Search Results (187)

A physics-informed neural network (PINN) framework is developed to model the large deformation and coupled electromechanical response of dielectric elastomer tubes for energy harvesting. The system integrates incompressible neo-Hookean elasticity with radial electric loading and compressible gas inflation, leading to nonlinear equilibrium equations with deformation-dependent boundary conditions. By embedding the governing equations and boundary conditions directly into its loss function, the PINN enables accurate, mesh-free solutions without requiring labeled data. It captures realistic pressure–volume interactions that are difficult to address analytically or through conventional numerical methods. The results show that internal volume increases by over 290% during inflation at higher reference pressures, with residual stretch after deflation reaching 9.6 times the undeformed volume. The axial force, initially tensile, becomes compressive at high voltages and pressures due to electromechanical loading and geometric constraints. Harvested energy increases strongly with pressure, while voltage contributes meaningfully only beyond a critical threshold. To ensure stable training across coupled stages, the network is optimized using the Optuna algorithm. Overall, the proposed framework offers a robust and flexible tool for predictive modeling and design of soft energy harvesters. Full article

This research paper delves into the application of optimality results in orthogonal fuzzy metric spaces to demonstrate the existence and uniqueness of solutions of nonlinear differential equations with boundary conditions and nonlinear integral equations, emphasizing the importance of orthogonal fuzzy metric spaces in extending fixed-point theory. Through introducing this innovative concept, the study provides a theoretical framework for analyzing mappings in diverse scenarios. In this study, we introduce the concept of best proximity point (BPP) within the framework of orthogonal fuzzy metric spaces by employing orthogonal fuzzy proximal contractive mappings. Moreover, this research explores the implications of the established results, considering both self-mappings and non-self mappings that share the same parameter set. Additionally, some examples are provided to illustrate the practical relevance of the proven results and consequences in various mathematical contexts. The findings of this study can open up avenues for further exploration and application in solving real-world problems. Full article

A method for selecting the optimal shape of holes, taking into account the strength anisotropy of composites, is proposed. The methodology includes the following: an algorithm for stress determination based on singular integral equations and Green’s solutions; a strength criterion for the boundary of unloaded holes, which takes into account the anisotropic mechanical and strength properties of composites; an algorithm for determining hole shapes by a formulated nonlinear programming problem. The results of the research are presented for holes of various shapes, including single- and double-periodic hole systems. It is established that the calculated allowable loads for composite plates with holes based on stress concentration factors can be significantly overestimated. At the same time, by designing holes of optimal shape, the allowable loads can be many times greater than those for circular holes. Full article

The objective of this work is to investigate the global existence, general decay and blow-up results for a class of p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions. The global existence of solutions has been obtained by potential well theory and the general decay result of energy has been established, in which the exponential decay and polynomial decay are only special cases, by using the multiplier techniques combined with a nonlinear integral inequality given by Komornik. Finally, the blow-up of solutions is established with positive initial energy. To our knowledge, the global existence, general decay, and blow-up result of solutions to p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions has not been studied. Full article

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability of our main theorems, we provide several examples in which we compute values of the two-parameter Mittag–Leffler functions. The proposed approach is particularly effective for addressing a wide range of integral and fractional nonlinear differential equations with initial or boundary conditions—especially those involving variable coefficients, which are typically challenging to treat using classical integral transform methods. Finally, we demonstrate a significant application of the inverse operator approach by solving a Caputo fractional convection partial differential equation in ${\mathbb{R}}^n$ with an initial condition. Full article

This study proposes an intelligent prediction framework integrating native sparse attention (NSA) with the Chen-Guan (CHG) algorithm to optimize tunnel boring machine (TBM) operations in heterogeneous geological environments. The framework resolves critical limitations of conventional experience-driven approaches that inadequately address the nonlinear coupling between the spatial heterogeneity of rock mass parameters and mechanical system responses. Three principal innovations are introduced: (1) a hardware-compatible sparse attention architecture achieving O(n) computational complexity while preserving high-fidelity geological feature extraction capabilities; (2) an adaptive kernel function optimization mechanism that reduces confidence interval width by 41.3% through synergistic integration of boundary likelihood-driven kernel selection with Chebyshev inequality-based posterior estimation; and (3) a physics-enhanced modelling methodology combining non-Hertzian contact mechanics with eddy field evolution equations. Validation experiments employing field data from the Pujiang Town Plot 125-2 Tunnel Project demonstrated superior performance metrics, including 92.4% ± 1.8% warning accuracy for fractured zones, ≤28 ms optimization response time, and ≤4.7% relative error in energy dissipation analysis. Comparative analysis revealed a 32.7% reduction in root mean square error (p < 0.01) and 4.8-fold inference speed acceleration relative to conventional methods, establishing a novel data–physics fusion paradigm for TBM control with substantial implications for intelligent tunnelling in complex geological formations. Full article

