Multiple Solutions Research Articles

Multimodal multi-objective hierarchical distributed consensus method for multimodal multi-objective economic dispatch of hierarchical distributed power systems

As the scale of power systems increases, the multi-layer distributed multi-objective consensus algorithm (MLDMOCA) can speed up computation and enhance privacy. However, the MLDMOCA needs to face: (1) the problems of error enlargement of hierarchy bringing deteriorating results; (2) over-dependence of certain weights on the choice of power balance adjustment factors. Moreover, multimodal multi-objective economic dispatch has not been performed for either small or large systems, resulting in poor diversity of economic scheduling solutions. Therefore, this study proposes a multimodal multi-objective hierarchical distributed consensus approach. The multimodal multi-objective hierarchical distributed consensus approach contains multimodal increments, and the proposed multimodal search method can obtain multimodal results. By changing the multimodal factor and multimodal critical threshold, the multimodal results and multimodal decision solutions of the proposed method are adjusted. In this study, the multimodal multi-objective hierarchical distributed consensus approach is simulated in IEEE 118-bus and IEEE 3059-bus cases. The simulation results show that the multimodal multi-objective hierarchical distributed consensus approach can achieve multiple multimodal solutions quickly and can adapt the power balance adjustment factor to almost all weights. Moreover, multiple linear-weighted multimodal objective values of the proposed approach are smaller than the objective value of the existing hierarchical distributed method.

Multiple Positive Solutions for Kirchhoff-Type Problems Involving Supercritical and Critical Terms

Multiple Positive Solutions for Kirchhoff-Type Problems Involving Supercritical and Critical Terms

Rayleigh–Bénard Convection With Multiple Solutions in Trapezoidal Closed Cavities

Abstract Rayleigh–Bénard convection (RBC) in symmetric trapezoidal closed cavities with cavity angle ϕ=70°−110°, filled with air, is studied using numerical simulations where inclined side walls are adiabatic. In contrast to rectangular cavities, where no flow exists below a threshold value, there is a weak convection even at a low Rayleigh number (Ra) due to the fact that there is a component of thermal gradient in the horizontal direction in these cavities. Interestingly, these cavities show sudden and significant jumps in the convection, similar to square cavities (Rac = 2585.02 for ϕ=90°), as Ra increases beyond a critical value (Rac). It is noted here that these Rac represent symmetry-breaking pitchfork bifurcations. These bifurcations are seen in both acute (Rac = 8000 for ϕ=70°) and obtuse (Rac = 2300 for ϕ=110°) angle trapezoidal cavities. Moreover, it is observed that multiple steady-state solutions (MSSS) exist as Ra is further increased. A forward and backward continuation approach for numerical simulations is used to track the co-existence of MSSS. These steady-states have co-existing one-roll and two-roll convective patterns beyond another threshold value of Ra. Here, two types of critical Ra have been identified for different cavity angles; one shows the sudden jump in the convection, and the other is the one beyond which MSSS co-exist. Furthermore, a codimension two bifurcation analysis is carried out with Ra and ϕ as two parameters. The bifurcation analysis divides the parameter space into different regions based on the multiplicity of the solutions.

On singular φ−Laplacian BVPs of nonlinear fractional differential equation

This paper investigates the existence of multiple positive solutions for a class of φ−Laplacian boundary value problem with a nonlinear fractional differential equation and fractional boundary conditions. Multiple solutions are proved under slight conditions on a possibly degenerating source term. Approximation techniques together with the fixed-point index theory a on cone of a Banach space are employed. Some illustrating examples of are also supplied. Mathematics Subject Classification (2010): 34A08, 34B15, 34B18, 47H10. Received 27 July; Accepted 13 December 2021

Mine seismic source location via forward modeling of spherical waves in a multilayered horizontal or inclined medium

