Search Results (16)

In this contribution, we propose and study long-time behaviors of a new class of N-dimensional delayed Kirchhoff–Carrier-type problems with variable transfer coefficients involving a logarithmic nonlinearity. We take into account the dependence of diffusion and damping coefficients on the position and direction, as well as the presence of different types of delays. This class of nonlocal anisotropic and nonlinear wave-type equations with multiple time-varying mixed delays and dampings, of a fairly general form, containing several arbitrary functions and free parameters, is of the following form: ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}-{div(\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }{(\mathbf{x})\nabla u)+\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)u-div(\zeta (t)A_{\sigma }(\mathbf{x})\nabla \frac{\partial u}{\partial t})+d_0(t)\frac{\partial u}{\partial t}+{\mathbb{D}}_r(\mathbf{x},t;\frac{\partial u}{\partial t})=\mathbb{G}(u),}$ where $u(\mathbf{x},t)$ is the state function, $\mathcal{M}$ and $\mathcal{K}$ are the nonlocal Kirchhoff operators and the nonlinear operator $\mathbb{G}(u)$ corresponds to a logarithmic source term. The symmetric tensor $A_{\sigma }$ describes the anisotropic behavior and processes of the system, and the operator ${\mathbb{D}}_r$ represents the multiple time-varying mixed delays related to velocity ${\displaystyle \frac{\partial u}{\partial t}}$. Our problem, which encompasses numerous equations already studied in the literature, is relevant to a wide range of practical and concrete applications. It not only considers anisotropy in diffusion, but it also assumes that the strong damping can be totally anisotropic (a phenomenon that has received very little mathematical attention in the literature). We begin with the reformulation of the problem into a nonlinear system coupling a nonlocal wave-type equation with ordinary differential equations, with the help of auxiliary functions. Afterward, we study the local existence and some necessary regularity results of the solutions by using the Faedo–Galerkin approximation, combining some energy estimates and the logarithmic Sobolev inequality. Next, by virtue of the potential well method combined with the Nehari manifold, conditions for global in-time existence are given. Finally, subject to certain conditions, the exponential decay of global solutions is established by applying a perturbed energy method. Many of the obtained results can be extended to the case of other nonlinear source terms. Full article

High-Voltage Alternating Current (HVAC) transmission systems form the backbone of modern power grids, enabling efficient long-distance and high-capacity power delivery. In Saudi Arabia, ongoing initiatives to modernize and strengthen grid infrastructure demand advanced solutions to ensure system reliability, operational stability, and the minimization of economic losses caused by faults. Traditional fault detection and classification methods often depend on the manual interpretation of voltage and current signals, which is both labor-intensive and prone to human error. Although data-driven approaches such as Artificial Neural Networks (ANNs) and Deep Learning have been applied to automate fault analysis, their performance is often constrained by the quality and size of available training datasets, leading to poor generalization and physically inconsistent outcomes. This study proposes a novel hybrid fault detection and classification framework for the 380 kV Marjan–Safaniyah HVAC transmission line by integrating Deep Learning with Physics-Informed Neural Networks (PINNs). The PINN model embeds fundamental electrical laws, such as Kirchhoff’s Current Law (KCL), directly into the learning process, thereby constraining predictions to physically plausible behaviors and enhancing robustness and accuracy. Developed in MATLAB/Simulink using the Deep Learning Toolbox, the proposed framework performs fault detection and fault type classification within a unified architecture. A comparative analysis demonstrates that the hybrid PINN approach significantly outperforms conventional Deep Learning models, particularly by reducing false negatives and improving class discrimination. Furthermore, this study highlights the crucial role of balanced and representative datasets in achieving a reliable performance. Validation through confusion matrices and KCL residual histograms confirms the enhanced physical consistency and predictive reliability of the model. Overall, the proposed framework provides a powerful and scalable solution for real-time monitoring, fault diagnosis, and intelligent decision-making in high-voltage power transmission systems. Full article

