Convergence Of Solutions Research Articles

First Derivative Approximations and Applications

In this paper, we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of the first derivative for the numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper.

The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

Significance of gyrotactic microorganism's analysis for magnetized convectively heat 3D Sisko fluid flow with bioconvection phenomenon

In this research article, the bioconvection of oxytactic microorganisms with magneto-hydrodynamic (MHD) 3D steady flow of Sisko nanomaterial over a stretching sheet through convective conditions is deliberated and analyzed. Possessions of thermophoresis, Brownian motion as well as mixed convection in the Sisko fluid are also deliberated. Sisko fluid is supposed to be electrically conducted over and done with a constant pragmatic magnetic field. The flow problem is communicated using governing relations and problem is transmuted into dimensionless form among non-similar procedure. To obtain convergent solutions we must solve nonlinear differential systems. The consequences of several physical parameters have been deliberate and discussed. Our results displayed that the higher microorganism difference parameter condensed the microorganism profile, while an opposite influence is established for magnetic parameter. Concentration proises of the Sisko fluid flow are improved by growing Biot number whereas contracted beside the Lewis number. In recent years, the scientists have shown great attention in the field of bioconvection owing to its countless applications such as amassing the cells, bioreactors, gas-bearing sedimentary, modeling oil, exclusion of living, and nonliving cells, and lots of additional.

Auto-Bäcklund transformation and exact solutions for a new integrable (2+1)-dimensional shallow water wave equation

Shallow water waves (SWWs) are often used to describe water flow and wave movement in shallow water areas. The article introduces a novel (2 + 1)-dimensional SWW equation. We prove that the equation is integrable and obtain an auto-Bäcklund transformation by truncating Painlevé expansion. Using the bilinear form of the equation, a new auto-Bäcklund transformation and some exact solutions are obtained. Besides, a convergent power series solution is derived using Lie symmetry method. These exact solutions can enrich mathematical modeling and help us understand nonlinear wave phenomena. Finally, conserved vectors are derived.

A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the (G′/G2)-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems.

A framework of integrated differential evolution variants based on adaptive relay mode for global optimization

Differential evolution (DE) is highly competitive in single-objective real parameter optimization. However, there are still some problems with differential evolutionary variants in optimizing complex multimode functions, such as poor solution accuracy and slow convergence. To address these problems, an optimization framework of integrated DE variants based on adaptive relay mode (fDE-ARM) is proposed. In this framework, different DE variants are integrated through two adaptive relay mechanisms to give full play to the advantages of the algorithms, thereby improving the performance of the whole algorithm. For the first adaptive relay mode, when the currently executed algorithm is judged to have converged, the population is updated by Gaussian random walk, and then the optimization is continued through the relay algorithm. For the second adaptive relay mode, the relay condition is determined by the average fitness improvement rate. If the condition is met, the relay algorithm takes over the optimization. At the same time, if the diversity is lower than the threshold, to improve the exploration, the roulette wheel method is used to select some individuals to perform the Gaussian random walk. In addition, a feedback mechanism is introduced in the relay process to avoid false switching. To verify the performance of the proposed algorithm, extensive simulations are performed on CEC2005, CEC2014, CEC2017, and CEC2021 benchmark functions. In addition, three engineering problems are used to test the performance. Compared with some newly proposed optimization algorithms, fDE-ARM is statistically superior to the comparison algorithms in terms of solution accuracy, convergence speed, and stability.

Convergence of solutions of the porous medium equation with reactions

Consider the Cauchy problem of one-dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as t\to \infty to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behavior of the solutions for PME with monostable, bistable and combustion types of nonlinearities.

