Search Results (1,539)

The Navier equations are reformulated to be third-order partial differential equations. New anti-Cauchy-Riemann equations can express a general solution in 2D space for incompressible materials. Based on the third-order solutions in 3D space and the Boussinesq–Galerkin method, a third-order method of fundamental solutions (MFS) is developed. For the 3D Navier equation in linear elasticity, we present three new general solutions, which have appeared in the literature for the first time, to signify the theoretical contributions of the present paper. The first one is in terms of a biharmonic function and a harmonic function. The completeness of the proposed general solution is proven by using the solvability conditions of the equations obtained by equating the proposed general solution to the Boussinesq–Galerkin solution. The second general solution is expressed in terms of a harmonic vector, which is simpler than the Slobodianskii general solution, and the traditional MFS. The main achievement is that the general solution is complete, and the number of harmonic functions, three, is minimal. The third general solution is presented by a harmonic vector and a biharmonic vector, which are subjected to a constraint equation. We derive a specific solution by setting the two vectors in the third general solution as the vectorizations of a single harmonic potential. Hence, we have a simple approach to the Slobodianskii general solution. The applications of the new solutions are demonstrated. Owing to the minimality of the harmonic functions, the resulting bases generated from the new general solution are complete and linearly independent. Numerical instability can be avoided by using the new bases. To explore the efficiency and accuracy of the proposed MFS variant methods, some examples are tested. Full article

In this paper, we delve into the following nonlinear fractional Kirchhoff-type problem ${(a+b||(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\left|\right|}_2^2{)(-\Delta )}^{s}u+\lambda u=g\left(u\right)+{\left|u\right|}^{2_s^{*}-2}u$ in ${\mathbb{R}}^3$ with prescribed mass ${\int }_{{\mathbb{R}}^3}{\left|u\right|}^{2}dx=\rho >0,$ where $s\in ({\displaystyle \frac{3}{4}},1),\lambda \in \mathbb{R},2_s^{*}={\displaystyle \frac{6}{3-2s}}$. Under some general growth assumptions imposed on g, we employ minimization of the energy functional on the linear combination of Nehari and Poho$\breve{z}$aev constraints intersected with the closed ball in the $L^2\left({\mathbb{R}}^3\right)$ of radius $\rho$ to prove the existence of normalized ground state solutions to the equation. Moreover, we provide a detailed description for the asymptotic behavior of the ground state energy map. Full article

In this article, we investigate the influence of viscosity on the evolution of the cosmos within the framework of the newly proposed $f(Q,L_m)$ gravity. We have considered a linear functional form $f(Q,L_m)=\alpha Q+\beta L_m$ with a bulk viscous coefficient $\zeta ={\zeta }_0+{\zeta }_{1}H$ for our analysis and obtained exact solutions to the field equations associated with a flat FLRW metric. In addition, we utilized Cosmic Chronometers (CC), CC + BAO, CC + BAO + GRB, and GRB data samples to determine the constrained values of independent parameters in the derived exact solution. The likelihood function and the Markov Chain Monte Carlo (MCMC) sampling technique are combined to yield the posterior probability using Bayesian statistical methods. Furthermore, by comparing our results with the standard cosmological model, we found that our considered model supports the acceleration of the universe in late time. Full article

Computational imaging spectrometers using broad-bandpass filter arrays with distinct transmission functions are promising implementations of miniaturization. The number of filters is limited by the practical factors. Compressed sensing is used to model the system as linear underdetermined equations for hyperspectral imaging. This paper proposes the following method: parallel dictionary reconstruction and fusion for spectral recovery in computational imaging spectrometers. Orthogonal systems are the dictionary candidates for reconstruction. According to observation of ground objects, the dictionaries are selected from the candidates using the criterion of incoherence. Parallel computations are performed with the selected dictionaries, and spectral recovery is achieved by fusion of the computational results. The method is verified by simulating visible-NIR spectral recovery of typical ground objects. The proposed method has a mean square recovery error of ≤1.73 × 10⁻⁴ and recovery accuracy of ≥0.98 and is both more universal and more stable than those of traditional sparse representation methods. Full article

To characterize a phenomenological model of a mechanical oscillator, it is important to know the properties of the envelope of the three main physical motion variables: deviation from equilibrium, velocity, and acceleration. Experimental data show that friction forces restrict the shape of these functions. A linear, exponential, or more abrupt decay can be observed depending on the different physical systems and conditions. This paper aimed to contribute to clarifying the role that some types of friction forces play in these shapes. Three types of friction—constant sliding friction, pressure drag proportional to the square of velocity, and friction drag proportional to velocity—were considered to characterize the line connecting the maxima and minima of displacement for a generic mechanical harmonic oscillator. The ordinary differential equation (ODE), describing the harmonic oscillator simultaneously containing the three types of dissipative forces (constant, viscous, and quadratic), was numerically solved to obtain energy dissipation, and the extrema of both displacement and velocity. The differential equation ruling the behavior of the amplitude, as a function of the friction force coefficients, was obtained from energy considerations. Solving this equation, we obtained analytical functions, parametrized by the force coefficients that describe the oscillator tail. A comparison between these functions and the predicted oscillator ODE extrema was made, and the results were in agreement for all the situations tested. Information from the velocity extrema and nulls was enough to obtain a second function that rules completely the ODE solution. The correlations obtained allow for the reverse operation: from the identified extremum data, it was possible to identify univocally the three friction coefficients fitting used in the model. Motion equations were solved, and some physical properties, namely energy conservation and work of friction forces, were revisited. Full article

