Numerous Problems Research Articles

A Study on Analysis of Numerical Methods

The present work reveals that numerical analysis is a branch of Mathematics that deals with devising efficient methods for obtaining numerical solutions to difficult Mathematical problems. We conducted this research paper by observing the different types of reviews, as well as conducting and evaluating literature review papers. Numerical analysis is mathematics which has developed efficient methods for obtaining numerical solutions to difficult mathematical problems. Most of the mathematical problems that arise in science and engineering are very difficult and sometimes impossible to solve properly. The method of numerical analysis is being used mainly in the fields of mathematics and computer science and is continuously creating and applying algorithms to solve numerical problems of mathematics

An efficient q-procedures to solve q-generalized quintic complex Ginzburg-Landau equations

Abstract In this work, q-Residual power series method (q-RPSM) is extended for the first time to solve q-generalized quintic complex Ginzburg-Landau (q-GCGL) equations. In 2019, O. A. Arqub [39], has investigated the solutions of time fractional Schrödinger equations without q-derivative by using RPSM. However, in this article, the RPSM procedure is extended to non-linear q-fractional partial differential
equations and thus some useful results are obtained under the definition of q-derivative, q-gamma func tion, q-Caputo derivative and q-factorial function. The solutions to few numerical examples under the q-Caputo operator are presented. In the current procedure, the targeted q-problems are first converted into system of two non-linear q-fractional partial differential equations and then solved by using q-RPSM. In particular, we considered two illustrative examples of fractional q-GCGL for analytical solutions using q-RPSM. The numerical simulations for the suggested numerical problems are done successfully. Various graphs such as Figures 1-12 are provided to compare the q-RPSM results with the exact solutions of the proposed problems. In this connection, Figures 1 and 4 show the 3D plot comparison between q-RPSM and exact solution of the 1st numerical example for œ and q=1. Figures 2 and 5 illustrate the q-graph
for various values of q and τ . In Figures 3 and 6, we discussed the fractional-solution graphs for different values of œ = 1, 0.7 and 0.5 at fix values of q=0.5, τ = 1 and 0.5. It is also investigated that higher terms series solution have the higher degree of accuracy. In the current q-RPSM simulations, the higher accuracy is achieved by taking sixth terms series solution. The fractional-order solutions are convergent toward integer-order solution as the fractional-order approaches to integer-order. The solutions curve at different q-values is also calculated and obtain the useful dynamics of the targeted problems. The q-analogy can be used effectively in the field of quantum physics and thus keep the higher norm for researchers working in the field.

Exploring the interplay between AI and human logic in mathematical problem-solving

This paper investigates the dynamic interplay between Artificial Intelligence (AI) and human logic in the domain of mathematical problem-solving. By critically examining a series of case studies, we compare the efficacy of AI-generated solutions, particularly those offered by ChatGPT, against traditional human problem-solving methods. The study employs various mathematical challenges, ranging from abstract logical puzzles to applied numerical problems, to evaluate AI's problem-solving approach and alignment with human cognitive processes. Our analysis highlights instances where AI's computational strategies complement or diverge from human reasoning, shedding light on AI's potential and limitations in deciphering mathematical problems. Furthermore, we explore the implications of integrating AI tools in educational contexts, specifically their role in enhancing students' mathematical problem-solving skills. The paper aims to contribute to the ongoing discourse on the optimal utilization of AI in education, proposing a balanced approach that leverages AI's computational power while fostering the depth and creativity of human logic. Through this comparative study, we advocate for a collaborative model where AI and human reasoning merge to enrich the educational landscape, particularly in the teaching and learning of mathematics.

Topology-informed Derivative-Free Metaheuristic Optimization Method

In this study, we propose a novel topology-informed search strategy for derivative-free metaheuristic optimization, enhancing both Simulated Annealing (SA) and Particle Swarm Optimization (PSO) through Topology Data Analysis (TDA). The proposed methods utilize persistence diagram and Gaussian kernel density estimation to guide search processes effectively. For SA, TDA evaluates neighboring points to generate persistence diagrams, guiding the search towards regions with significant topological features. This approach balances exploration and exploitation, reducing the risk of converging to suboptimal solutions. In PSO, dynamic adjustment of social and cognitive coefficients based on feature density in persistence diagrams ensures robust convergence. This strategy guides particles toward the best positions, promoting early exploration and later convergence. Lastly, nine continuous non-linear numerical problems and a cost optimization problem in pharmaceutical manufacturing illustrate the feasibility of the proposed methods. The study highlights the potential of TDA-informed metaheuristic strategies to improve derivative-free optimization methods, offering an advancement in navigating complex optimization landscapes.

