all-terrain vehicle. Analysis of the dependencies presented in the figures shows that the shape of the oscillations of the allterrain vehicle’s frame, as well as the amplitude and frequency, significantly depend on the speed of the vehicle (at a constant road profile). Changing the road profile leads to corresponding changes in the forms and characteristics of forced vibrations of a vehicle on a suspension, which makes it possible to build the necessary amplitude-frequency characteristics, optimize the elastic and dissipative parameters of suspensions, optimize their number and location, and also monitor the movements of arbitrary points at which various units are located.

Stability analysis of uncertain simple pendulum equation *

The law of motion of a simple pendulum system is described by an uncertain simple pendulum equation which is a second-order uncertain differential equation driven by Liu process (LP). The stability of a simple pendulum system refers to whether the system tends to the equilibrium state under small perturbation. In order to discuss the sensitivity of the uncertain simple pendulum equation to the perturbation in the initial state, we give the concept of many kinds of stability of the uncertain simple pendulum equation, including almost deterministic stability, distributional stability and exponential stability. And, the sufficient conditions of almost deterministic stability, distributional stability and exponential stability of the uncertain simple pendulum equation are proved respectively.

Topology and Dynamics of Transcriptome (Dys)Regulation.

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of these sequences. We have previously devised a mathematical approach for defining a "healthy" sequence. This sequence is characterized by having at most four distinct nucleotides (denoted as nt≤4). It serves as the generator of a group denoted as fp. The desired properties of this sequence are as follows: fp should be close to a free group of rank nt-1, it must be aperiodic, and fp should not have isolated singularities within its SL2(C) character variety (specifically within the corresponding Groebner basis). Now, let us explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as S24. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI Equation. In certain cases, S24 degenerates, in the sense that two punctures collapse, resulting in a "wild" dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find many applications including cancer research and the investigation of neurodegenative diseases.

A High-Resolution Exponential Spline Method and Its Convergence Analysis for Two-Point Mixed Boundary Value Problems

In this paper, we present a new numerical method for effectively solving general second-order ordinary differential equations with mixed boundary conditions. Our approach utilizes a quasi-variable mesh enabling us the flexibility to adapt the mesh density according to different boundary layer problems. By employing a discretization technique that incorporates the construction of exponential spline, we achieve third-order accuracy at internal grid points, while boundary points exhibit fourth-order accuracy. Since the method is based on off-step grid points, it eliminates the need to modify the method while employing it to singular problems. We provide computational results to various numerical problems which arise in different physical phenomena like Burger’s equation and Sturm–Liouville equation. A comparison with recent findings underscores the superior performance of our method.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Background of Second-Order Linear Differential Equations

The differential equations, usually the goal is to find a solution. In other words, we want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with which we are working. Second-order differential equations have several important characteristics that can help us determine which solution method to use. In this section, we examine some of these characteristics and the associated terminology. An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation (DE).

Attached Flows for Reaction–Diffusion Processes Described by a Generalized Dodd–Bullough–Mikhailov Equation

This paper uses the attached flow method for solving nonlinear second-order differential equations of the reaction–diffusion type. The key steps of the method consist of the following: (i) reducing the differentiability order by defining the first derivative of the variable as a new variable called the flow and (ii) a forced decomposition of the derivative-free term so that the flow appears explicitly in it. The resulting reduced equation is solved using specific balancing rules. Only step (i) would lead to an Abel-type equation with complicated integral solutions. Completed with (ii) and with the graduation procedure, the attached flow method used in the paper, without requiring such a great effort, allows for the obtaining of accurate analytical solutions. The method is applied here to a subclass of reaction–diffusion equations, the generalized Dodd–Bulough–Mikhailov equation, which includes a translation of the variable and nonlinearities up to order five. The equation is solved for each order of nonlinearity, and the solutions are discussed following the values of the parameters involved in the equation.

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

In this paper, a one-step block of optimized hybrid schemes for the numerical integration of second-order initial value problems (IVP) of ordinary differential equations (ODE) is constructed via collocation techniques. The developed scheme is obtained by considering two intra-step nodal points as hybrid points, which are chosen in order to achieve optimized errors of the main formulae approximating the solution such that 0 < v1 < v 2 < 1 where v1 and v2 are defined as hybrid points. The characteristics of the developed scheme are analyzed. Application of the new scheme on some second-order IVPs shows the accuracy and effectiveness of the scheme compared with some existing methods.

Double shooting method for FRCM reinforced systems in debonding problems

The debonding process in FRCM (Fiber Reinforced Cementitious Matrix) strengthened brittle substrates, such as concrete and masonry, is characterized by complex failure modes that include debonding at both the external and internal interfaces of the composite, as well as damage to the mortar matrix. This study addresses this problem classically as a mode-II fracture process by accounting for the one-dimensional non-linear interfacial shear stress-slip relationship combined with matrix axial non-linearity. The approach employed considers the longitudinal displacements of both the inner and outer layers of mortar, along with that of the fiber textile, as unknowns, and analytically solves the second-order ordinary differential equation (ODE) system that governs the debonding problem. Then, the boundary value problem (BVP) is transformed into an initial value problem (IVP) and a double shooting method is proposed to find displacements and stresses along the bonded length in the presence of non-linearity. The appropriate initial values of matrix layer displacements, satisfying the required boundary conditions, are efficiently determined through a two-dimensional bisection method working on rectangular and triangular patches. The softening behavior of both mortar layers and interfaces between mortar and fiber are taken into account using jagged constitutive relationships to enhance convergence. The numerical calculation speed, robustness, and key simulation parameters are thoroughly discussed. Moreover, the proposed numerical approach is compared with existing experimental data and models, on both concrete and masonry FRCM-reinforced specimens. The influence of mortar failure and interfacial strength is also explored. The model demonstrates its ability to effectively reproduce the global bond behavior observed in experimental studies, while capturing the local behavior and simulating various failure modes observed in the tests.

A Novel Approach to Linear and Nonlinear Time-History Analysis of Structures: Gauss–Lobatto–Hermite 4-Point (GLH-4P) Method

AbstractAn efficient time integration scheme has been devised for conducting time-history dynamic analysis of single-degree-of-freedom (SDOF) structural systems. The developed scheme belongs to Runge-Kuta family. The primary innovation of the new formulation lies in the constructive cooperation between the Hermite interpolation and the Gauss–Lobatto integration, linked to address the second-order ordinary differential equation of motion (DEOM). To put the idea into practice, several Hermite interpolators are generated to exploit the information of the internal points of Gauss-Lobatto integrator. The 4-point Gauss–Lobatto formula is employed to numerically integrate acceleration and velocity functions. This novel methodology has been designated the Gauss–Lobatto–Hermite 4-point (GLH-4P) method. GLH-4P furnishes a generally applicable implicit algorithm for conducting both linear and nonlinear analyses of structures subjected to seismic excitation. The advantage of the proposed algorithm over the conventional techniques is its higher accuracy level. Besides, it offers better stability compared with the conventional methods such as the Newmark-β and Wilson-θ methods. Unlike the other methods, GLH-4P can handle high-frequency systems without reducing the step size. Numerical examples reveal the efficacy of the GLH-4P method when juxtaposed against existing methodologies.

Behavior of a System of Singularly Perturbed Differential Equations with Generalized Inhomogeneous Parts

The work studies systems of singularly perturbed second-order differential equations. Heterogeneity is a singular generalized Dirac function that has only a value at one point. In this case, the phenomenon of prolonged loss of stability takes place, and the estimate of solutions to the singularly perturbed problem depends on the point a.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.