Search Results (9,765)

In this study, the influence of thermal loading in a supersonic flight environment on the mechanical stiffness of elastic structures and the corresponding aeroelastic stability limits is investigated analytically. Recognizing that elevated temperatures inherently alter constituent elastic properties, a temperature-dependent continuous elasticity framework is incorporated directly into the governing differential operators of the structural domain. The macro-mechanical behavior of representative panel- and wing-type elements is modeled utilizing the Euler–Bernoulli beam formulation, while high-speed supersonic aerodynamic effects are represented through linearized first-order piston theory. The continuous spatial displacement fields are discretized by means of a modal expansion, and the coupled aeroelastic system is subsequently transformed into a finite set of dynamic state-space equations using the Ritz–Galerkin truncation method. The numerical and analytical outputs demonstrate that aerothermal softening not only induces continuous erosion in the material stiffness but also directly modulates the aeroelastic pole trajectories, thereby prematurely contracting the safe supersonic flight envelope. The primary novelty of the proposed framework lies in the derivation of explicit analytical expressions that directly map temperature-dependent stiffness variations onto supersonic aeroelastic instability boundaries. Because this approach is formulated in a generalized analytical form, it can be applied across diverse material systems, geometric profiles, and thermal conditions with reduced computational overhead compared to full fluid–structure interaction solvers, thereby providing a theoretical basis for preliminary stability assessment of supersonic aerospace configurations operating under high-temperature conditions. Full article

This article shows the geometric decay rate of the Euler–Maruyama scheme for a one-dimensional stochastic differential equation towards its invariant probability measure under total variation distance. Firstly, the existence and uniqueness of invariant probability measure and the uniform geometric ergodicity of the chain are studied through the introduction of non-atomic Markov chains. Secondly, the equivalent conditions for uniform geometric ergodicity of the chain are discovered by constructing a split Markov chain based on the original Euler–Maruyama scheme. Full article

Accurate modeling of bi-flux anomalous diffusion presents significant computational challenges in engineering. This paper investigates the effectiveness of physics-informed neural networks as surrogate models for the bi-flux anomalous diffusion equation. We investigate one-dimensional linear and nonlinear cases. Optimal hyperparameter configurations are determined using a modified differential evolution algorithm, guided by an objective function that leverages a combination of loss values. This optimization approach enables a rigorous evaluation of different neural network setups, providing valuable insights and practical guidance for researchers working with bi-flux anomalous diffusion phenomena. A comparison between physics-informed neural networks and conventional multilayer perceptrons is presented for the analyzed model. Finally, the capability of the best-performing models to act as virtual sensors is evaluated. This work provides guidance on the use of neural networks to efficiently and accurately tackle complex bi-flux anomalous diffusion problems, potentially accelerating research and development in fields where such processes are critical. Full article

This paper investigates the asymptotic stability of a one-dimensional porous elastic system subject to a single boundary control of the fractional derivative type. The system consists of two coupled hyperbolic equations describing the displacement of an elastic solid and the volume fraction, with boundary conditions that include a generalized Caputo fractional derivative of order $\alpha \in (0,1)$ at $x=L$. This configuration has not been previously addressed in the literature. Using semigroup theory, we first reformulate the system as an abstract Cauchy problem and prove that the associated operator generates a $C_0$-semigroup of contractions on a suitable energy space, establishing global well-posedness. Under explicit and generic conditions on the physical parameters and the length L, we prove strong stability via the Arendt–Batty criterion, showing that all solutions tend to zero in the energy norm as $t\to \infty$. The main result is a polynomial decay rate: there exists $c>0$ such that $\parallel S_{\mathcal{A}}\left(t\right)U_0{\parallel }_{\mathcal{H}}\le c{t}^{-1/6}{\parallel U_0\parallel }_{D\left(\mathcal{A}\right)}$ for all initial data in the domain of the generator. The proof relies on the Borichev–Tomilov theorem and a detailed contradiction argument based on asymptotic expansions of the resolvent operator. Notably, the decay rate is independent of any relation between the wave propagation speeds, which distinguishes our result from many previous studies on porous elastic or Timoshenko systems. Full article

This paper addresses the input-to-state stability (ISS) in different $L^q$-norms for a class of coupled nonlinear partial differential equations of parabolic type subject to both in-domain disturbances and Dirichlet boundary disturbances, where $q\in [1,+\infty )$. Specifically, we first prove the continuous dependence of solutions to the system on initial data and disturbances in different $L^q$-norms by using the generalized Lyapunov method, and subsequently derive ISS estimates via a density argument. The main challenge arises in handling the nonlinear coupling terms and deriving ISS small-gain conditions within the generalized Lyapunov framework, as each coupling term depends on all other state variables of the system. Full article