Seismic source location is a classic inverse problem in seismology. In mathematical physics, inverse problems have multiple natural solutions. The objective of this study was to develop a generic theory and method of seeking the true solution from multiple solutions for the location of a coal mine seismic source in an idealized velocity structure model of a coal mine with a small scale and complex geological environment. Starting from the simplest velocity structure model, the complexity of the model gradually increased, until it approached the real velocity structure model, i.e., the multilayered horizontal and inclined velocity structure model, in order to find a generic method for solving the multi-solution inverse problem of coal mine seismic source location. Specifically, the wavefront distribution equation in a two-layer horizontal medium was derived and then expanded to any multi-layer horizontal medium. Based on this equation, a positive definite nonlinear equation system was established from the perspective of any observation system. The equation system contained four unknown variables of the spatiotemporal position of the seismic source. To determine the spatiotemporal parameters of the seismic source, nonlinear equations for four stations were required. To solve the nonlinear equation system, an initial iteration value was determined. In order to reduce the difficulty of determining the initial iteration value, the variable substitution method was used to reduce the number of location parameters. By rotating the original geodetic coordinate system of the station to be parallel and orthogonal to the medium interface, the wavefront method was extended to inclined medium. In conclusion, in this study, the problem of coal mine seismic source location in a multi-layer horizontal or inclined medium was effectively solved. The method proposed in this study provides a reference for solving the true solution from multiple solutions for the location of a coal mine seismic source in small-scale coal mines with complex geological environments.

New (3+1)-dimensional integrable generalized KdV equation: Painlevé property, multiple soliton/shock solutions, and a class of lump solutions

The present work aims to examine a newly proposed (3+1)-dimensional integrable generalized Korteweg-de Vries (gKdV) equation. By employing the Weiss- Tabor-Carnevale technique in conjunction with Kruskal ansatz, we establish the com- plete integrability of the suggested model by demonstrating its ability to satisfy the Painlev´e property. The bilinear form of the (3+1)-dimensional gKdV equation is em- ployed to construct multiple soliton solutions. By manipulating the various values of the corresponding parameters, we generate a category of lump solutions that exhibit localization in all dimensions and algebraic decay.

Multiple soliton, soliton molecules and the other diverse wave solutions to the (2+1)-dimensional Kadomtsev–Petviashvili equation

The central purpose of this paper is to explore the nonlinear dynamics of the (2+1)-dimensional Kadomtsev–Petviashvili equation (KPE). The multiple soliton solutions (MSSs) are constructed via applying the Hirota method. Then the soliton molecules on the ([Formula: see text]-, ([Formula: see text]- and ([Formula: see text]-planes are extracted via imposing the velocity resonance conditions to the MSSs. Eventually, two effective techniques, the sub-equation approach (SEA) and the variational approach (VA), are employed to probe some other diverse wave solutions, which are the bright wave, dark wave, singular wave and the singular periodic wave solutions. The dynamics of the extracted solutions are unveiled graphically to exhibit the physical attributes. The attained solutions in this paper can enlarge the exact solutions of the (2+1)-dimensional KPE and enable us to understand the nonlinear dynamic behaviors better.

Analytical optimization and performance evaluation of non-traditional tuned mass damper for stochastic base excited structures considering dual solutions

Compared with traditional tuned mass damper (TMD), non-traditional tuned mass damper (NTMD) has higher damping efficiency in some conditions. Nevertheless, the analytical optimization problem of NTMD has not been well solved; especially, multiple optimal solutions are not discussed. In this study, the closed-form optimal expressions of the NTMD using tuning stiffness are derived based on H 2 norm optimization. And the vibration mitigation performance of the NTMD is evaluated considering dual-solutions situation subjected to typical ground motion and impulse excitation. Different from solving the unique optimal frequency ratio, there exist two sets of optimal stiffness and damping ratio derived using tuned spring elements for given mass ratio, and the effective mass ratio is also limited as 1/3. The vibration mitigation or isolation performance and robustness of H 2 optimized NTMD with higher stiffness ratio are far superior to NTMD with lower stiffness ratio. NTMD using larger optimal tuning stiffness ratio also can be seen as an optimized form of Maxwell model using additional mass. Hence, using a large stiffness, namely, a large frequency ratio to design NTMD is recommended, which is different from traditional TMD. The case of NTMD with large stiffness can be adopted when the demands of practical engineering are not met using the optimal parameters of small stiffness. And a compromise scheme can also be provided, which are the combination with a larger mass ratio but is less than the maximum limit value 1/3 and a not very large stiffness ratio but is exceeded 0.5 according to demands of practical engineering.

Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

Abstract In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation ( − Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u ) in R N , ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

A collective incentive strategy to manage ridership rebound and consumer surplus in mass transit systems

This paper models and analyzes a collective incentive strategy with passenger departure time equilibrium to maximize the total passenger surplus, which could further enhance the sustainability of transit systems in the post-pandemic era. In the proposed collective incentive strategy, passengers with transit passes have dynamic discounts where the discount value depends on the total number of passengers enrolled in the incentive program. A bi-level optimization model is established by considering passengers' demand elasticity and responses to the strategy, which eventually determines the starting time and duration of the collective incentive strategy during the time of the day. We further analyze the properties of the proposed collective incentive model to identify the analytical solutions and performance. Analytical results indicate that there exists a threshold demand elasticity below which the total surplus of commuters with and without transit passes can be improved simultaneously. Properties also show that there could exist multiple local optimal solutions of incentive strategy duration for a given starting time. A case study is conducted with the metro system in Toronto: the results show that the proposed collective incentive strategy could improve both the fare revenues and ridership by 33.31 % and 83.56 %, respectively.

Physicochemical Targeting of Lipid Nanoparticles to the Lungs Induces Clotting: Mechanisms and Solutions.

Lipid nanoparticles (LNPs) have become the dominant drug delivery technology in industry, holding the promise to deliver RNA to up or down-regulate any protein of interest. LNPs have mostly been targeted to specific cell types or organs by physicochemical targeting in which LNP's lipid compositions are adjusted to find mixtures with the desired tropism. Here lung-tropic LNPs are examined, whose organ tropism derives from containing either a cationic or ionizable lipid conferring a positive zeta potential. Surprisingly, these LNPs are found to induce massive thrombosis. Such thrombosis is shown in the lungs and other organs, and it is shown that it is greatly exacerbated by pre-existing inflammation. This clotting is induced by a variety of formulations with cationic lipids, including LNPs and non-LNP nanoparticles, and even by lung-tropic ionizable lipids that do not have a permanent cationic charge. The mechanism depends on the LNPs binding to and then changing the conformation of fibrinogen, which then activates platelets and thrombin. Based on these mechanisms, multiple solutions are engineered that enable positively charged LNPs to target the lungs while ameliorating thrombosis. The findings illustrate how physicochemical targeting approaches must be investigated early for risks and re-engineered with a careful understanding of biological mechanisms.

Relative Motion Estimation Algorithm for Noncooperative Targets Considering Multiple Solutions of Rotational Parameters.

On-orbit servicing using a space robot is gaining popularity among the space community for both economic and safety aspects. In particular, the estimation of the relative motion of a noncooperative target is a challenging problem. This study presents a relative motion estimation scheme based on stereovision for noncooperative targets considering multiple solutions of rotational parameters. Specifically, the mass distribution of the target is identified based on the least-square method and the principle of conservation of angular momentum. Then, the determination of a unique principal axis coordinate frame of the target is employed to resolve the multiple-solution problem. In addition, an EKF (extended Kalman filter)-based filter with global observability is designed to estimate the full motion states and inertia parameters of the target. The convergence performance of the proposed method is verified by numerical simulation. The results also demonstrate that the method is robust to occlusion.

Multiple Solutions to a Transmission Problem with a Critical Hardy-Sobolev Exponential Source Term

Multiple Solutions to a Transmission Problem with a Critical Hardy-Sobolev Exponential Source Term

The Use of Syntactic Information in fake news detection: A Systematic Review

Fake news has been a critical problem for society, to the extent that its damaging effects can already be seen in several areas, such as democracy and health. However, as fake news grow in number, manual fact-checking becomes impractical for identifying them, which makes automatic detection a compelling alternative. In this sense, this study gathers multiple solutions for the problem of automatically detecting fake news, through the usage of both lexical and syntactic information. This study consists of a systematic review on fake news detection through linguistic patterns, focusing on the use of syntax to aid in the task. Solving complex problems by capturing linguistic patterns is mostly explored in the Natural Language Processing (NLP) area. In general, the use of shallow syntax representations, such as Parts of speech, only marginally increases the performance of classifiers in this task. However, relying on deeper syntactic representations, such as context-free grammars or syntactic dependency trees, present more promising results.

