Convergence Of Solutions Research Articles

Exploring the role of nanoparticles in enhancing thermal performance of fractional Carreau fluid flow under magnetic field and hall current

Nanofluids garner significant scientific interest due to their exceptional heat-conducting abilities and potential to enhance heat transfer efficiency. This manuscript investigates the behavior of a two-dimensional fractional Carreau fluid model on a stretched sheet over time, incorporating convection, magnetic fields, nanoparticles, diffusion, and thermal radiations. The model elucidates viscoelastic nanofluids’ memory and inheritance properties, employing non-integer Caputo fractional derivatives innovatively. Fundamental equations are transformed into dimensionless form and solved using an explicit finite difference approach. Essential parameters like Skin friction coefficient, Nusselt number, and Sherwood number are accurately determined. Rigorous stability and convergence criteria ensure effective solution convergence. Visual representations demonstrate the significant impact of each parameter on fluid flow. It is noted that temperature gradient increases 49.38% with the increase of fractional exponent α while it decreases 13.39% with the increase of magnetic parameter M. Moreover, concentration gradient increases 66.48% with the increase of thermophoresis parameter Nt and 94.71% with the increase of pedesis parameter Nb.

Long time existence of the non-isentropic slightly compressible Navier-Stokes equations with boundary conditions

We investigate the long time existence of smooth solutions to the initial boundary value problem for the non-isentropic slightly compressible Navier–Stokes equations with slip or non-slip boundary conditions on the velocity. We verify that the compressible Navier–Stokes equations with boundary conditions admit a unique smooth solution on the time interval where the smooth solution of the incompressible Navier–Stokes equations exists, when the Mach number is sufficiently small. Moreover, we obtain the uniform convergence of smooth solutions for the compressible system toward those for the corresponding incompressible system on that time interval.

Novel criteria for stability and convergence of solutions for non-autonomous nonlinear systems

In this paper, we give a new integral inequality which is used to study the asymptotic behaviour of solutions for a class of nonlinear dynamic systems with small perturbation using a numerical approach. We provide some new results on the stability of perturbed systems where necessary and sufficient condition is derived. We show that the perturbed nonlinear system can be globally uniformly practically asymptotically stable provided that the bound of perturbation is small enough. A numerical example is presented to illustrate the validity of the main result.

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

We compute the first twenty moments of three convergent quartic bi-tracial 2-matrix ensembles in the large N limit. These ensembles are toy models for Euclidean quantum gravity originally proposed by John Barrett and collaborators. A perturbative solution is found for the first twenty moments using the Schwinger-Dyson equations and properties of certain bi-colored unstable maps associated to the model. We then apply a result of Guionnet et al. to show that the perturbative and convergent solution coincide for a small neighbourhood of the coupling constants. For each model we compute an explicit expression for the free energy, critical points, and critical exponents in the large N limit. In particular, the string susceptibility is found to be γ = 1/2, hinting that the associated universality class of the model is the continuous random tree.

Highly accurate wavelet solution for the two-dimensional Bratu's problem

A modified wavelet interpolation Galerkin method (WIGM) with improved flexibility of nodal distribution is formulated and utilized to solve the Bratu's problem. Such a method treats the nonlinear term as an independent function and directly approximates it through the proposed wavelet interpolation, resulting in a low-cost discretization of the Bratu's equation with exponential nonlinearity. An efficient implementation of Newton's method to find the lower and upper branches of solution as well as the turning point is then presented. Numerical results demonstrate that the present WIGM can accurately estimate both two solution branches and turning point for the Bratu's problem. Compared with many other existing methods, the proposed WIGM also offers marked superiority in terms of accuracy when using the same node distribution. Moreover, a convergent solution to the resulting nonlinear discrete system obtained by the WIGM can be observed in a few iterations even when the initial guess significantly deviates from the exact solution. Finally, a highly accurate approximation of both two-branched solutions and the turning point for the two-dimensional Bratu's problem is obtained by the proposed WIGM.