In this paper, we investigate the existence and positivity of solutions for a class of twelfth-order nonlinear boundary value problems that naturally arise in the mathematical modeling of elastic and micro-mechanical systems. The considered model incorporates higher-order derivatives to account for nonlocal and gradient effects that commonly appear in the analysis of micro- and nano-scale elastic structures. By employing the Leray–Schauder nonlinear alternative and fixed point theorems, we establish sufficient conditions for the existence of at least one positive solution. The analysis relies on the explicit construction and properties of the associated Green’s function, which plays a fundamental role in deriving upper and lower bounds for the nonlinear term. The obtained results extend and generalize earlier works on sixth, eighth and tenth-order problems to the twelfth-order case. Finally, numerical examples are presented to illustrate the applicability and accuracy of the theoretical findings. The results provide a rigorous analytical foundation for the study of high-order elastic models and micro-scale structural stability. Full article

Studying Bingham flows in permeable media under a periodic magnetic field enhances the understanding of yield-stress fluids for applications like oil recovery and filtration. This study combines non-Newtonian behavior with porous-medium resistance and magnetic variations, facilitating the analysis of complex flow phenomena, including oscillatory yielding and improved flow control in porous structures. The viscous potential theory is employed to streamline the mathematical processes. The utilization of linear governing partial differential equations of motion, along with appropriate nonlinear boundary conditions, yields additional simplifications. The investigation yields a nonlinear Mathieu oscillator that governs the interfacial displacement. A non-perturbative method is used to convert this nonlinear ordinary differential equation into a linear equation. A non-dimensional formulation minimizes the fundamental variables required to characterize the system by establishing a collection of dimensionless physical characteristics. The study analyzes a nonlinear Mathieu oscillator with complex coefficients to explore system dynamics related to elevation. By simplifying the variable coefficients, it enhances the examination of stability and resonance behavior. Despite inherent complexities, the work effectively clarifies fundamental concepts, contributing to a more coherent understanding of the subject. The Hartman number, magnetic field, and magnetic permeability ratio exert a destabilizing effect. Conversely, the Bingham parameter, Weber number, and periodic frequency exert a stabilizing influence. Full article

This paper proposes a localized Hermite method of approximate particular solutions (LHMAPS) for solving the 2D inhomogeneous Helmholtz-type equations. Building on the local scheme of the localized method of approximate particular solutions (LMAPS) for the Helmholtz-type differential operator, LHMAPS employs Hermite-type local approximations involving both the solution values and their Laplacian to improve the accuracy of LMAPS. The polyharmonic spline (PS) radial basis functions and polynomial basis functions are considered in the formulation of LHMAPS. Numerical experiments are presented to demonstrate the enhanced accuracy achieved by employing Hermite-type local approximations. Full article

The well-known effect of stabilization by noise for Ito’s scalar linear stochastic differential equation was proven by R.Z. Khasminskii more than 50 years ago. Here, a similar statement is obtained for a system of linear stochastic differential equations. The obtained result is illustrated on the system of two linear stochastic differential equations via several special examples with numerical simulations and figures. Full article

In transient stability preventive control of power systems, time-domain simulation is computationally intensive. In addition, the initial operating feature data often contain abundant redundant and irrelevant information. These factors may adversely affect the assessment performance of machine learning models. To address these issues, a transient stability preventive control method based on the sample convolution and interaction network (SCINet) is proposed. First, a feature selection algorithm based on the orthogonal maximal information coefficient and information gain (OMICIG) is developed to extract the key operating features of the system. Second, the SCINet model is employed to learn the nonlinear mapping relationship between the selected key operating features and the transient stability index (TSI). Then, the trained SCINet model is embedded into the transient stability constrained optimal power flow (TSCOPF) model as a surrogate transient stability constraint. In this way, the complicated computation associated with nonlinear differential-algebraic equations (DAE) in the conventional TSCOPF model is avoided. Furthermore, an improved dung beetle optimizer (IDBO) algorithm is used to iteratively solve the resulting model, thereby deriving a preventive control strategy that ensures transient stability while maintaining system operating economy. Finally, simulation studies on the New England 10-machine 39-bus and the IEEE 118-bus system demonstrate the effectiveness of the proposed method. Full article