Nano-structures play a crucial role in advancing technology due to their unique properties and applications in various fields. This study examines the forced vibration behavior of an orthotropic nano-system consisting of an elastically connected nanoplate and a doubly curved shallow nano-shell. Both nano-elements are simply supported and embedded in a Winkler-type elastic medium. Utilizing the Eringen constitutive elastic relation, Kirchhoff–Love plate theory, and Novozhilov’s linear shallow shell theory, we derive a system of four coupled nonhomogeneous partial differential equations (PDEs) describing the forced transverse vibrations of the system. We perform forced vibration analysis using modal analysis. The developed model is a novel approach that has not been extensively researched by other authors. Therefore, we provide insights into the nano-system of an elastically connected nanoplate and a doubly curved shallow nano-shell, offering a detailed analytical and numerical analysis of the PDEs describing transverse oscillations. This includes a clear insight into natural frequency analysis and the effects of the nonlocal parameter. Additionally, damping proportional coefficients and external excitation significantly influence the transverse displacements of both the nanoplate and nano-shell. The proposed mathematical model of the ECSNPS aids in developing new nano-sensors that respond to transverse vibrations based on the geometry of the nano-shell element. These sensors are often used to adapt to curved surfaces in medical practice and gas sensing. Full article

Considering the solutions of a class of noncooperative Kirchhoff-type $p\left(x\right)$-Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the ${\left(PS\right)}_c^{*}$ condition using another method, diverging from the approaches cited in the previous literature. Full article

In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. Full article

This paper combines two types of energy storage components, the battery and supercapacitor (SC), to form a fully active hybrid energy storage system (HESS) as a power source for electric vehicles (EVs). At the same time, a hierarchical coordinated energy management strategy based on model predictive control (HCEMS-MPC) is presented. Firstly, the mathematical model of the fully active HESS is obtained based on Kirchhoff’s law and state-space modeling technology. Secondly, considering the state of charge (SOC) of the battery, a fuzzy-control-based upper-level energy management strategy (EMS) is proposed to optimize power allocation and to generate a reference current for a lower-level current controller. Then, a lower-level current predictive controller is designed to achieve accurate current tracking. Finally, a lower-level voltage sliding mode controller is designed to stabilize the bus voltage. Compared with previous works, the HCEMS-MPC strategy only needs to adjust the weight matrix and the reaching term to avoid the problem of excessive controller parameters. The simulation results, under different driving conditions, show that the HCEMS-MPC strategy has a better performance with respect to its fast response, error reduction, and robust stability. In addition, the SOC of the battery decreases more slowly, and the final SOC value significantly increases, thereby extending the single-discharge cycle time of the battery and improving the service life of the battery. Full article

We modeled and implemented a joint airborne system integrating ground penetrating radar (GPR) and magnetometer (MAG) models specifically for landmine detection applications. We conducted both simulations and experimental analyses of the joint airborne GPR and MAG models, with a focus on detecting the metallic components of different types of landmines, including antitank (AT) M15 metallic, antipersonnel (AP) M16 metallic, and AT M19 plastic (minimum-metal) landmines. The GPR model employed the finite-difference time-domain (FDTD) method and was evaluated using a singular value decomposition (SVD) and Kirchhoff migration (KM) with matched filtering (MF). These advanced techniques enabled the automatic identification and precise focusing of the reflected hyperbolic signals emitted by the landmines while considering cross-range resolution. Additionally, the MAGs were utilized based on the magnetic dipole model with a de-trend and a spatial median filtering method to estimate the magnetic anomaly of the landmines while considering various data spatial intervals. The joint airborne GPR and MAG system was implemented by combining and integrating the GPR and MAG models for experimental validation. Through this comprehensive approach, which included experiments, simulations, and data processing, the design parameters of the final system were obtained. These design parameters can be used in the development and application of landmine detection systems based on airborne GPR and MAG technology. Full article

This paper deals with the existence and multiplicity of solutions for a perturbed nonlocal fourth-order class of $p(·)\&q(·)$-Kirchhoff elliptic systems under Navier boundary conditions. By using the variational method and Ricceri’s critical point theorem, we can find a proper conditions to ensure that the perturbed fourth-order of $\left(p\right(x),q(x\left)\right)$-Kirchhoff systems has at least three weak solutions. We have extended and improved some recent results. The complexity of the combination of variable exponent theory and fourth-order Kirchhoff systems makes the results of this work novel and new contribution. Finally, a very concrete example is given to illustrate the applications of our results. Full article

First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, we define weak left fractional derivatives and demonstrate that they coincide with the left Riemann–Liouville ones on time scales. Next, we prove the equivalence of two kinds of norms in the introduced space and derive its completeness, reflexivity, separability, and some embedding. Finally, as an application, by constructing an appropriate variational setting, using the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary conditions is studied, and three results of the existence of weak solutions for this problem is obtained. Full article