Multiple solutions and conserved vectors of a shallow water wave equation arising in fluid mechanics; Lie group analysis

Investigation has shown that shallow water wave model equations constitute a class of hyperbolic partial differential equations describing the movement below a pressure surface in a fluid, and this makes them highly useful to be studied. Therefore, in this paper we analytically investigate a one-dimensional shallow water wave equation that is highly applicable in fluid dynamics. We leverage the robust nature of the Lie group technique to provide symmetries to the equation. Commutator and adjoint representations of symmetries were generated and presented in tabular form. Sequel to that, we perform reductions of the shallow water wave model under study via its point symmetries so that its exact solutions can be found. Thus, to achieve this, various combinations of the symmetries are made. Besides, we examine the travelling wave solutions of the model. The direct integration method is used to obtain elliptic integral functions as well as elliptic solutions in terms of the Weierstrass function. In addition, we utilize standard techniques such as (G′/G)-expansion, power series, and Kudryashov’s approaches to generate more closed-form solutions to the model. Other interesting solutions achieved include hyperbolic, trigonometric, rational, and exponential functions. In search of the physical meaning associated with the results, graphical demonstrations are included, and these are given in 2D, 3D, and density plots. Thereafter, the construction of conserved vectors for the aforementioned model is done with the use of the standard multiplier approach and Ibragimov’s theorem. These techniques give the opportunity to furnish various forms of conserved currents.

A novel high-resolution imaging method based on sparsity in the time domain and spectral fitting in the frequency domain

During the propagation of seismic waves underground, the high-frequency seismic response of thin reservoir is absorbed and attenuated, which poses a challenge in seismic thin reservoir prediction. The high-resolution processing techniques have the capability to significantly expand the frequency range of the seismic data, so it becomes a key technique for thin reservoir prediction. Most of these techniques necessitate the extraction of seismic wavelets. However, the spatial and temporal variations of seismic data result in multiple solutions for wavelet extraction. Simultaneously, the majority of techniques fail to consider the influence of spatial tectonic features on the high-resolution processing. In this paper, we propose a novel solution to address these two fundamental challenges by utilizing seismic spectral expansion, sparse reflection coefficients, and spatial continuity constraints. First, we propose an innovative spectral fitting method that aims to expand the frequency bandwidth while adhering to the desired wavelet constraints. This method allows us to fully utilize the effective frequency information. It not only obtains broadband seismic data but also captures precise wavelets. Then, sparse deconvolution is employed to further extend the frequency range by utilizing the accurately expected wavelet and obtaining a high-resolution reflection coefficient. Finally, the Hessian matrix regularization is employed to constrain the spatial continuity of the reflection coefficient. This method is validated in both the model and real seismic data. Compared to traditional sparse deconvolution and spectral modeling deconvolution with spatial constraints, this method not only expands the frequency bandwidth and enhances seismic resolution but also preserves operational frequency information and improves the spatial continuity of seismic data. It has been verified that this approach can be used to forecast thin reservoir and reconstruct spatial tectonic characteristics.

Non‐singular complexiton, singular complexiton and complex multiple soliton solutions to the (3 + 1)‐dimensional nonlinear evolution equation

This research mainly concerned with some new solutions of the (3 + 1)‐dimensional nonlinear evolution equation (NEE). First, we extract the resonant multiple soliton solutions (RMSSs) by taking advantage of the linear superposition principle (LSP) and weight algorithm (WA). Then the non‐singular complexiton and singular complexiton solutions are developed by introducing pairs of the conjugate parameters. Besides, the complex multiple‐soliton solutions are also explored with the aid of the bilinear approach. The graph descriptions of the attained solutions are paraded to show the dynamical properties. The outcomes of this work are all new and are desirous to bring some new perspective to the investigation of the complexiton solutions to the other partial differential equations (PDEs).