Enhanced LSTM‐DQN algorithm for a two‐player zero‐sum game in three‐dimensional space

AbstractTo tackle the challenges presented by the two‐player zero sum game (TZSG) in three‐dimensional space, this study introduces an enhanced deep Q‐learning (DQN) algorithm that utilizes long short term memory (LSTM) network. The primary objective of this algorithm is to enhance the temporal correlation of the TZSG in three‐dimensional space. Additionally, it incorporates the hindsight experience replay (HER) mechanism to improve the learning efficiency of the network and mitigate the issue of the “sparse reward” that arises from prolonged training of intelligence in solving the TZSG in the three‐dimensional. Furthermore, this method enhances the convergence and stability of the overall solution.An intelligent training environment centred around an airborne agent and its mutual pursuit interaction scenario was designed to proposed approach's effectiveness. The algorithm training and comparison results show that the LSTM‐DQN‐HER algorithm outperforms similar algorithm in solving the TZSG in three‐dimensional space. In conclusion, this paper presents an improved DQN algorithm based on LSTM and incorporates the HER mechanism to address the challenges posed by the TZSG in three‐dimensional space. The proposed algorithm enhances the solution's temporal correlation, learning efficiency, convergence, and stability. The simulation results confirm its superior performance in solving the TZSG in three‐dimensional space.

An improved slime mould algorithm using multiple strategies

Aiming at the defects of standard slime mould algorithm (SMA), such as local optima stagnation, slow convergence and improper balance between exploitation and exploration, we propose an improved SMA that contains the adaptive t-distributed variation strategy, improved location update formula and chaotic opposition-based learning strategy, that is, the MISMA. Utilizing comparative experiments and ablation studies on classical benchmark and CEC2020 benchmark suite, we proved that MISMA outperforms other state-of-the-art rival algorithms on convergence speed, solution accuracy, and robustness, each component of MISMA achieves an improvement on the defects of SMA at each stage and exhibits synergistic effects.

An online transfer learning based multifactorial evolutionary algorithm for solving the clustered Steiner tree problem

The rapid increase in the number and computing power of devices interacting through the network over the past few years has led to a growing trend of decentralized clustering network architecture. The Clustered Steiner Tree Problem (CluSteiner) is a recently introduced NP-Hard problem and is assessed to be the core of efficient multicast routing in such networks. However, research on this problem in terms of theory and algorithmic design is still in its infancy as current solving approaches are limited in the literature. Lately, the Multifactorial Evolutionary Algorithm (MFEA) has been applied to solve several clustering-structure network design problems for the reason that these problems rarely exist independently and are often deployed concurrently in practice. To effectively transfer knowledge between problems while solving them simultaneously, this paper proposes an approach based on the novel data-driven MFEA-II to deal with the CluSteiner problem. In the proposal, an efficient encoding and a two-level decoding mechanism are introduced, combined with a module capable of online learning the amount of knowledge transferred between tasks that help the algorithm to explore potential regions of the search space. Experiments are undertaken on various test instances to evaluate the efficacy of the proposed algorithm. The empirical results indicate that our proposal can perform well on both metric and non-metric graphs with the statistical tests used to verify its efficiency over other baseline algorithms regarding the solution quality and convergence trend.

Similarity reduction, group analysis, conservation laws, and explicit solutions for the time-fractional deformed KdV equation of fifth order

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of [Formula: see text] which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail.

