Search Results (84)

While recent advances have successfully integrated neural networks with physical models to derive numerical solutions, there remains a compelling need to obtain exact analytical solutions. The ability to extract closed-form expressions from these models would provide deeper theoretical insights and enhanced predictive capabilities, complementing existing computational techniques. In this paper, we study the nonlinear Gardner equation and the (2+1)-dimensional Zabolotskaya–Khokhlov model, both of which are fundamental nonlinear wave equations with broad applications in various physical contexts. The proposed models have applications in fluid dynamics, describing shallow water waves, internal waves in stratified fluids, and the propagation of nonlinear acoustic beams. This study integrates a modified generalized Riccati equation mapping approach and a novel generalized $\left({\displaystyle \frac{G^{\prime }}{G}}\right)$-expansion method with neural networks for obtaining exact solutions for the suggested nonlinear models. Researchers are currently investigating potential applications of these neural networks to enhance our understanding of complex physical processes and to develop new analytical techniques. The proposed strategies incorporate the solutions of the Riccati problem into neural networks. Neural networks are multi-layer computing approaches including activation and weight functions among neurons in input, hidden, and output layers. Here, the solutions of the Riccati equation are allocated to each neuron in the first hidden layer; thus, new trial functions are established. We evaluate the suggested models, which lead to the construction of exact solutions in different forms, such as kink, dark, bright, singular, and combined solitons, as well as hyperbolic and periodic solutions, in order to verify the mathematical framework of the applied methods. The dynamic properties of certain wave-related solutions have been shown using various three-dimensional, two-dimensional, and contour visualizations. This paper introduces a novel framework for addressing nonlinear partial differential equations, with significant potential applications in various scientific and engineering domains. Full article

Let $\Omega$ be a homogeneous function of degree zero on ${\mathbb{R}}^n$, $n\ge 2$, integrable and having mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$, and let $T_{\Omega }$ be the homogeneous convolution singular integral operator with kernel ${\displaystyle \frac{\Omega \left(x\right)}{{\left|x\right|}^n}}$. By introducing reasonable refined decomposition and approximation techniques, together with sparse domination and variable measure interpolation methods, we establish the quantitative $A_1-A_{\infty }$ weighted estimates for $T_{\Omega }$ under the rough condition $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ for some $q\in (1,\infty )$. The results of the paper improve the previous works for the case $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$. We also give the quantitative $A_1-A_{\infty }$ weighted estimates for the commutator $[b,T_{\Omega }]$ with $BMO$ symbol b. Full article

This paper investigates the existence and multiplicity of positive solutions for a system of generalized Laplacian problems. By analyzing the asymptotic behavior of nonlinearity, we establish conditions for the existence of positive solutions and the presence of multiple positive solutions. Our main results reveal how the norm of the positive solutions behaves as the parameter $\lambda$ approaches 0 or ∞, specifically that the norm tends to either 0 or ∞. Full article

Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations (PDEs) by embedding physical laws into the learning process. However, standard PINNs often suffer from training instabilities and unbalanced optimization when handling multi-term loss functions, especially in problems involving singular perturbations, fractional operators, or multi-scale behaviors. To address these limitations, we propose a novel gradient variance weighting physics-informed neural network (GVW-PINN), which adaptively adjusts the loss weights based on the variance of gradient magnitudes during training. This mechanism balances the optimization dynamics across different loss terms, thereby enhancing both convergence stability and solution accuracy. We evaluate GVW-PINN on three representative PDE models and numerical experiments demonstrate that GVW-PINN consistently outperforms the conventional PINN in terms of training efficiency, loss convergence, and predictive accuracy. In particular, GVW-PINN achieves smoother and faster loss reduction, reduces relative errors by one to two orders of magnitude, and exhibits superior generalization to unseen domains. The proposed framework provides a robust and flexible strategy for applying PINNs to a wide range of integer- and fractional-order PDEs, highlighting its potential for advancing data-driven scientific computing in complex physical systems. Full article

For a non-conservative nonlinear oscillator (NCNO) having a periodic solution, the existence of the first integral is a certain symmetry of the nonlinear dynamical system, which signifies the balance of kinetic energy and potential energy. A first-order nonlinear ordinary differential equation (ODE) is used to derive the first integral, which, equipped with a right-end boundary condition, can determine an implicit potential function for computing the period by an exact integral formula. However, the integrand is singular, which renders a less accurate value of the period. A generalized integral conservation law endowed with a weight function is constructed, which is proved to be equivalent to the exact integral formula. Minimizing the error to satisfy the periodicity conditions, the optimal initial value of the weight function is determined. Two non-iterative methods are developed by integrating three first-order ODEs or two first-order ODEs to compute the period. Very accurate value of the period can be observed upon testing five examples. For the NCNO without having the first integral, the integral-type period formula is derived. Four examples belong to the Liénard equation, involving the van der Pol equation, are evaluated by the proposed iterative method to determine the oscillatory amplitude and period. For the case with one or more limit cycles, the amplitude and period can be estimated very accurately. For the NCNO of a broad type with or without having the first integral, the present paper features a solid theoretical foundation and contributes integral-type formulations for the determination of the oscillatory period. The development of new numerical algorithms and extensive validation across a diverse set of examples is given. Full article

