Variational Method Research Articles

Abstract The effect of the generalised exponential cosine screened Coulomb potential (GEC-SCP) on the stability of a model two-electron atom (Ze+e+e) interacting via this type of potential has been investigated. GECSCP is taken in the form: V(ρ: μ,θ) = exp[-(μ cosθ)ρ] cos(μ sinθ)ρ]/ρ (in a.u.), where µ (0 ≤ µ < ∞) and θ (0 ≤ θ ≤ π/2) are two adjustable parameters. An extensive wavefunction, having 715 number of terms, is utilised to calculate the ground state energy of the model atom (for Z = 1, 2, 3) within the framework of the Ritz's variational method. Convergence of the computed results with respect to the number of terms in the wavefunction is corroborated. An inclusive study is made on the variation of the ground state energy and other related quantities with respect to the parameters µ and θ. Special emphasis is given on the determination of the critical values of the parameters which correspond to the critical bound-continuum limits of the model atoms.

Analyzing the application of fractional-order brusselator system via the variational iteration method

Consistent reduced order modeling for wind turbine wakes using variational multiscale method and actuator line model

Development and verification of variational nodal method for arbitrary unstructured triangular prism mesh based on VINUS

Variational Holo-spectral analysis method of high-dimensional feature representation for underwater acoustic target recognition

In this work, we deal with the analysis of some qualitative properties of solutions to a class of fractional pseudo-parabolic equations involving nonlocal nonlinearity. We first establish the existence of local solutions to the problem. Then, we prove the existence and uniqueness of global solutions, as well as the long-time behavior of these global solutions. Next, we prove the blow-up phenomenon at finite time occurs when the initial datum lies in the unstable set, that is, I ( u 0 ) < 0 , without requiring the subcritical initial energy condition J ( u 0 ) ≤ d . Furthermore, we establish an upper bound for the blow-up time and derive both upper and lower bounds for the decay rate of the solutions. Finally, we show the existence of ground-state solutions to the corresponding stationary equation and investigate the relationship between global solutions and ground-state solutions. Our arguments are based on the Galerkin approximation method, the contraction mapping principle, variational methods combined with the Hardy–Littlewood–Sobolev inequality, and a modified differential inequality.

In this paper, we deal with the generalized extensible beam equation of the form { Δ 2 u − M ( ‖ ∇u ‖ L p p ) Δ p u + μW ( x ) u = h ( x , u ) + βg ( x ) | u | ν − 2 u , x ∈ R N , u ∈ H 2 ( R N ) . Δ 2 := Δ ( Δ ) is the biharmonic operator, N is an integer such that 4 $ ]]> N > 4 and M ( t ) = a t δ + b with a, b, 0 $ ]]> δ > 0 , W and h satisfy suitable conditions stated later. Unlike existing works on the generalized extensible beam equations, we replace the Laplacian with a p-Laplacian. In addition, the nonlinearity property combines both convex and concave terms. The proof is based on variational methods. Our main result extends and improves those of Sun et al. (Electron J Differ Equ. 2019;41:1–23) and F. Wu (Appl Math Lett. 2020;132:108197).

We study Rydberg excitons in two-dimensional transition-metal dichalcogenide semiconductors, using the Rytova-Keldysh potential to account for nonlocal dielectric screening effects. We determine exciton binding energies in monolayer WSe\(_2\) and WS\(_2\) by the variational method. We perform calculations for different dielectric environments (isolation, hBN encapsulation, and SiO\(_2\) substrate support) to investigate the influences of the surrounding dielectric environment. Our theoretical results agree reasonably with experimental measurements and previous numerical calculations. The study shows a strong dependence of exciton binding energy and spatial extent on both the principal and angular quantum numbers, as well as on the surrounding dielectric environment.

Abstract The detailed knowledge of bathymetry pattern represents a key factor in the deep understanding of ocean processes, physical oceanography, biology, ecohydraulics, and marine geology. However, the accuracy of bathymetry modeling is still low from satellite altimetry, gravity model, and shipborne gravity data. In this paper, a novel scheme is proposed based on black-box theory for regional bathymetry modeling in the Persian Gulf and the Oman Sea via geodetic data sources such as satellite altimetry, gravity model, and shipborne gravity data. Multi-Layer Perceptron (MLP), Adaptive Neuro-Fuzzy Inference System (ANFIS), and Local Linear Model Tree (LOLIMOT) algorithms are used as nonlinear black-box tools to identify the basic mathematical model. The geoid height, gravity gradient, and gravity anomaly are used as inputs to these artificial intelligence models, with the GEBCO bathymetry model as the output. The derived basic model is further improved by assimilating with the shipborne bathymetry measurements using the 3D variational optimization method to determine the final bathymetry model. The model is validated by the shipborne bathymetry in control tracks of regions Chabahar, Genaveh, and Alamshah, and the results show high accuracy and reliability with root mean square errors (RMSEs) of about 4, 0.8, and 0.92 m, respectively. The proposed approach is valuable for various uses in marine science.