This paper addresses the problem of optimal stabilization for stochastic dynamical systems characterized by Markov switches and concentration points of jumps, which is a scenario not adequately covered by classical stability conditions. Unlike traditional approaches requiring a strictly positive minimal interval between jumps, we allow jump moments to accumulate at a finite point. Utilizing Lyapunov function methods, we derive sufficient conditions for exponential stability in the mean square and asymptotic stability in probability. We provide explicit constructions of Lyapunov functions adapted to scenarios with jump concentration points and develop conditions under which these functions ensure system stability. For linear stochastic differential equations, the stabilization problem is further simplified to solving a system of Riccati-type matrix equations. This work provides essential theoretical foundations and practical methodologies for stabilizing complex stochastic systems that feature concentration points, expanding the applicability of optimal control theory. Full article

A well-known model of infectious diseases, described by a nonlinear system of delay differential equations, is investigated under the influence of stochastic perturbations. Using the general method of Lyapunov functional construction combined with the linear matrix inequality (LMI) approach, we derive sufficient conditions for the stability of the equilibria of the considered system. Numerical simulations illustrating the system’s behavior under stochastic perturbations are provided to support the thoretical findings. The proposed method for stability analysis is broadly applicable to other systems of nonlinear stochastic differential equations across various fields. Full article

The objective of the present study is to detect chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. By virtue of the traveling wave hypothesis, the discussed model is reduced to a simple form known as an elliptic equation. The latter equation, which is a second-order ordinary differential equation, is handled by the undetermined coefficient method of two forms expressed in terms of the hyperbolic secant and tangent functions. Additionally, the auxiliary equation method is applied to derive several miscellaneous solutions. Various types of chirped solitons are revealed such as W-shaped, bright, dark, gray, kink and anti-kink waves. Taking into consideration the existence conditions, the dynamical behaviors of optical solitons and their corresponding chirp are illustrated. The modulation instability of the perturbed CLL equation is examined by means of the linear stability analysis. It is found that all solutions are stable against small perturbations. These entirely new results, compared to previous works, can be employed to understand pulse propagation in optical fiber mediums and dynamic characteristics of waves in plasma. Full article

A meshless, quasi-convex reproducing kernel particle framework for three-dimensional steady-state thermomechanical coupling problems is presented in this paper. A meshfree, second-order, quasi-convex reproducing kernel scheme is employed to approximate field variables for solving the linear Poisson equation and the elastic thermal stress equation in sequence. The quasi-convex reproducing kernel approximation proposed by Wang et al. to construct almost positive reproducing kernel shape functions with relaxed monomial reproducing conditions is applied to improve the positivity of the thermal matrixes in the final discreated equations. Two numerical examples are given to verify the effectiveness of the developed method. The numerical results show that the solutions obtained by the quasi-convex reproducing kernel particle method agree well with the analytical ones, with a slightly better-improved numerical accuracy than the element-free Galerkin method and the reproducing kernel particle method. The effects of different parameters, i.e., the scaling parameter, the penalty factor, and node distribution on computational accuracy and efficiency, are also investigated. Full article

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. Full article

This paper establishes some links between Sturm–Liouville problems and the well-known controllability property in linear dynamic systems, together with a control law design that allows any prefixed arbitrary final state finite value to be reached via feedback from any given finite initial conditions. The scheduled second-order dynamic systems are equivalent to the stated second-order differential equations, and they are used for analysis purposes. In the first study, a control law is synthesized for a forced time-invariant nominal version of the current time-varying one so that their respective two-point boundary values are coincident. Afterward, the parameter that fixes the set of eigenvalues of the Sturm–Liouville system is replaced by a time-varying parameter that is a control function to be synthesized without performing, in this case, any comparison with a nominal time-invariant version of the system. Such a control law is designed in such a way that, for given arbitrary and finite initial conditions of the differential system, prescribed final conditions along a time interval of finite length are matched by the state trajectory solution. As a result, the solution of the dynamic system, and thus that of its differential equation counterpart, is subject to prefixed two-point boundary values at the initial and at the final time instants of the time interval of finite length under study. Also, some algebraic constraints between the eigenvalues of the Sturm–Liouville system and their evolution operators are formulated later on. Those constraints are based on the fact that the solutions corresponding to each of the eigenvalues match the same two-point boundary values. Full article

