Search Results (99)

This study advances a hierarchical bilevel optimization paradigm to rigorously characterize the intertwined processes of strategic bidding and regulatory market participation in electricity systems increasingly dominated by renewable resources. At the upper tier, a central regulatory authority orchestrates participation rules, renewable integration mandates, and incentive mechanisms with the overarching aim of maximizing system-wide social welfare while driving decarbonization and reliability objectives. At the subordinate level, profit-maximizing generation firms—each managing heterogeneous renewable portfolios—pursue strategic bidding under deep uncertainty, conceptualized as a multi-agent game governed by imperfect and asymmetric information. The interaction between these tiers is formalized as a bilevel Stackelberg game that encapsulates price-responsive demand, intertemporal reserve adequacy, and policy-driven incentive structures. To ensure both computational tractability and robustness against strategic indeterminacy, the lower-level equilibrium is reformulated into a mathematical program with equilibrium constraints (MPEC), enabling a hybrid solution procedure that combines penalty-based regularization with exact decomposition algorithms. The framework’s efficacy is validated through a stylized multi-zone case study featuring diverse renewable assets and strategic participants, revealing how policy signals, capacity ceilings, and market power asymmetries reshape efficiency frontiers and bidding equilibria. A set of high-resolution post-processing visualizations is further employed to illustrate the dynamic evolution of marginal prices, equilibrium trajectories, and regulatory impacts under uncertainty. Full article

This study introduces a high-order numerical scheme for solving 1D second-order hyperbolic telegraph equations (HTEs) with variable coefficients. We employ a generalized temporal discretization (TD) of order p via the Rothe approach, combined with a spatial spectral collocation (SCM) method using generalized shifted Jacobi polynomials (GSJPs). By utilizing a Galerkin-type basis that structurally satisfies homogeneous boundary conditions (HBCs)—including Dirichlet or Neumann types—we achieve a global error bound of $\mathcal{O}({(\Delta \tau )}^p+N^{-s})$, where $\Delta \tau$ denotes the temporal step size and s represents the spatial regularity of the exact solution (ExaS). The proposed algorithm, Rothe-GSJP, allows for an optimal balance between the temporal and spatial parameters, minimizing computational effort for high-precision engineering applications such as Phase-Locked Loop (PLL) modeling. Numerical experiments performed on an i9-10850 workstation show that the scheme always reaches the machine precision floor of ${10}^{-16}$. While the framework supports temporal orders up to $p=6$, the results indicate that $p\in \{2,3,4\}$ provides an optimal balance between high-order precision and absolute stability. The Rothe-GSJP method proves to be a robust, efficient, and highly accurate alternative to traditional solvers for hyperbolic systems. Full article

The present study models the deflection of nonlinear waves over a Kirchhoff plate underlying a Pasternak-like elastic foundation. A promising version of the tanh expansion analytical method has been deployed for the construction of regular exact solutions for the model, including the application of certain ansatz functions for validations and yet construction of more solutions. The resulting frequency equation and the modulation instability spectrum have been obtained for the linearized model, including the expressions for the related phase and group velocities. In addition, the study examines the equilibrium status of the resulting dynamical system with the help of the bifurcation analysis. Numerically, nonlinear deflection and dispersion of waves have been simulated through the acquired expressions and equations. Notably, the study notes that increasing both the Pasternak-like nonlinear parameter $\eta$ and time variation (for $x>0$) decreases the nonlinear deflection in the plate, while increasing the stiffness of the Winkler foundation increases deflection in the medium. In addition, the study establishes, concerning the determined frequency equation, that increasing the Winkler foundation stiffness increases the dispersion of nonlinear waves in the medium, while an opposite trend has been noted concerning the imposed Pasternak-like nonlinear foundation. In addition, both phase and group velocities, the gain function for modulation instability, and the resulting dynamical system have been noted to be greatly affected by the variation of the imposed foundational parameters. Lastly, this study has potential applications in various engineering fields while modeling and analysis of mechanical structures supported by additional structures. Full article

Based on the hydrodynamic description, the dynamics of nonlinear cylindrical waves in an isentropic plasma are investigated. The problem is considered in an electrostatic formulation for a two-dimensional plasma medium where ions form a stationary background. Proceeding from the particular, exact solution of hydrodynamic equations, we obtain the system of differential equations which describes the electron’s dynamics, taking into account the finite temperature of electrons. Moreover, we find the conditions when this system is reduced to the generalized Ermakov–Pinney equation which was used for analyzing electron dynamics. In the present calculations, a parabolic-in-radius temperature profile was used, associated with an electron density varying only with time. In the framework of the model that worked out, the influence of initial conditions and thermal effects on the regular and singular dynamics of excited waves are discussed. It is shown that the development of singular behavior due to intrinsic nonlinearity is avoided by taking into account thermal effects and the initial rotation of the electron flow. Full article