A novel reinforcement learning based Heap-based optimizer

The Heap-Based Optimization (HBO), a novel meta-heuristic algorithm, constructs a hierarchical heap resembling a company structure to facilitate interaction among search agents. This enables the search agent to learn various levels of information from the population to obtain improved solutions, but it has drawbacks such as limited search capability and slow convergence speed when tackling complex optimization problems. To address these challenges, we propose an enhanced heap optimizer based on reinforcement learning(RLHBO), which demonstrates robust stability, rapid convergence, and high solution accuracy. Firstly, fixed operators are replaced with a reinforcement learning dynamic control search strategy in RLHBO to regulate the search range of agents. This prevents agents from being restricted in their search when encountering local or global optima, thus enhancing their search capability. Additionally, RLHBO introduces a dimensional self-learning strategy and a convergence strategy based on search direction similarity to address issues such as invalid and discrete solutions, accelerating convergence. Furthermore, an initialization strategy based on quasi-oppositional learning is employed to increase the diversity of the algorithm-generated population, surpassing the limitations of random initialization and further boosting convergence speed. Extensive testing on various complex functions from CEC2017 and CEC2022 demonstrates that RLHBO, as presented in this paper, outperforms HBO and numerous other advanced MAs in terms of search capability and convergence speed.

Convergence of the block Lanczos method for the trust‐region subproblem in the hard case

SummaryThe trust‐region subproblem (TRS) plays a vital role in numerical optimization, numerical linear algebra, and many other applications. It is known that the TRS may have multiple optimal solutions in the hard case. In [Carmon and Duchi, SIAM Rev., 62 (2020), pp. 395–436], a block Lanczos method was proposed to solve the TRS in the hard case, and the convergence of the optimal objective value was established. However, the convergence of the KKT error as well as that of the approximate solution are still unknown for this method. In this paper, we give a more detailed convergence analysis on the block Lanczos method for the TRS in the hard case. First, we improve the convergence speed of the approximate objective value. Second, we derive the speed of the KKT error tends to zero. Third, we establish the convergence of the approximation solution, and show theoretically that the projected TRS obtained from the block Lanczos method will be close to the hard case more and more as the block Lanczos process proceeds. Numerical experiments illustrate the effectiveness of our theoretical results.

A pareto strategy based on multi-objective optimal integration of distributed generation and compensation devices regarding weather and load fluctuations

In this study, we present a comprehensive optimization framework employing the Multi-Objective Multi-Verse Optimization (MOMVO) algorithm for the optimal integration of Distributed Generations (DGs) and Capacitor Banks (CBs) into electrical distribution networks. Designed with the dual objectives of minimizing energy losses and voltage deviations, this framework significantly enhances the operational efficiency and reliability of the network. Rigorous simulations on the standard IEEE 33-bus and IEEE 69-bus test systems underscore the effectiveness of the MOMVO algorithm, demonstrating up to a 47% reduction in energy losses and up to a 55% improvement in voltage stability. Comparative analysis highlights MOMVO's superiority in terms of convergence speed and solution quality over leading algorithms such as the Multi-Objective Jellyfish Search (MOJS), Multi-Objective Flower Pollination Algorithm (MOFPA), and Multi-Objective Lichtenberg Algorithm (MOLA). The efficacy of the study is particularly evident in the identification of the best compromise solutions using MOMVO. For the IEEE 33 network, the application of MOMVO led to a significant 47.58% reduction in daily energy loss and enhanced voltage profile stability from 0.89 to 0.94 pu. Additionally, it realized a 36.97% decrease in the annual cost of energy losses, highlighting substantial economic benefits. For the larger IEEE 69 network, MOMVO achieved a remarkable 50.15% reduction in energy loss and improved voltage profiles from 0.89 to 0.93 pu, accompanied by a 47.59% reduction in the annual cost of energy losses. These results not only confirm the robustness of the MOMVO algorithm in optimizing technical and economic efficiencies but also underline the potential of advanced optimization techniques in facilitating the sustainable integration of renewable energy resources into existing power infrastructures. This research significantly contributes to the field of electrical distribution network optimization, paving the way for future advancements in renewable energy integration and optimization techniques for enhanced system efficiency, reliability, and sustainability.