Abstract. A comprehensive understanding and accurate modelling of the terrestrial carbon cycle are of paramount importance to improve projections of the global carbon cycle and more accurately gauge its impact on global climate systems. Land surface models, which have become an important component of weather and climate applications, simulate key aspects of the terrestrial carbon cycle, such as photosynthesis and respiration. These models rely on parameterisations that require careful calibration. In this study we explore the assimilation of atmospheric CO2 concentration data for parameter calibration of the ORganizing Carbon and Hydrology In Dynamic EcosystEms (ORCHIDEE) land surface model using an EnVarDA method, an adjoint-free ensemble-variational data assimilation method. By circumventing the challenges associated with developing and maintaining tangent linear and adjoint models, the EnVarDA method offers a very promising alternative. Using synthetic observations generated through a twin experiment, we demonstrate the ability of EnVarDA to assimilate atmospheric CO2 concentrations for model parameter calibration. We then compare the results to a VarDA method that uses finite differences to estimate tangent linear and adjoint models, which reveals that EnVarDA is superior in terms of computational efficiency, fit to the observations, and parameter recovery.

ABSTRACTThis paper is concerned with the existence of positive solutions for a class of Schrödinger‐Bopp‐Podolsky systems in three‐dimensional Euclidean space. The system couples a nonlinear Schrödinger equation with a fourth‐order elliptic equation governing the electrostatic potential. We establish existence under assumptions on an external potential, which is allowed to vanish at infinity, and on a nonlinearity with subcritical growth. A key feature of our work is demonstrating that existence hinges on a delicate interplay between the decay rate of the potential and the local behavior of the nonlinearity near the origin. The proof relies on variational methods. We employ a penalization technique to overcome the lack of compactness, establish the existence of a solution for the resulting auxiliary problem via the Mountain Pass Theorem, and use crucial ‐estimates to show this solution solves the original system for a suitable range of parameters.

In this paper, we study the existence of solutions to the following Hartree equation \begin{align*} \begin{cases} -\Delta u+\lambda V(x) u+\mu u=\left(\int_{\mathbb{R}^N}\frac{|u|^p}{|x-y|^{N-\alpha}}\right)|u|^{p-2}u,\ \text{in}\ \mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2=\omega, \end{cases} \end{align*} Where $N\geq 3$ , $\omega,\lambda \gt 0$ , $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$ and µ will appear as a Lagrange multiplier. We assume that $0\leq V\in L^{\infty}_{loc}(\mathbb{R}^N)$ has a bottom $int V^{-1}(0)$ composed of $\ell_0$ $(\ell_{0}\geq1)$ connected components $\{\Omega_i\}_{i=1}^{\ell_0}$ , where $int V^{-1}(0)$ is the interior of the zero set $V^{-1}(0)=\{x\in\mathbb{R}^N| V(x)=0\}$ of V. It is worth pointing out that the penalization technique is no longer applicable to the local sublinear case $p\in \left(\frac{N+\alpha}{N},2\right)$ . Therefore, we develop a new variational method in which the two deformation flows are established that reflect the properties of the potential. Moreover, we find a critical point without introducing a penalization term and give the existence result for $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$ . When ω is fixed and satisfies $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}$ sufficiently small, we construct a $\ell$ -bump $(1\leq\ell\leq \ell_{0})$ positive normalization solution, which concentrates at $\ell$ prescribed components $\{\Omega_i\}^{\ell}_{i=1}$ for large λ. We also consider the asymptotic profile of the solutions as $\lambda\rightarrow\infty$ and $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}\rightarrow 0$ .

An incremental variational method to the coupling between gradient damage, thermoelasticity and heat conduction

In this study, we compare two novel hybrid data assimilation (DA) methods: Localized Weighted Ensemble Kalman filter (LWEnKF) and Implicit Equal-Weights Variational Particle Smoother (IEWVPS). These methods integrate a particle filter (PF) with traditional DA methods. LWEnKF combines the PF with EnKF, while IEWVPS integrates the PF with the four-dimensional variational (4DVAR) method. These hybrid DA methods not only overcome the limitations of linear or Gaussian assumptions in traditional assimilation methods but also address the issue of filter degeneracy in high-dimensional models encountered by pure PFs. Using the Regional Ocean Model System (ROMS), the effects of different DA methods for mesoscale eddies in the northern South China Sea (SCS) are examined using simulation experiments. The hybrid DA methods outperform the linear deterministic variational and Kalman filter methods: compared to the control experiment (no assimilation), EnKF, LWEnKF, IS4DVar and IEWVPS reduce the sea level anomaly (SLA) root-mean-squared error (RMSE) by 55%, 65%, 65% and 80%, respectively, and reduce the sea surface temperature (SST) RMSE by 77%, 78%, 74% and 82%, respectively. In the short-term assimilation experiment, IEWVPS exhibits superior performance and greater stability compared to 4DVAR, and LWEnKF outperforms EnKF (LWEnKF’s posterior SLA RMSE is 0.03 m, lower than EnKF’s value of 0.04 m). Long-term forecasting experiments (16 days, starting on 20 July 2017) are also conducted for mesoscale eddy prediction. The variational methods (especially IEWVPS) perform better in simulating the flow field characteristics of eddies (maintaining accurate eddy structure for the first 10 days, with an average SLA RMSE of 0.05 m in the studied AE1 eddy region), while the filters are more advantageous in determining the total root-mean-squared error (RMSE), as well as the temperature under the sea surface. Overall, compared to EnKF and 4DVAR, the hybrid DA methods better predict mesoscale eddies across both short- and long-term timescales. Although the computational costs of hybrid DA are higher, they are still acceptable: specifically, IEWVPS takes approximately 907 s for a single assimilation cycle, whereas LWEnKF only takes 24 s, and its assimilation accuracy in the later stage can approach that of IEWVPS. Given the computational demands arising from increased model resolution, these hybrid DA methods have great potential for future applications.