Analyses of the tail-[formula omitted] minimization for fast and enhanced sparse selections

We investigate the effectiveness and efficiency of the iterative tail-ℓ2 minimization (tail-ℓ2-min) technique for its sparse selection capabilities. We conduct profile analyses on the tail-ℓ2-min, establishing the equivalence of the tail-ℓ2-min problem to a two-stage profile ℓ2 formulation, both featuring analytical solutions. The tail null space property (NSP) of sensing matrix A is shown to be equivalent to the NSP of the newly defined profile matrix Ã. Besides the error bound analysis for the tail-ℓ2-min under the typical tail-NSP condition, a novel error bound of the tail-ℓ2-min formulation is also established without relying on NSP or restricted isometry property (RIP) assumptions. It merely contains tractable coefficients of A, and offers insights into successful recovery, with the observation of the convergent iterative procedure. Numerical studies and the applications to image reconstruction demonstrate the superiority and fast convergence of the tail-ℓ2 sparse solution over state-of-the-art sparse selection methodologies. The sparsity level of a signal that the tail-ℓ2 profile algorithm guarantees the recovery is around 41% higher than that of the basis pursuit algorithm. The analytical solutions of the tail-ℓ2 method at each iteration also ensure that the tail-ℓ2 sparse recovery process is notably fast, especially for high dimensions and high sparsity levels.

A Comparison of Free and Mapped Meshes for Static Structural Analysis

Abstract This study addresses the challenge faced by Finite Element Analysts when choosing between free and mapped meshes, especially in terms of convergence stability and solution accuracy. The investigation focuses on 3D solid models under static structural loading, analyzed using Ansys® and MSC Patran®. Both free and mapped mesh types, employing equivalent 3D solid elements, are used to assess an aircraft structural component under design load conditions, with fixed boundaries. For free meshes, Tet10 elements in Patran (equivalent to Solid 72 in Ansys) are used, whereas for mapped meshes, CPENTA / CHEXA elements in Patran (equivalent to Wed6 / Hex8 in Ansys) are employed. Mesh convergence studies ensure that discretization does not affect the numerical solution. Notably, a significant stress increase is observed with successive refinement of free meshes, while mapped meshes achieve mesh independence at coarser refinement levels. Comparison of fringe plots indicates the same location for maximum deformation and equivalent stress in both free and mapped mesh models. The findings demonstrate that free meshes tend to underpredict maximum deformation and equivalent stress compared to mapped meshes, with both meshes showing deformation and stress at consistent locations. The findings underscore the importance of carefully choosing the appropriate mesh type, particularly when analyzing critical structural components, to ensure reliability and accuracy in FEA simulations.

Hydrodynamic limit for the non-cutoff Boltzmann equation

This work deals with the non-cutoff Boltzmann equation for all types of potentials, in both the torus \mathbf{T}^{3} and in the whole space \mathbf{R}^{3} , under the incompressible Navier–Stokes scaling. We first establish the well-posedness and decay of global mild solutions to this rescaled Boltzmann equation in a perturbative framework, that is, for solutions close to the Maxwellian, obtaining in particular integrated-in-time regularization estimates. We then combine these estimates with spectral-type estimates in order to obtain the strong convergence of solutions to the non-cutoff Boltzmann equation towards the incompressible Navier–Stokes–Fourier system.

Improving the Kepler optimization algorithm with chaotic maps: comprehensive performance evaluation and engineering applications

The Kepler Optimisation Algorithm (KOA) is a recently proposed algorithm that is inspired by Kepler’s laws to predict the positions and velocities of planets at a given time. However, although promising, KOA can encounter challenges such as convergence to sub-optimal solutions or slow convergence speed. This paper proposes an improvement to KOA by integrating chaotic maps to solve complex engineering problems. The improved algorithm, named Chaotic Kepler Optimization Algorithm (CKOA), is characterized by a better ability to avoid local minima and to reach globally optimal solutions thanks to a dynamic diversification strategy based on chaotic maps. To confirm the effectiveness of the suggested approach, in-depth statistical analyses were carried out using the CEC2020 and CEC2022 benchmarks. These analyses included mean and standard deviation of fitness, convergence curves, Wilcoxon tests, as well as population diversity assessments. The experimental results, which compare CKOA not only to the original KOA but also to eight other recent optimizers, show that the proposed algorithm performs better in terms of convergence speed and solution quality. In addition, CKOA has been successfully tested on three complex engineering problems, confirming its robustness and practical effectiveness. These results make CKOA a powerful optimisation tool in a variety of complex real-world contexts. After final acceptance, the source code will be uploaded to the Github account: nawal.elghouate@usmba.ac.ma.