Deep and wide search assisted evolutionary algorithm with reference vector guidance for many-objective optimization

Many-objective optimization problems appear in a large many practical cases, but maintaining the convergence and diversity of solutions becomes a big challenge. In response to this, a deep and wide search assisted evolutionary algorithm with reference vector guidance (RVEA-DWC) is proposed. In the algorithm, a novel environmental selection criterion based on substituting Iɛ+ indicator for distance is introduced to enhance the convergence, and once the selected individuals exceed the population size after multiple traversals, those with poor convergence or diversity are randomly eliminated. In addition, to tackle the irregular Pareto front shapes, the invalid reference vectors are deleted regularly, and a certain number of reference vectors with local diversity and global diversity are added, so as to conduct a overall search that combines deep search and wide search to balance exploitation and exploration. An experimental study of 5, 10, 15, 20 objectives is conducted on 60 test instances. The results demonstrate that the proposed algorithm is superior to the other twelve many-objective evolutionary algorithms.

A novel derivative search political optimization algorithm for multi-area economic dispatch incorporating renewable energy

Multi-area economic dispatch (MAED) incorporating renewable energy has become an important issue in the power system optimization. Existing intelligent optimization algorithms often suffer from poor solution accuracy or slow convergence when dealing with MAED problems. In this paper, a novel derivative search-based political optimization (DSPO) algorithm is proposed to handle the MAED problem incorporating renewable energy including wind and solar energy. In the renewable energy modeling, the Weibull and log-normal probability density functions are used to calculate available wind and solar power respectively. In order to improve the search performance, DSPO adopts two strategies: leader guide strategy and derivative search mechanism. The former adds the leader’s global optimal information which can direct candidate solutions to more promising regions and speed up convergence. The latter derives neighborhood solutions around some high-quality solutions to improve the exploitation ability. The DSPO algorithm is applied to solve four MAED problems which take into account valve point effect, prohibited operating zone, power loss and so on. The simulation results show that the DSPO algorithm achieves the overall best results in terms of convergence speed, solution accuracy and stability when compared with several well-established algorithms.

Robust cooperative control strategy for a platoon of connected and autonomous vehicles against sensor errors and control errors simultaneously in a real-world driving environment

In a real-world driving environment, a platoon of connected and autonomous vehicles (CAVs) is subject to many internal and external disturbances, resulting in uncertain vehicle dynamics. In general, the disturbances can be categorized into two types: disturbances due to vehicle sensor errors (e.g., GPS error) and disturbances due to vehicle control errors (e.g., actuator delay). In the literature, many control strategies have been proposed to improve the robustness of the CAV platoon against uncertain vehicle dynamics induced by these disturbances. However, most of these strategies only consider one type of disturbance and cannot tackle both types of disturbances simultaneously. Furthermore, they are designed to maximize the benefits of each vehicle in the platoon independently, which can deteriorate the performance of the platoon. To address these problems, this study proposes a robust cooperative control (RCC) strategy to maneuver the vehicles in the platoon cooperatively to counteract the impacts of both types of disturbances. The RCC strategy is developed based on a minimax problem, where the maximization subproblem seeks to find the worst inputs for the uncertainty terms in the vehicle dynamics equation to minimize the platoon performance, while the minimization subproblem seeks to find the optimal control decisions for all subsequent vehicles to maximize the platoon performance in the worst case. To solve the minimax problem, this study proposes a globally convergent solution algorithm. It can solve the minimax problem very efficiently to enable real time deployment of the RCC strategy. Numerical application indicates that compared to the existing methods, the RCC strategy can dramatically improve the robustness of the CAV platoon against the uncertain vehicle dynamics induced by both vehicle state detection errors and vehicle control errors. Therefore, it can maneuver the CAV platoon safely and efficiently in a real-world driving environment.