Bivariate drought risk assessment under uncertainty using variational bayesian monte carlo-based maximum entropy-copula method

We consider a semilinear 2n-order problem with nonconstant coefficients. We deduce existence results by using variational methods in two directions. We primarily treat the existence when the nonlinearity has asymptotic linear behaviour at infinity and is either asymptotically sublinear or linear at zero. Secondly, we discuss the superlinear case at infinity and prove three existence results showing that our problem has at least one or two nonzero solutions.

This work deals with the numerical solution of a time-fractional heat equation where a Caputo fractional derivative of order 0 is used in place of the traditional first-order time derivative. This change improves the model's capacity to represent anomalous diffusion behavior and memory effects, which are frequently seen in intricate engineering and physical systems. Applying and evaluating the Fractional Reduced Differential Transform Method (FRDTM) to solve this fractional-order partial differential equation is the aim of this work. The Fractional Variational Iteration Method (FVIM) was used to validate the findings. For different fractional orders, namely, and the classical case where with a known exact solution, two numerical examples were performed. The findings demonstrate that FRDTM offers extremely stable and accurate solutions that closely match the exact solution in the classical case. When it comes to capturing the change from rapid decay at lower fractional orders to more sustained solution profiles as the order increases, the FRDTM performs better than the FVIM. The differences between the two methods demonstrate FRDTM's superior convergence and accuracy across all cases considered. Finally, this study demonstrates the effectiveness of FRDTM as a reliable semi-analytical tool for solving fractional heat problems, and it contributes to advancing computational approaches for solving partial differential equations in science and engineering.

Abstract This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a (p, N)-Laplace Schrödinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space R N ${\mathbb{R}}^{N}$ . Through the use of smooth variational methods, penalization techniques, and the application of the Lusternik–Schnirelmann category theory, we establish a connection between the number of positive solutions and the topological properties of a set in which the potential function achieves its minimum values.

Non-perturbative approaches to linear and nonlinear responses (NLR) of atoms, molecules, and molecular aggregates are reviewed in relation to low and high harmonic generations (HG) by laser fields. These response properties are effective for the generation of entangled light pairs for quantum information processing by spontaneous parametric downconversion (SPDC) and stimulated four-wave mixing (SFWM). Quasi-energy derivative (QED) methods, such as QED Møller–Plesset (MP) perturbation, are reviewed as time-dependent variational methods (TDVP), providing analytical expressions of time-dependent linear and nonlinear responses of open-shell atoms, molecules, and molecular aggregates. Numerical Liouville methods for the low HG (LHG) and high HG (HHG) regimes are reviewed to elucidate the NLR of molecules in both LHG and HHG regimes. Three-step models for the generation of HHG in the latter regime are reviewed in relation to developments of attosecond science and spectroscopy. Orbital tomography is also reviewed in relation to the theoretical and experimental studies of the amplitudes and phases of wave functions of open-shell atoms and molecules, such as molecular oxygen, providing the Dyson orbital explanation. Interactions between quantum lights and molecules are theoretically examined in relation to derivations of several distribution functions for quantum information processing, quantum dynamics of molecular aggregates, and future developments of quantum molecular devices such as measurement-based quantum computation (MBQP). Quantum dynamics for energy transfer in dendrimer and related light-harvesting antenna systems are reviewed to examine the classical and quantum dynamics behaviors of photosynthesis. It is shown that quantum coherence plays an important role in the well-organized arrays of chromophores. Finally, applications of quantum optics to molecular quantum information and quantum biology are examined in relation to emerging interdisciplinary frontiers.

Abstract We prove the existence of a positive radial solution in a unit ball centered at the origin for some classes of Hénon equations involving supercritical nonlinearity. More precisely, we study how Hénon's weight impacts the variable supercritical exponent in the context of the work by do Ó, Ruf, and Ubilla. For this purpose, we combine variational methods with a new Sobolev–Hardy type embedding for radial functions into variable exponent Lebesgue spaces. The suitable use of the Radial Lemma allows us to arrive at the necessary estimate with fewer assumptions than those found in the existing literature.