Geometric image transformations are fundamental to image processing, computer vision and graphics, with critical applications to pattern recognition and facial identification. The splitting-integrating method (SIM) is well suited to the inverse transformation $T^{-1}$ of digital images and patterns, but it encounters difficulties in nonlinear solutions for the forward transformation T. We propose improved techniques that entirely bypass nonlinear solutions for T, simplify numerical algorithms and reduce computational costs. Another significant advantage is the greater flexibility for general and complicated transformations T. In this paper, we apply the improved techniques to the harmonic, Poisson and blending models, which transform the original shapes of images and patterns into arbitrary target shapes. These models are, essentially, the Dirichlet boundary value problems of elliptic equations. In this paper, we choose the simple finite difference method (FDM) to seek their approximate transformations. We focus significantly on analyzing errors of image greyness. Under the improved techniques, we derive the greyness errors of images under T. We obtain the optimal convergence rates $O\left(H^2\right)+O(H/N^2)$ for the piecewise bilinear interpolations ($\mu =1$) and smooth images, where $H(\ll 1)$ denotes the mesh resolution of an optical scanner, and N is the division number of a pixel split into $N^2$ sub-pixels. Beyond smooth images, we address practical challenges posed by discontinuous images. We also derive the error bounds $O\left(H^{\beta }\right)+O(H^{\beta }/N^2)$, $\beta \in (0,1)$ as $\mu =1$. For piecewise continuous images with interior and exterior greyness jumps, we have $O\left(\sqrt{H}\right)+O(\sqrt{H}/N^2)$. Compared with the error analysis in our previous study, where the image greyness is often assumed to be smooth enough, this error analysis is significant for geometric image transformations. Hence, the improved algorithms supported by rigorous error analysis of image greyness may enhance their wide applications in pattern recognition, facial identification and artificial intelligence (AI). Full article

Sustainability research encompasses diverse theories and frameworks focused on promoting sustainable economic ($E$), social ($S$), and environmental ($Env$) systems. However, integrated approaches to sustainability challenges have been impeded due to the absence of a unified analytical framework in the field. This study investigated how foundational and emerging theories, including resilience thinking, systems theory, and planetary boundaries, could be synthesized to develop an Integrated Sustainability Model (ISM) that captures nonlinear feedback, adaptive capacities $A_i\left(t\right)$, and threshold effects across these domains. The ISM model employs a system dynamics approach, where the rates of change for $E$, $S$, and $Env$ are governed by coupled differential equations, each influenced by cross-domain feedback (${\alpha }_i$ and ${\beta }_i$), adaptive capacity functions, and depletion rates (${\gamma }_i$). The model explicitly incorporates boundary constraints and adaptive capacity, operationalizing the dynamic interplay and co-evolution of sustainability dimensions. Grounded in an integrative perspective, this research introduces the Synergistic Resilience Theory (SRT), which proposes optimal sustainability arises from managing economic, social, and environmental systems as interconnected, adaptive components of a resilient system. Theoretical analysis and conceptual simulations demonstrated that high adaptive capacity and positive cross-domain reinforcement foster resilient, synergistic growth, while reduced capacity or breaches of critical thresholds ($En{v}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $S_{\mathrm{m}\mathrm{i}\mathrm{n}}$) can lead to rapid decline and slow recovery. These insights illuminate the urgent need for integrated, preventive policy interventions that proactively build adaptive capacity and maintain system resilience. This research, by advancing a mathematically robust and conceptually integrative framework, provides a potent new lens for developing empirically validated, holistic sustainability strategies within sustainability research. Full article