Efficiency high-order discontinuous Galerkin method for low speeds flows with time-derivative preconditioning

A local preconditioning high-order discontinuous Galerkin (DG) method is introduced to enhance the efficiency and accuracy of the density-based approach for low-speed flows. The method is based on the time-derivative preconditioning technique typically employed in finite volume methods. Traditional approximation methods that only precondition low-order derivatives lead to incompatibility of the equations. To extend the preconditioning technique to the DG method, we employ an analytic preconditioning mass matrix derived from the DG formulation of the preconditioned Navier–Stokes equations. Modified numerical flux functions and a self-adaptive local velocity truncation parameter are employed for DG systems with preconditioning. We demonstrate the efficiency and accuracy of the preconditioned DG method using explicit and implicit schemes for steady flows and a dual-time scheme for unsteady flows. The method has been implemented and used to compute a variety of flow problems, including inviscid and laminar flows, utilizing various degrees of polynomial approximations. Computations with and without preconditioning are performed to analyze the influence of spatial discretization on the accuracy and convergence of the DG solutions at low Mach numbers. Numerical results underscore the enhanced accuracy and convergence properties of the proposed method for low-speed flows, indicating significant performance improvements over traditional methods.

Effects of new nonlinear curvature models and non-local elasticity on buckling investigation of FG tapered nano-beams locating on elastic foundations

New nonlinear curvature models are proposed to show their effects on the compressive critical buckling load on functionally graded (FG) tapered nano-beams locating on elastic foundations in the presence of non-local elasticity. Fourth and fifth orders finite difference methods (bvp4c and bvp5c) are applied on the governing differential equation which is a highly non-linear, due to curvature and has variable coefficients, due to properties. Dimensional analyses have been applied on the governing system together with the problem conditions to produce their dimensionless forms. For different considered parameters, the critical buckling load is computed using the eigenvalue theorem. The effects of nonlinear curvature, variable Young modulus, variable moment inertia and foundation parameters are studied and displayed in tables and figures. The stability and convergence of the present solution are tested using more than one strategy as: comparison with available results, comparison between bvp4c and bvp5c with different mesh intervals and tolerances of truncation errors. These comparisons show that, the new curvature model affects the critical buckling with different ratios depending on variability of properties and foundation parameters. The new results show that the nonlinear curvature model reduces the critical buckling load in comparison with the classical linear model, which, leads to decreasing in factor of safety in design of structures. The new results show that the design of structures will be influenced with the new reduction of the compressive critical buckling load, since it may affect the factor of safety in design of structures. The new results can lead, generally, to more reduction of critical buckling load for large or small input parameters and it will affect other critical values such that frequencies. The introduced Tables and Figures of the present problem with comparisons of previously exact, analytical and numerical works establish the higher accuracy and stability of current findings using bvp4c or bvp5c with preferable of the later. This study is related to some practical applications such as aeronautical and structural engineering.

From technology opportunities to solutions generation via patent analysis: Application of machine learning-based link prediction

Technology convergence represents a significant mode of technological innovation that is widely prevalent across various industries. This innovative approach integrates multiple technologies to develop new integrated solutions, thereby fostering a competitive advantage for enterprises. Anticipating future potential technology convergence is of paramount importance for businesses. However, previous research has predominantly relied on the topological information of convergence networks, overlooking the nodal attributes and inter-nodal relationships that have an impact on the emergence of technology convergence. To enhance existing studies, this paper employs three types of features: node attributes and inter-node relationships based on the drivers of technology convergence, along with link prediction similarity indices. Additionally, we utilize Graph Convolutional Neural Network (GCN) for node embedding to leverage node attributes. Machine learning models are utilized for link prediction based on these features to identify potential technology opportunities. To guide research and development (R&D) efforts, we recommend high-value patents for each node using entropy weighting across five metrics that objectively quantify patent value, and transform patent abstracts into vectors using Doc2Vec. Patents with high similarity in abstract text between nodes are utilized to extract technical solutions and fuse ideas for technology convergence. A case study is conducted within the autonomous driving industry, leveraging comprehensive information including node attributes, inter-node relationships, and topology-based similarities to identify technology opportunities and guide the generation of R&D ideas through the convergence of technical solutions.