As low-voltage distribution networks incorporate increasingly diverse loads, series arc faults exhibit weak characteristics that are easily masked by load currents, leading to high misjudgment rates in traditional detection methods. This paper proposes a series arc fault detection method based on an improved Light Gradient Boosting Machine (LightGBM) model. First, a test platform containing 12 household loads was built to collect arc data from both individual and composite loads. Composite loads refer to composite load conditions where multiple devices are running simultaneously and arcing occurs on some loads. To address the challenge of feature extraction, Variational Mode Decomposition (VMD) is employed to isolate the fundamental frequency component. To enhance high-frequency arc characteristics, singular value decomposition (SVD) is then applied. A multidimensional statistical feature set—comprising peak-to-peak value, kurtosis, and other indicators—is constructed. Finally, the LightGBM algorithm is used to identify arc faults based on these features. To overcome the LightGBM model’s limited ability to focus on hard-to-classify samples, a dynamic weighted hybrid loss function is developed. Experiments demonstrate that the proposed method achieves 98.9% accuracy across 223,615 sample groups. When deployed on STM32H723VGT6 hardware, the average fault alarm time is 83.8 ms, meeting requirements. Full article

Recently we studied a collocation–quadrature method in weighted ${\mathbf{L}}^2$ spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form $u\left(x\right)-{\int }_{\alpha x}^{1}h(x-\alpha y)u\left(y\right)dy=f\left(x\right),0<x<1,$ where $h\left(x\right)$ (with a possible singularity at $x=0$) and $f\left(x\right)$ are given (in general complex-valued) functions, and $\alpha \in (0,1)$ is a fixed parameter. Here, we want to investigate the same method for the case when $\alpha =1.$ More precisely, we consider (in general weakly singular) Volterra integral equations of the form $u\left(x\right)-{\int }_0^{x}h(x,y){(x-y)}^{\kappa }u\left(y\right)dy=f\left(x\right),0<x<1,$ where $\kappa >-1$, and $h:D⟶\mathbb{C}$ is a continuous function, $D=\left\{(x,y)\in {\mathbb{R}}^2:0<y<x<1\right\}.$ The passage from $0<\alpha <1$ to $\alpha =1$ and the consideration of more general kernel functions $h(x,y)$ make the studies more involved. Moreover, we enhance the family of interpolation operators defining the approximating operators, and, finally, we ask if, in comparison to collocation–quadrature methods, the application of the Nyström method together with the theory of collectively compact operator sequences is possible. Full article

Runoff prediction plays a critical role in water resource management and flood mitigation. Traditional runoff prediction methods often rely on single-layer optimization frameworks that process the data without decomposition and employ relatively simple prediction models, leading to suboptimal performance. In this study, a novel two-layer optimization framework is proposed that integrates data decomposition techniques with multi-model combination strategies, establishing a closed-loop feedback mechanism between decomposition and prediction processes. The framework employs the Snow Ablation Optimizer (SAO) to optimize combination weights across both layers. Its adaptive fitness function incorporates three evaluation metrics—Mean Absolute Percentage Error (MAPE), Relative Root Mean Square Error (RRMSE), and Nash–Sutcliffe Efficiency (NSE)—to enable adaptive data processing and intelligent model selection. We validated the framework using observational data from Huangzhuang Hydrological Station in the Hanjiang River Basin. The results demonstrate that, at the decomposition layer, optimal performance was achieved by combining non-decomposition, Singular Spectrum Analysis (SSA), and Complementary Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) (with coefficients 0.4061, 0.6115, and −0.0063), paired with the long short-term memory (LSTM) model. At the prediction layer, the proposed algorithm achieved a 32.84% improvement over the best single decomposition method and a 30.21% improvement over the best single combination optimization approach. These findings confirm the framework’s effectiveness in enhancing runoff data decomposition and optimizing multi-model selection. Full article

A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence and uniqueness of the $R_{\nu }$-generalized solution, its coercive and differential properties of biharmonic boundary value problems with a corner singularity. This paper establishes estimates of a function in the neighborhood of a singularity point in the norms of weighted Lebesgue spaces through its norms in weighted Sobolev spaces over the entire domain, with a minimum weight exponent. In addition, we obtain an estimate of the function norm in a boundary strip for the degeneration of a function on the entire boundary of the domain. These estimates will be useful not only for studying differential problems with singularity, but also in estimating the convergence rate of an approximate solution to an exact one in the weighted finite element method. Full article

Hyperparameter optimization (HPO), which is also called hyperparameter tuning, is a vital component of developing machine learning models. These parameters, which regulate the behavior of the machine learning algorithm and cannot be directly learned from the given training data, can significantly affect the performance of the model. In the context of relevance vector machine hyperparameter optimization, we have used zero-mean Gaussian weight priors to derive iterative equations through evidence function maximization. For a general Gaussian weight prior and Bayesian linear regression, we similarly derive iterative reestimation equations for hyperparameters through evidence function maximization. Subsequently, after using relative entropy and Bayesian optimization, the aforementioned non-closed-form reestimation equations can be partitioned into E and M steps, providing a clear mathematical and statistical explanation for the iterative reestimation equations of hyperparameters. The experimental result shows the effectiveness of the EM algorithm of hyperparameter optimization, and the algorithm also has the merit of fast convergence, except that the covariance of the posterior distribution is a singular matrix, which affects the increase in the likelihood. Full article