In this paper, we consider a modified Benjamin–Bona–Mahony (BBM) equation, which, for example, arises in shallow-water models. We discuss the well-posedness of the Dirichlet initial-boundary-value problem for the BBM equation. Our focus is on identifying a time-dependent source based on integral observation. First, we reformulate this inverse problem as an equivalent direct (forward) problem for a nonlinear loaded pseudoparabolic equation. Next, we develop and implement two efficient numerical methods for solving the resulting loaded equation problem. Finally, we analyze and discuss computational test examples. Full article

This paper focuses on a transmission problem describing the scattering of a TE-wave on a slab having an absolutely conducting wall at the bottom and covered with graphene at the top, accounting for the optical nonlinearity of graphene. This problem is reduced to a nonlinear hypersingular boundary integral equation defined on $\mathbb{R}$. To find an approximate solution to this equation, we develop a novel mathematical approach that combines the collocation method using Chebyshev series to represent a solution (it allows to calculate hypersingular integrals analytically) with an iterative scheme (it allows to account for the nonlinearity of graphene). Using this approach, we numerically simulate the scattering of TE-wave at 3 THz by a ten-micron graphene-coated slab filled with silica. It is shown that by tuning the chemical potential of graphene, one can modulate both the phase and amplitude of the reflected wave. The presented simulation results also demonstrate the effect of the nonlinearity of graphene on the reflected wave. Full article

The primary objective of this study is to explore sufficient conditions for the existence, uniqueness, and optimal stability of positive solutions to a finite system of Hilfer–Hadamard fractional differential equations with two-point boundary conditions. Our analysis centers around transforming fractional differential equations into fractional integral equations under minimal requirements. This investigation employs several well-known special control functions, including the Mittag–Leffler function, the Wright function, and the hypergeometric function. The results are obtained by constructing upper and lower control functions for nonlinear expressions without any monotonous conditions, utilizing fixed point theorems, such as Banach and Schauder, and applying techniques from nonlinear functional analysis. To demonstrate the practical implications of the theoretical findings, a pertinent example is provided, which validates the results obtained. Full article

The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data. Full article

High-Static–Low-Dynamic Stiffness (HSLDS) isolators have been extensively studied, primarily considering continuous stiffness and viscous damping, often overlooking stiffness discontinuities and dry friction forces. This paper aims to provide a more accurate model of real systems by investigating the dynamic behavior of HSLDS isolators, including piecewise nonlinear–linear stiffness, viscous damping, and dry friction. The equation of motion is analyzed using the Krylov–Bogoliubov–Mitropolsky (KBM) averaging method, deriving approximate analytical expressions to evaluate the frequency response curves and stability boundaries near primary resonance conditions. The model is validated by comparing the approximate solution with direct numerical integration and Den Hartog’s closed-form solution. A parametric analysis explores the impact of key parameters through amplitude–frequency diagrams and critical forcing boundaries. A numerical example is presented, demonstrating how the present method can be used to identify critical dynamic conditions, such as saddle-node bifurcations and activation of the piecewise restoring force nonlinearity. Results confirm the reliability of the KBM method in dealing with piecewise restoring forces while highlighting its limitations in case of high dry friction. This study offers an approximate yet effective approach for evaluating the system’s dynamic behavior, providing insights that could facilitate the design of isolation mounts and serve as benchmarks for future research. Full article

To address the boundary value problem associated with a class of third-order nonlinear differential equations with variable coefficients, this study integrates three key methods: the elastic transformation method (ETM), the similar construction method (SCM), and the elastic inverse transformation method (EITM). Firstly, ETM is employed to transform the original high-order nonlinear differential equations into the Tschebycheff equation, successfully reducing the order of the problem. Subsequently, SCM is applied to determine the general solution of the Tschebycheff equation under boundary conditions, thereby ensuring a structured and systematic approach. Ultimately, the EITM is used to reconstruct the solution of the original third-order nonlinear differential equation. The accuracy of the obtained solution is further validated by analyzing the corresponding solution curves. The synergy of these methods introduces a novel approach to solving nonlinear differential equations and extends the application of Tschebycheff equations in nonlinear systems. Full article

We discuss the existence criteria and Ulam–Hyers stability for solutions to a nonlocal integral boundary value problem of nonlinear coupled Hilfer–Hadamard-type fractional Langevin equations. Our results rely on the Leray–Schauder alternative and Banach’s fixed point theorem. Examples are included to illustrate the results obtained. Full article