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in the data can lead to significant errors in the solution and lack stability. Secondly, the determination of the parameters of the classical Laplace equation is difficult to adapt to the requirements of complex applications. For this purpose, in this paper, the Laplace equation with uncertain parameters is defined, and the uncertainty is represented by fuzzy numbers. In the case of granular differentiability, it is transformed into a granular differential equation, proving its serious ill-posedness. To overcome the ill-posedness, the Fourier regularization method is used to stabilize the numerical solution, and the stability estimation and error analysis between the regularization solution and the exact solution are given. Finally, numerical examples are given to illustrate the effectiveness and practicability of this method. Full article

The paper investigates the regularization of solutions to nonlinear Volterra integral equations of the first kind, under the assumption that a solution exists and belongs to the space of continuous functions. The kernel of the equation is a differentiable function and vanishes on the diagonal at an interior point of the integration interval. By applying an appropriate differential operator (with respect to x), the Volterra integral equation of the first kind is reduced to a Volterra integral equation of the third kind, equivalent with respect to solvability. The subdomain method is employed by partitioning the integration interval into two subintervals. Within the imposed constraints, a compatibility condition for the solutions is satisfied at the junction point of the partial subintervals. A Lavrentiev-type regularizing operator is constructed that preserves the Volterra structure of the equation. The uniform convergence of the regularized solution to the exact solution is proved, and conditions ensuring the uniqueness of the solution in Hölder space are established. Full article

The two-fluid model (TFM) has become a foundational tool in numerical codes used for engineering analyses of two-phase flows in energy systems. However, its completeness remains a topic of debate because improper modeling of interfacial inertial coupling can render the momentum conservation equations elliptic. This issue leads to short wavelength perturbations growing at an infinite rate. This paper demonstrates the practical feasibility of incorporating variational inertial-coupling terms into an industrial CFD TFM code to ensure it is well-posed without the need for regularization. For verification, two special cases with exact analytical solutions of the TFM equations are utilized, exhibiting convergence at a mesh resolution of 1 mm. Full article

We investigate a numerical method for approximating stochastic Caputo fractional differential equations driven by multiplicative noise. The nonlinear functions f and g are assumed to satisfy the global Lipschitz conditions as well as the linear growth conditions. The noise is approximated by a piecewise constant function, yielding a regularized stochastic fractional differential equation. We prove that the error between the exact solution and the solution of the regularized equation converges in the $L^2((0,T)\times \mathrm{\Omega })$ norm with an order of $O(\Delta t^{\alpha -1/2})$, where $\alpha \in (1/2,1]$ is the order of the Caputo fractional derivative, and $\Delta t$ is the time step size. Numerical experiments are provided to confirm that the simulation results are consistent with the theoretical convergence order. Full article

This paper presents a semi-analytical, mesh-free space–time Collocation Trefftz Method (SCTM) for solving two-dimensional (2D) wave equations. Given prescribed initial and boundary data, collocation points are placed on the space–time (ST) boundary, reformulating the initial value problem as an equivalent boundary value problem and enabling accurate reconstruction of wave propagation in complex domains. The main contributions of this work are twofold: (i) a unified ST Trefftz basis that treats time as an analytic variable and enforces the wave equation in the full ST domain, thereby eliminating time marching and its associated truncation-error accumulation; and (ii) a Multiple-Scale Characteristic-Length (MSCL) grading strategy that systematically regularizes the collocation linear system. Several numerical examples, including benchmark tests, validate the method’s feasibility, effectiveness, and accuracy. For both forward and inverse problems, the solutions produced by the method closely match exact results, confirming its accuracy. Overall, the results reveal the method’s feasibility, accuracy, and stability across both forward and inverse problems and for varied geometries. Full article

In complicated scenarios, due to the low precision of float solutions and poor reliability of fixed solutions, it is challenging to achieve a balance between accuracy and reliability of the integer least squares (ILS) estimation. To address this dilemma, the best integer equivariant (BIE) estimation, which makes a weighted sum of all possible candidates, has recently been attached great importance. The BIE solution approaches the float solution at a low ILS success rate, maintaining positioning reliability. As the success rate increases, it converges to the fixed solution, facilitating high-precision positioning. Furthermore, the posterior variance of BIE estimation provides the capability of reliability evaluation. However, in environments with a limited number or a deficient configuration of available satellites, there is a sharp decline in the strength of the GNSS precise positioning model. In this case, the exactness of weight allocation for integer candidates in BIE estimation will be severely compromised by unmodeled errors. When the ambiguity is incorrectly fixed, the wrongly determined optimal candidate is probably assigned an excessively high weight. Therefore, the BIE solution in a weak GNSS model always exhibits a significant positioning error consistent with the fixed solution. Moreover, the posterior variance of BIE estimation approximately resembles that of a fixed solution, losing error warning ability. Consequently, the BIE estimation may exhibit lower reliability compared to the ILS estimation employing a validation test with a loose acceptance threshold. To improve the positioning performance in weak GNSS models, a BIE ambiguity resolution (AR) method based on Tikhonov regularization is proposed in this paper. The method introduces Tikhonov regularization into the least squares (LS) estimation and the ILS ambiguity search, mitigating the serious impact of unmodeled errors on the BIE estimation under weak observation conditions. Meanwhile, the regularization factors are appropriately selected by utilizing an optimized approach established on the L-curve method. Simulation experiments and field tests have demonstrated that the method can significantly enhance the positioning accuracy and reliability in weak GNSS models. Compared to the traditional BIE estimation, the proposed method achieved accuracy improvements of 73.6% and 69.3% in the field tests with 10 km and 18 km baselines, respectively. Full article