Cattaneo-Christov fluxes for nonlinear mixed convective entropy generated Reiner-Rivlin material flow

Thermal and solutal transportation processes involve in tremendous useful applications related to heat exchanger, scientific, micro-electronic device technologies and industry sectors like nuclear reactors cooling, marine engineering, nanotechnology, thin-film technology, biomedical applications and thermal conduction in soft tissue. In view of such useful applications the Cattaneo-Christov analysis for nonlinear mixed convective Reiner-Rivlin material flow in presence of constant applied magnetic field is considered. Heat source, magnetohydrodynamics and radiation are incorporated in energy equation. Cattaneo-Christov flux is employed to discuss the thermal and mass transport characteristics. Entropy calculation in radiating flow is discussed. The proposed non-linear systems are transformed into dimensionless ordinary systems through appropriate transformation. Non-linear ordinary expressions are computed for convergent series solutions through Optimal homotopy analysis technique (OHAM). Graphical interpretations for quantities under interest against pertinent variables are explored. Large magnetic parameter has opposite impact for entropy and fluid flow. An increase of temperature through thermal relaxation time parameter is noticed. Velocity has same trend for both mixed convection and Reiner-Rivlin fluid variables. Reduction in concentration is seen for solutal relaxation time parameter. Temperature enhancement is witnessed in presence of heat generation. Entropy rate enhancement for radiation is observed. Decrease in concentration is noticed for reaction. Entropy rate increases against diffusion variable is enhanced.

Multi-energy synergistic planning of distributed energy supply system: Wind-solar-hydrogen coupling energy supply

This study is to improve the efficiency of energy utilization with the continuous growth of global energy demand and the increasingly severe environmental problem. A hydrogen-containing distributed energy supply system is proposed, which includes a hydrogen energy subsystem. The main driver of the system is renewable energy, while hydrogen energy is used as a carrier to assist the flow of energy. This study proposes an improved following electric load (FEL) strategy based on a comprehensive consideration of energy interactions within the system, and designs an optimal configuration model aimed at improving its economic, environmental and energy benefits. In addition, an improved rime algorithm (IRIME) is proposed, whose convergence performance and solution stability are greatly improved. A multi-objective optimization algorithm with better convergence performance, multi-objective improved rime algorithm (MO-IRIME), is proposed for solving the optimal configuration model to obtain the optimal capacity of the equipment inside hydrogen-containing distributed energy supply system. The total annual cost is reduced by 44.80%, the primary energy consumption by 68.47% and the pollutant gas emissions by 75.79% compared to the reference system. In addition, the way of interaction and coordination of multiple energy sources in the daily cycle is analyzed.

An improved whale algorithm for real-time boundary processing and its convergence analysis

Abstract To tackle the deficiencies in the whale algorithm, including its weak stability, sluggish convergence rate, and tendency to get trapped in local optima, an adapted whale optimization algorithm has been created, which introduces quasi-opposition learning and a real-time boundary processing mechanism. In the enhanced algorithm, a quasi-opposition learning mechanism is implemented to broaden the search scope and augment the variety of the population during the initial phases of iteration. This approach also helps to guarantee the convergence of the algorithm in the later phases of iteration. The unified boundary processing method of delay concentration is improved to solve the problem that a large amount of convergence damages diversity if a large number of transgressive individuals are unified and centralized. The algorithm flow is given and the asymptotic fitness and the asymptotic global optimal solution are proved. The time complexity is analyzed theoretically. Ultimately, six recognized algorithms were utilized to simulate the CEC 2017 function set across different scales. Findings indicate that the enhanced algorithm significantly boosts convergence rate, optimization precision, and solution robustness, demonstrating strong convergence characteristics.