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article

The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. It is known that the numerical integration of definite integrals is of great importance in basic and applied sciences. In this paper we consider the optimization problem of weighted quadrature formulas with derivatives in Sobolev space. Using the extremal function, the square of the norm of the error functional of the considered quadrature formula is calculated. Then, minimizing this norm by coefficients, we obtain a system to find the optimal coefficients of this quadrature formula. The uniqueness of solutions of this system is proved, and an algorithm for solving this system is given. The proposed algorithm is used to obtain the optimal coefficients of the derivative weight quadrature formulas. It should be noted that the optimal weighted quadrature formulas constructed in this work are optimal for the approximate calculation of regular, singular, fractional and strongly oscillating integrals. The constructed optimal quadrature formulas are applied to the approximate solution of linear Fredholm integral equations of the second kind. Finally, the numerical results are compared with the known results of other authors. Full article

With the continual expansion of global urbanization and population growth, urban energy demands have intensified, and anthropogenic activities have precipitated profound shifts in the global climate. These climatic changes directly alter urban environmental conditions, which in turn exert indirect effects on human physiological function. Consequently, the comfort of outdoor community environments has emerged as a critical metric for assessing the quality of human habitation. Although existing studies have focused on improving singular environmental factors—such as wind or thermal comfort—they often lack an integrated, multi-factor coupling mechanism, and adaptive strategy systems tailored to hot-summer, cold-winter regions remain underdeveloped. This study examines the Minhe Community in Qian’an City to develop a performance evaluation framework for outdoor spaces grounded in subjective comfort and to close the loop from theoretical formulation to empirical validation via an interdisciplinary approach. We first synthesized 25 environmental factors across eight categories—including wind, thermal, and lighting parameters—and applied the Analytic Hierarchy Process (AHP) to establish factor weights, thereby constructing a comprehensive model that encompasses both physiological and psychological requirements. Field surveys, meteorological data collection, and ENVI-met (V5.1.1) microclimate simulations revealed pronounced issues in the community’s wind distribution, thermal comfort, and acoustic environment. In response, we proposed adaptive interventions—such as stratified vegetation design and permeable pavement installations—and validated their efficacy through further simulation. Post-optimization, the community’s overall comfort score increased from 4.64 to 5.62, corresponding to an efficiency improvement of 21.3%. The innovative contributions of this research are threefold: (1) transcending the limitations of single-factor analyses by establishing a multi-dimensional, coupled evaluation framework; (2) integrating AHP with ENVI-met simulation to realize a fully quantified “evaluation–simulation–optimization” workflow; and (3) proposing adaptive strategies with broad applicability for the retrofit of communities in hot-summer, cold-winter climates, thereby offering a practical technical pathway for urban microclimate enhancement. Full article

In this study, an optimization framework for turbomachinery blades using a hybrid surrogate model assisted by proper orthogonal decomposition (POD) is introduced and then applied to the aero-structural multidisciplinary design optimization of a transonic fan rotor, NASA Rotor 67. The rotor blade is optimized through blade sweeping controlled by Gaussian radial basis functions. Calculations of aerodynamic and structural performance are achieved through computational fluid dynamics and computational structural mechanics. With a number of performance snapshots, singular value decomposition is employed to extract the basis modes, which are then used as the kernel functions in training the POD-based hybrid model. The inverse multi-quadratic radial basis function is adopted to construct the response surfaces for the coefficients of kernel functions. Aerodynamic design optimization is first investigated to preliminarily explore the impact of blade sweeping. In the aero-structural optimization, the aerodynamic performance, and von Mises stress are considered equally important and incorporated into one single objective function with different weight coefficients. The results are given and compared in detail, demonstrating that the average stress is dependent on the aerodynamic loading, and the configuration with forward sweeping on inner spans and backward sweeping on outer spans is the most effective for increasing the adiabatic efficiency while decreasing the average stress when the total pressure ratio is constrained. Through this study, the optimization framework is validated and a practical configuration for reducing the stress in a transonic fan rotor is provided. Full article

This study investigates the nonlocal boundary value problem for generalized Laplacian equations involving a singular, possibly non-integrable weight function. By analyzing the asymptotic behaviors of the nonlinearity $f=f\left(s\right)$ near both $s=0$ and $s=\infty$, we establish the existence, nonexistence, and multiplicity of positive solutions for all positive values of the parameter $\lambda$. Our proofs employ the fixed-point theorem of cone expansion and compression of norm type, a powerful tool for demonstrating the existence of solutions in cones, as well as the Leray–Schauder fixed-point theorem, which offers an alternative approach for proving the existence of solutions. Illustrative examples are provided to concretely demonstrate the applicability of our main results. Full article