Meta-analytic deconvolution seeks to recover the distribution of true effects from noisy site-specific estimates. While Efron’s log-spline prior provides an elegant empirical Bayes solution with excellent point estimation properties, its plug-in nature yields severely anti-conservative uncertainty quantification for individual site effects—a critical limitation for what Efron terms “finite-Bayes inference.” We develop a fully Bayesian extension that preserves the computational advantages of the log-spline framework while properly propagating hyperparameter uncertainty into site-level posteriors. Our approach embeds the log-spline prior within a hierarchical model with adaptive regularization, enabling exact finite-sample inference without asymptotic approximations. Through simulation studies calibrated to realistic meta-analytic scenarios, we demonstrate that our method achieves near-nominal coverage (88–91%) for 90% credible intervals while matching empirical Bayes point estimation accuracy. We provide a complete Stan implementation handling heteroscedastic observations—a critical feature absent from existing software. The method enables principled uncertainty quantification for individual effects at modest computational cost, making it particularly valuable for applications requiring accurate site-specific inference, such as multisite trials and institutional performance assessment. Full article

The statistical mechanics of structured particles with arbitrary size and shape adsorbed onto discrete lattices presents a longstanding theoretical challenge, mainly due to complex spatial correlations and entropic effects that emerge at finite densities. Even for simplified systems such as hard-core linear k-mers, exact solutions remain limited to low-dimensional or highly constrained cases. In this review, we summarize the main theoretical approaches developed by our research group over the past three decades to describe adsorption phenomena involving linear k-mers—also known as multisite occupancy adsorption—on regular lattices. We examine modern approximations such as an extension to two dimensions of the exact thermodynamic functions obtained in one dimension, the Fractional Statistical Theory of Adsorption based on Haldane’s fractional statistics, and the so-called Occupation Balance based on expansion of the reciprocal of the fugacity, and hybrid approaches such as the semi-empirical model obtained by combining exact one-dimensional calculations and the Guggenheim–DiMarzio approach. For interacting systems, statistical thermodynamics is explored within generalized Bragg–Williams and quasi-chemical frameworks. Particular focus is given to the recently proposed Multiple Exclusion statistics, which capture the correlated exclusion effects inherent to non-monomeric particles. Applications to monolayer and multilayer adsorption are analyzed, with relevance to hydrocarbon separation technologies. Finally, computational strategies, including advanced Monte Carlo techniques, are reviewed in the context of high-density regimes. This work provides a unified framework for understanding entropic and cooperative effects in lattice-adsorbed polyatomic systems and highlights promising directions for future theoretical and computational research. Full article

This study investigates the influence of multiplicative noise—modeled by a Wiener process—and spatial-fractional derivatives on the dynamics of the space-fractional stochastic Regularized Long Wave equation. By employing a complete discriminant polynomial system, we derive novel classes of fractional stochastic solutions that capture the complex interplay between stochasticity and nonlocality. Additionally, the variational principle, derived by He’s semi-inverse method, is utilized, yielding additional exact solutions that are bright solitons, bright-like solitons, kinky bright solitons, and periodic structures. Graphical analyses are presented to clarify how variations in the fractional order and noise intensity affect essential solution features, such as amplitude, width, and smoothness, offering deeper insight into the behavior of such nonlinear stochastic systems. Full article

This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function method with the complete discriminant polynomial system. This method proves to be effective in discovering precise solutions for stochastic fractional partial differential equations (SFPDEs), including special cases. Applying this method to the SFRLWE yields new exact solutions, offering fresh insights. We investigated how noise affects stochastic solutions and discovered that more intense noise can result in flatter surfaces. We note that multiplicative noise can stabilize the solution, and we show how fractional derivatives influence the dynamics of noise. We found that the noise strength and fractional derivative affect the width, amplitude, and smoothness of the obtained solutions. Additionally, we conclude that multiplicative noise impacts and stabilizes the behavior of SFRLWE solutions. Full article

In this paper, the inverse problem of identifying the source term of the time fractional diffusion-wave equation is studied. This problem is ill-posed, i.e., the solution (if it exists) does not depend on the measurable data. Under the priori bound condition, the condition stable result and the optimal error bound are all obtained. The fractional Landweber iterative regularization method is used to solve this inverse problem. Based on the priori regularization parameter selection rule and the posteriori regularization parameter selection rule, the error estimation between the regularization solution and the exact solution is obtained. Moreover, the error estimations are all order optimal. At the end, three numerical examples are given to prove the effectiveness and stability of this regularization method. Full article