Further study on oil/water relative permeability ratio model and waterflooding performance prediction model for high water cut oilfields sustainable development

The accuracy of predicting waterflooding performance is crucial in determining the scale of investment for oilfield development. However, existing common waterflooding prediction models often relies on assumptions that may not apply universally or lack theoretical derivation through statistical analysis. This has led to unsatisfactory prediction accuracy and multiple potential solutions. To address these limitations, it is proposed to incorporate the oil/water relative permeability ratio model into the derivation process of waterflooding prediction models. Initially, an evaluation of prevalent oil/water relative permeability ratio models is conducted, along with an analysis of their primary constraints. Additionally, the applicability of the analytical relative permeability model is thoroughly examined. Building upon the analytical relative permeability model and a modified Welge equation, a new waterflooding model is formulated, encompassing all pertinent physical coefficients. Notably, this model aligns seamlessly with the commonly used Arps’ decline curve, while extending its applicability to a broader range of conditions. Moreover, it can be simplified to generate typical water drive curves under suitable circumstances. The semi-log relationship between oil/water relative permeability ratio and water saturation is further simplified into a linear relationship or a multi-term formula. Compared with the traditional waterflooding model, the new model proposed in this research has a wider application range and can be applied to oilfield at high water cut. At the same time, the new model clarifies the coefficient of waterflooding curve A and the physical meaning of parameter 7.5 in Tong’s chart method for the first time. The new model proposed in this research further enriches the connotation of waterflooding theory and has certain application significance.

Boundary Layer Stagnation Point Flow and Heat Transfer over a Nonlinear Stretching/Shrinking Sheet in Hybrid Carbon Nanotubes: Numerical Analysis and Response Surface Methodology under the Influence of Magnetohydrodynamics

The present study aims to offer new numerical solutions and optimisation strategies for the fluid flow and heat transfer behaviour at a stagnation point through a nonlinear sheet that is expanding or contracting in water-based hybrid nanofluids. Most hybrid nanofluids typically use metallic nanoparticles. However, we deliver a new approach by combining single- and multi-walled carbon nanotubes (SWCNTs-MWCNTs). The flow is presumptively steady, laminar, and surrounded by a constant temperature of the ambient and body walls. By using similarity variables, a model of partial differential equations (PDEs) with the magnetohydrodynamics (MHD) effect on the momentum equation is converted into a model of non-dimensional ordinary differential equations (ODEs). Then, the dimensionless first-order ODEs are solved numerically using the MATLAB R2022b bvp4C program. In order to explore the range of computational solutions and physical quantities, several dimensionless variables are manipulated, including the magnetic parameter, the stretching/shrinking parameter, and the volume fraction parameters of hybrid and mono carbon nanotubes. To enhance the originality and effectiveness of this study for practical applications, we optimise the heat transfer coefficient via the response surface methodology (RSM). We apply a face-centred central composite design (CCF) and perform the CCF using Minitab. All of our findings are presented and illustrated in tabular and graphic form. We have made notable contributions in the disciplines of mathematical analysis and fluid dynamics. From our observations, we find that multiple solutions appear when the magnetic parameter is less than 1. We also detect double solutions in the shrinking region. Furthermore, the increase in the magnetic parameter and SWCNTs-MWCNTs volume fraction parameter increases both the skin friction coefficient and the local Nusselt number. To compare the performance of hybrid nanofluids and mono nanofluids, we note that hybrid nanofluids work better than single nanofluids both in skin friction and heat transfer coefficients.

Multiplicity and concentration of positive solutions to the double phase Kirchhoff type problems with critical growth

The aim of this paper is to study the multiplicity and concentration of positive solutions to the $(p,q)$ Kirchhoff-type problems involving a positive potential and a continuous nonlinearity with critical growth at infinity. Applying penalization techniques, truncation methods and the Lusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its minimum values.