This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the time-reversal symmetry present in linear oscillatory systems. Applying the LADM and HPM, we derive approximate solutions for the Rayleigh oscillator. Due to the absence of exact analytical solutions in the literature, these approximations are benchmarked against high-precision numerical results obtained using Mathematica’s NDSolve function. We perform a detailed error analysis across different damping parameter values ε and time intervals. Our results reveal how the asymmetric damping influences the accuracy and convergence behavior of each method. This study highlights the role of nonlinear asymmetry in shaping the solution dynamics and provides insight into the suitability of the LADM and HPM under varying conditions. Full article

This research developed a complex physical and mathematical model of the flat rolling theory problem. This model takes into account the influence of many parameters affecting the roll’s gripping capacity and the overall stability of the entire rolling process. It is important to emphasize that the method of the argument of functions of a complex variable does not rely on simplifying assumptions commonly associated with: the linearized theory of plasticity; or the decoupled solution of stress and strain fields. Furthermore, it does not utilize the rigid-plastic material model. Within this method, solutions are developed based on the complete formulation of the system of equations in terms of stresses and strains, incorporating constitutive relations, thermal effects, and boundary conditions that define a well-posed problem in the theory of plasticity. The presented applied problem is closed in nature, yet it accounts for the effects of mechanical loading and satisfies the system of equation. For this purpose, such factors as roll geometry, physical and mechanical properties of the rolled metal (including its fluidity, hardness, plasticity, and structure heterogeneity), rolling speed, metal temperature, roll lubrication, and many other parameters that can influence the process have been taken into account. Based on the developed mathematical model, a new, previously undescribed force factor significantly affecting the capture of metal by rolls and the stability of the rolling process was identified and investigated in detail. This factor is associated with force stretching of metal in the lagging zone—the area behind the rolls, where the metal has already left the deformation zone, but continues to experience residual stress. It was shown that this stretching, depending on the process parameters, can both contribute to the rolling stability and, on the contrary, destabilize it, causing oscillations and non-uniformity of deformation. The qualitative indicators of transient regime stability have been determined for various values of the parameter α. Specifically, for α = 0.077, the ratio f/α ranges from 1.10 to 1.95; for α = 0.129, the ratio f/α ranges from 1.19 to 1.95; and for α = 0.168, the ratio f/α ranges from 1.28 to 1.95. Full article

Settlement analyses of foundation soils are very important for the investigation, design, and construction of buildings. However, due to complex natural sedimentary processes, soil-forming environments, and geological tectonic stress histories, settlement properties show obvious spatial variability and autocorrelation. Moreover, measurement data on the physical and mechanical parameters of building foundation soils are limited. This limits the accuracy of formation stability analyses and safety evaluations. In this study, a series of field tests of building foundation soils were carried out, and the statistical physical and mechanical properties of the clay strata were obtained. A random field method and copula functions of uncertain geotechnical properties with limited survey data are proposed. A dual-yield surface constitutive model of the soil properties and a stability analysis method for uncertain deformation were developed. The detailed analytical procedures for soil deformation and stratum settlement are presented. The reliability functions and failure probabilities of variable settlement processes are calculated and analyzed. The impact of the spatial variation and cross-correlation of geotechnical properties on the probabilistic stability of variable land subsidence is discussed. This work presents an innovative analysis approach for evaluating the variable settlement properties of building foundation soils. The results show that the four different mechanical parameters can be regressed to linear equations. The horizontal fluctuation scale is significantly larger than the vertical scale. Copula theory provides a powerful framework for modeling limited geotechnical parameters. The bootstrap approach avoids parametric assumptions, leveraging empirical data to enhance the reliability analysis of variable settlement. The variability parameter exerts a greater influence on land subsidence processes than the correlation structure. The failure probabilities of variable stratum settlement for different cross-correlations of building foundation soils are different. These results provide an important reference for the safety of building engineering. Full article

This paper presents a modeling framework for hysteretically nonlinear piezoelectric composite beams using functional differential equations (FDEs). While linear piezoelectric models are well established, they fail to capture the complex nonlinear behaviors that emerge at higher electric field strengths, particularly history-dependent hysteresis effects. This paper develops a cascade model that integrates a high-dimensional linear piezoelectric composite beam representation with a nonlinear Krasnosel’skii–Pokrovskii (KP) hysteresis operator. The resulting system is formulated using a state-space model where the input voltage undergoes a history-dependent transformation. Through modal expansion and discretization of the Preisach plane, we derive a tractable numerical implementation that preserves essential nonlinear phenomena. Numerical investigations demonstrate how system parameters, including the input voltage amplitude, and hysteresis parameters significantly influence the dynamic response, particularly the shape and amplitude of limit cycles. The results reveal that while the model accurately captures memory-dependent nonlinearities, it depends on numerous real and distributed parameters, highlighting the need for efficient reduced-order modeling approaches. This work provides a foundation for understanding and predicting the complex behavior of piezoelectric systems with hysteresis, with potential applications in vibration control, energy harvesting, and precision actuation. Full article