Physical Models for Sustainability using Fredholm Integro-Differential Equations: Applicability and Analysis of Chebyshev Polynomial Method

Numerical analysis is concerned with the mathematical derivation, explanation and evaluation/analysis of algorithms, models and methods used to obtain numerical solutions for mathematical problems. This paper explores the reliability of the Chebyshev Polynomial Method (CPM) for solving a specific class of equations known as the second-order Fredholm Integro-Differential Equations (FIDEs). A series expansion of the Chebyshev polynomial is derived, used in solving these integral equations, and later on examined in terms of accuracy and convergence of solutions. The evaluation process involves a hybrid approach, combining manual methods and mathematical programs like MAPLE and MATLAB. In addition, three numerical examples were solved in which two truncation points are considered per each example. Furthermore, the performance of the CPM is reported in terms of accuracy, convergence, suitability, reliability and effectiveness in the context of the exact solution.

Numerical analysis of a nonlinear age-structured HBV model with saturated incidence and spatial diffusion

In this paper, the numerical properties of a nonlinear age-structured hepatitis B virus model with saturated incidence and spatial diffusion are studied. Applying linearly implicit Euler method in time integration, a numerical scheme which can preserve the biological meanings is constructed. The convergence of the numerical solution in finite time is explored. In stability analysis, a threshold is proposed, which is called numerical basic reproduction number and denoted by R0h. It is proved that the numerical solution is locally asymptotically stable at the disease-free equilibrium when R0h<1. Moreover, it is proved the numerical basic reproduction number converges to the exact basic reproduction number of the model with first order accuracy. Furthermore, it is shown that a numerical space independent equilibrium exists and is asymptotically stable if R0h>1, which implies the threshold stability of the model can be preserved by numerical solution proposed. Eventually, our conclusions are tested through numerical experiments.

An inverse front tracking method for modeling solidification process in a smelting furnace

An inverse process based on the conjugate gradient and adjoint problem is developed for predicting the time-varying bank thickness during the solidification/melting process in a smelting furnace. The direct problem rest on a single-phase moving boundary model to simulate the solid bank and refractory brick, simultaneously. The time-varying heat flux at solid/liquid interface, initial bank thickness and time-varying heat transfer coefficient at outer surface of the furnace are considered as the unknowns. Unlike the methods developed so far based on the enthalpy model, the current method does not require any prior information regarding the thermophysical properties of the liquid-phase and complex physical phenomena that occur in it. This improves the uniqueness of the solution that is inherent in the ill-posed inverse methods. To avoid the so-called “inverse crime”, a two-phase enthalpy method is used to generate the simulated measurement. These measurements are used in the present single-phase method for reconstruction of bank history. The model has shown good performance for non-intrusive temperature obtained from the outer surface of the furnace. The inverse model has been validated through an experiment conducted in a metallurgical reactor. A reasonable agreement between the estimated and measured molten salt bank indicates the existence and convergence of the inverse solution.

Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile

We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct the unique self-similar solution, and show that starting from arbitrary initial data, solution orbits converge to the self-similar solution.

A fast multi-objective pelican optimizer for optimal coordination of DGs and DSTATCOM considering risk penetration level of wind

In this study, a new variant, the fast pelican optimizer (FPO), is proposed to improve the performance of the radial distribution network (RDEN). The proposed variant is characterized by creating a dynamic interaction between two phases, exploration and exploitation, during the search process. The modifications introduced within the standard algorithm allow the proposed new variant, namely FPO, to be fast and adaptive to efficiently solve various complex optimization problems. In the first stage, the proposed FPO is newly adapted and applied to solve the optimal locations of various types of distributed generation based renewable sources and multi shunt compensators, namely the DSTATCOM devices based FACTS technology, and in the second stage, the proposed FPO is applied to optimize the active power of DG units in coordination with the reactive power of multi DSTATCOM. Three objective functions, such as the total power losses, the total voltage deviation, and the margin stability, are optimized individually and in coordination to enhance the performances of the practical radial distribution network (RDEN) 33-bus. A deep comparative study in terms of solution quality and convergence accuracy based statistical analysis is elaborated to demonstrate the competitive aspect of the proposed FPO.