Hybrid rigid–flexible mechanisms are a type of compliant mechanism that combines rigid and flexible elements, being that their mobility is due to rigid-body joints and the relative flexibility of bendable rods. Two of the modeling methods of flexible rods are the Cosserat rod model and its simplification, the Kirchhoff rod model. Both of them present a system of differential equations that must be solved in conjunction with the boundary constraints of the rod, leading to a boundary value problem (BVP). In this work, two methods to solve this BVP are applied to analyze the influence of external loads in the movement of hybrid compliant mechanisms. First, a shooting method (SM) is used to integrate directly the shape of the flexible rod and the forces that appear in it. Then, an integration with elliptic integrals (EI) is carried out to solve the workspace of the compliant element, considering its buckling mode. Applying both methods, an algorithm that obtains the locus of all possible trajectories of the mechanism’s coupler point, and detects the buckling mode change, is developed. This algorithm also allows calculating all possible circuits of the mechanism. Thus, the performance of this method within the path analysis of mechanisms is demonstrated. Full article

We consider the behavior of a viscous fluid within a container that has an elastic upper, free boundary. The movement of the upper boundary is described by a combination of a plate equation and a boundary condition of friction type that quantifies the elasticity of the boundary. We show the local existence of weak solutions to this coupled system in three dimensions, by applying the Galerkin method to a regularized version of the problem and using a fixed-point argument afterwards. Full article

This paper describes a methodology for implementing the state estimation and enhancing the accuracy in large-scale power systems that partially depend on variable renewable energy resources. To determine the actual states of electricity grids, including those of wind and solar power systems, the proposed state estimation method adopts a fast-decoupled weighted least square approach based on the architecture of application common database. Renewable energy modeling is considered on the basis of the point of data acquisition, the type of renewable energy, and the voltage level of the bus-connected renewable energy. Moreover, the proposed algorithm performs accurate bad data processing using inner and outer functions. The inner function is applied to the largest normalized residue method to process the bad data detection, identification and adjustment. While the outer function is analyzed whether the identified bad measurements exceed the condition of Kirchhoff’s current law. In addition, to decrease the topology and measurement errors associated with transformers, a connectivity model is proposed for transformers that use switching devices, and a transformer error processing technique is proposed using a simple heuristic method. To verify the performance of the proposed methodology, we performed comprehensive tests based on a modified IEEE 18-bus test system and a large-scale power system that utilizes renewable energy. Full article

Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of elliptic functional differential equations subject to functional boundary conditions. We obtain a localization of the corresponding non-negative eigenfunctions in terms of their norm. Under additional growth conditions, we also prove the existence of an unbounded set of eigenfunctions for these systems. The class of equations that we study is fairly general and our approach covers some systems of nonlocal elliptic differential equations subject to nonlocal boundary conditions. An example is presented to illustrate the theory. Full article

The present paper proposes a linearised hybrid finite element-statistical energy analysis (FE-SEA) formulation for built-up systems with nonlinear joints and excited by random, as well as harmonic, loadings. The new formulation was validated via an ad-hoc developed stochastic benchmark model. The latter was derived through the combination of the Lagrange-Rayleigh-Ritz method (LRRM) and the Monte Carlo simulation (MCS). Within the build-up plate systems, each plate component was modelled by using the classical Kirchhoff’s thin-plate theory. The linearisation processes were carried out according to the loading-type. In the case of random loading, the statistical linearisation (SL) was employed, while, in the case of harmonic loading, the method of harmonic balance (MHB) was used. To demonstrate the effectiveness of the proposed hybrid FE-SEA formulation, three different case studies, made-up of built-up systems with localized cubic nonlinearities, were considered. Both translational and torsional springs, as joint components, were employed. Four different types of loadings were taken into account: harmonic/random point and distributed loadings. The response of the dynamic systems was investigated in terms of ensemble average of the time-averaged energy. Full article

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter $\lambda$ . By mountain pass theorem, we obtain that system has at least one non-trivial weak solution $u_{\lambda }$ under some local conditions for each given large parameter $\lambda$ . We get a concrete lower bound of the parameter $\lambda$ , and then obtain two estimates of weak solutions $u_{\lambda }$ . We also obtain that $u_{\lambda }\to 0$ if $\lambda$ tends to ∞. Finally, we present an example as an application of our results. Full article