This work investigates the controllability and stability properties of nonlinear piecewise dynamical systems subject to integral boundary conditions studied in arbitrary time domains. The study employs powerful mathematical ideas like fixed point theorems, Gramian-type matrices and the unified approach given by the theory of time scales. This unified approach provides results applicable seamlessly in various settings of continuous-time systems, discrete-time systems, and systems with hybrid behavior of both. This paper expands existing findings in the literature, providing a more complete view. To show the application of the theoretical results, a detailed example is presented and supported by numerical simulations to confirm the efficiency of the proposed methods. Full article

In our research, we developed a Distributed Relaxation Spectrum Delay Differential Equation (DRSDDE) model to simulate viscoelastic responses exhibited by materials with multiple-scale relaxation mechanisms and finite delay times. Our model expanded upon traditional integer-order viscoelastic models to include a continuum relaxation process using a log-time-space Gaussian distribution representing a continuum of relaxation processes, including a direct representation of the effect of delayed feedback via an explicit time delay term. Consequently, the resultant model can be viewed as a generalized Maxwell-type formulation where the viscoelastic behavior exhibits distributed relaxation dynamics and has finite signal propagation characteristics. We then used experimental data obtained from three representative materials: PDMS Sylgard 184, bovine brain white matter, and polyurethane foam to calibrate the model. Calibration was achieved by estimating model parameters through the use of Gauss-Legendre quadrature combined with non-linear optimization of the relaxation spectrum. The results indicate that the coefficients of determination for each of the materials exceeded R² > 0.83. Therefore, the proposed DRSDDE model outperformed the classical Zener model when simulating materials that exhibit a wide relaxation spectrum. The parameter values estimated for each of the examined materials provided additional insight into their physical behaviors. Specifically, the characteristic relaxation times for the studied materials were determined based upon \(\tau\)_c = 10^µ ranging from about 63 s to 158 s. These results illustrate different dominant relaxation regimes for the investigated materials. Additionally, both characteristic equations and frequency domain analyses were utilized to study the stability and bifurcation properties of the DRSDDE model. A significant finding resulted from identifying a delay-insensitive stability regime for materials with \(\tilde{K} < 1\) (as illustrated by bovine brain white matter). For materials with \(\tilde{K} > 1\), the analysis revealed Hopf bifurcation results illustrating critical delay thresholds and frequencies for the onset of oscillations. Further, it was established that all calibrated delay values were significantly less than these threshold values. This indicates that all identified models functioned well below the oscillation thresholds at realistic delay times. Ultimately, the proposed DRSDDE model represents a physically intuitive, robust, and flexible method for modeling complex viscoelastic systems. Future research will involve investigating temperature-dependent effects, nonlinear bifurcations, and experimental validations of predicted oscillatory dynamics Full article

A non-homogeneous fractional differential equation with a variable coefficient involving a Caputo fractional derivative is considered. The equation is first transformed into an integral equation and then solved using the method of successive approximations. The obtained general solution involves a generalized Mittag–Leffler-type function and Meijer G-functions. Example solutions corresponding to particular choices of the non-homogeneous term are presented. As an application of the considered non-homogeneous equation, direct and inverse source problems are studied. The solutions are expressed in the form of series expansions using an orthogonal basis obtained through separation of variables. Illustrative examples for the direct and inverse problems are also presented for specific choices of the initial and final time data and the source function. Full article

In this work, we develop two new classes of rational contraction mappings along with the corresponding fixed point theorems for these types of contractions in suprametric spaces. Furthermore, we use the obtained results to investigate two nonlinear systems, namely, a fractional chaotic financial system and a nonlinear fractional differential equation under integral boundary conditions. Both these nonlinear problems are transformed into fixed point problems in appropriate suprametric spaces, thereby demonstrating the applicability of the developed rational contraction results to nonlinear systems with memory effects. Full article

This research explores the existence of solutions for a class of random fractional differential equations characterized by bounded delay, specifically within the context of Fréchet spaces. Random fractional differential equations serve as powerful mathematical tools for modeling complex phenomena subjected to stochastic perturbations and hereditary effects. Despite their significance, establishing solution existence in infinite-dimensional spaces remains a challenging task. By integrating the properties of the noncompactness measures with a generalized Darbo fixed point approach, we establish new existence results for the associated Darboux-type problem under milder compactness conditions. To illustrate the practical utility of these analytical results and demonstrate the validity of our theoretical framework, a representative example is provided. Full article