Neural network algorithm with transfer learning and dropout for using a UAV to search the lost target in motion

When performing the rescue mission, how to find a lost target in motion as soon as possible is a very valuable and interesting research topic. To find the optimal flight path of the UAV used to search the lost target in motion under different prior knowledge and interference intensity, this paper proposes a novel optimization technique called neural network algorithm with transfer learning and dropout (TLDNNA), which is a variant of neural network algorithm (NNA) inspired by artificial neural networks. To improve the global search ability and convergence performance of NNA, the multiple transfer mechanism, the generalized mean transfer position, and the random dropout operator driven by the principles of transfer learning and dropout are introduced to TLDNNA. To investigate the performance of TLDNNA, TLDNNA and 12 powerful population-based optimization algorithms are applied to the problem of using a UAV to search the lost target in motion in the designed 9 scenarios. Experimental results show that TLDNNA is remarkably better than the other 12 algorithms in most of the scenarios in terms of solution quality, stability, and convergence performance. In addition, TLDNNA can improve the overall search performance by 18.14% and overall computational efficiency by 3.41 times compared to NNA. The source code of TLDNNA can be obtained by https://github.com/jsuzyy/TLDNNA.

Integrating machine learning with dynamic multi-objective optimization for real-time decision-making

Real-time decision-making in dynamic multi-objective optimization problems (DMOPs) is challenging due to constantly changing objectives and constraints. This paper integrates machine learning with Non-dominated Sorting Genetic Algorithm II (NSGA-II) to solve DMOPs and make real-time decisions. Learning-based methods have gained popularity for predicting solutions in new environments and capturing changing patterns in optimal solutions. However, existing approaches often struggle with training difficulty and reduced prediction accuracy due to irrelevant or redundant variables. Therefore, we introduce a new interdependent prediction (IDP) technique to identify correlations between variables and prediction targets and select significant variables for a predictive model. In this way, a better initial population is predicted. The IDP strategy is integrated within the dynamic NSGA-II, introducing a new algorithm called IDP-DNSGA-II. This integration facilitates rapid convergence, finding optimal or near-optimal solutions. The proposed method is evaluated against standard benchmarks, demonstrating superior performance in convergence speed and solution diversity with the changes in the problem environment. The IDP-DNSGA-II is validated through real-world optimization challenges in sustainable automobile production distribution in order-to-delivery systems to enhance environmental sustainability and operational efficiency. This study identifies the minimum frequency of change required in real-world problems to adequately track the optimal decision in real-time.

Novel hybrid arithmetic-equilibrium optimizer as robust tuner for power system stabilizer (PSS) parameters in interconnected electrical power system

Nowadays, the metaheuristic approaches become popular in the optimal tuning parameters of power system controllers, especially Power System Stabilizers (PSS). Besides the easiness, the stochastic behavior in metaheuristic approaches needs further investigation to increase the quality of the offered optimal parameters. Thus, this paper proposes A Novel Hybrid Arithmetic-Equilibrium Optimizer (HAEO) as a robust tuner for PSS in the interconnected electrical power system. The algorithms are hybridized in unique ways to utilize the features in both Arithmetic Optimizer (AO) and Equilibrium Optimizer (EO). Hybridization is performed to improve the exploration ability by providing the widest optimal parameters possible. In benchmarking by the Friedman Ranking Test, the proposed algorithm shows a significant improvement in consistency and accuracy over the basic AO and EO. A wider-distributed optimal solution and better convergence are also offered by the proposed algorithm. Besides that, the application in the optimal tuning shows that the proposed algorithm can provide the best PSS parameters in different objective functions with better average and standard deviation values. The HAEO-PSS gives a significant overshoot suppression of 35 % to 57 %, while the settling time of the system is also improved by 7 % to 13 %. Overall, the HAEO-PSS is proven to improve the stability response of the interconnected electrical power system both in local and interarea responses.