Enabling FEM-based absolute permeability estimation in giga-voxel porous media with a single GPU

The characterization of porous media via digital testing usually relies on intensive numerical computations that can be parallelized in GPUs. For absolute permeability estimation, Stokes flow simulations are carried out at the micro-structure to recover velocity fields that are used in upscaling with Darcy’s law. Digital models of samples can be obtained via micro-computed tomography (μCT) scans. As μCT data is three-dimensional, meshes grow cubically with image dimensions, causing the numerical problem at hand to become compute- and memory-bound as either resolution improves or larger fields-of-view are considered. While the usual focus is on accelerating solvers, memory usage continues to be a significant limitation for analyses of representative volumes in relatively accessible hardware. In this work, we explore the possibility of implementing MINRES solvers in GPU that favor a reduction in memory allocation. These solvers are applied to matrix-free FEM-based permeability characterization of μCT images. Our goal is to enable the study of 10003 voxel images in single GPU machines. Implementations that only require five, three, or two n-sized vectors of variables are presented, with n being the number of unknowns. Further, we employ a mesh numbering strategy that enables node-by-node massively parallel operations within a non-monolithic voxel-based pore space without storing connectivity tables. The proposed solvers, available through the open-source chfem software, are verified against analytical models for simple three-dimensional micro-structures, then are validated against numerical Digital Petrophysics benchmarks. A consumer-grade graphics card with 12GB of RAM is employed for the characterization of images with up to roughly 540 million voxels in a matter of tens of minutes. Stokes flow FEM-based simulations in meshes with 449 million degrees-of-freedom (DOFs) are carried out in 9 to 15 min, allocating less than 10GB in global memory. Finally, simulations on three 10003 carbon fiber domains, amounting to more than 3.7 billion DOFs, were run on a high-end GPU with 80GB of RAM in under 2.5 h, achieving very close agreement with flow-tube permeability experiments.

Analysis of Junior High School Students' Misconceptions Based on Constructivism Theory in Solving Numeracy Literacy Problems

This study aims to use constructivist principles to find out how misunderstandings arise when solving arithmetic problems among students of grade 7 C of SMPN 2 Ciruas and to identify the factors responsible for them so that this problem can be presented to students and teachers as a reference for more effective learning. The subjects of this study were students of grade 7 C with high, medium and low comprehension skills. This study is qualitative and uses descriptive methods. It uses test instruments in the form of AKM sample questions and interview guides related to constructivist principles. The data analysis techniques used are data reduction, data display and inference. The results of this study explain the misunderstandings that students, especially those with low comprehension, experience when presented with numerical problems in a mathematical format. According to constructivist theory, factors that cause misunderstandings include: 1) lack of additional study time; 2) poor prior knowledge; 3) lack of understanding of the problem; 4) not taking an active role in the classroom; and 5) incorrect interpretation of the problem.

AMBWO: An Augmented Multi-Strategy Beluga Whale Optimization for Numerical Optimization Problems

Beluga whale optimization (BWO) is a swarm-based metaheuristic algorithm inspired by the group behavior of beluga whales. BWO suffers from drawbacks such as an insufficient exploration capability and the tendency to fall into local optima. To address these shortcomings, this paper proposes augmented multi-strategy beluga optimization (AMBWO). The adaptive population learning strategy is proposed to improve the global exploration capability of BWO. The introduction of the roulette equilibrium selection strategy allows BWO to have more reference points to choose among during the exploitation phase, which enhances the flexibility of the algorithm. In addition, the adaptive avoidance strategy improves the algorithm’s ability to escape from local optima and enriches the population quality. In order to validate the performance of the proposed AMBWO, extensive evaluation comparisons with other state-of-the-art improved algorithms were conducted on the CEC2017 and CEC2022 test sets. Statistical tests, convergence analysis, and stability analysis show that the AMBWO exhibits a superior overall performance. Finally, the applicability and superiority of the AMBWO was further verified by several engineering optimization problems.

Comments on “Intuitionistic fuzzy TOPSIS for non-linear multi-objective transportation and manufacturing problem”

Chhibber et al. (2022) proposed a method to solve a special type of non-linear multi-objective intuitionisticfuzzy transportation problems (NLMOIFTPs). Based on the results of a numerical problem, Chhibber et al. (2022) claimed that their proposed method is better than several existing methods. In this paper, it is pointed out thatChhibber et al. (2022)’s method isnot valid as Chhibber et al. (2022) have considered several mathematicallyincorrectresults to propose their method. Also, it is pointed out that in theactualcase, the numerical problem, considered by Chhibber et al. (2022), is a combination of three independent non-linear single-objective intuitionisticfuzzy transportation problems (IFTPs). However, Chhibber et al. (2022) have solved the numerical problem as NLMOIFTP which is mathematically incorrect.

Insight into thermal dynamic of stagnated flow of MHD Jeffery fluid when the joule heating, viscous dissipation and Soret effect are present: A multistep Milne's approach

The purpose of this work is to use the RK approach in conjunction with Milne's Predictor Corrector method to investigate the numerical solution of the magnetohydrodynamic Jeffery fluid passing over the upper horizontal parabolic surface. Milne's predictor-corrector methodology is highly important since it increases result accuracy when compared to either approach alone. The physical aspects are investigated by using the variable fluid properties, and consider the stagnation point and boundary layer flow of the MHD Jeffery fluid, which flows on the upper horizontal parabolic surface. The influence of Jeffery fluid flowing the paraboloid surface is ascertained by measuring the velocity slip and melting surface effects. We examined both the heat and mass transfer rates with the effects of viscous dissipation, Joule heating, heat source/sink, activation energy, and the Soret effect because these rates are important in chemical reactions, electronic devices, climate change, separation and distillation processes, and water and air pollution. The governing equations in the form of PDEs are obtained by implementing all of the assumptions on the fundamental conservation laws. These equations are then converted into ODEs by using the similarity variables. The "Milne's Predictor-Corrector Method" is a numerical strategy used to solve numerical problems and the results are analyzed both graphically and numerically manners with this numerical technique. Analysis is done on the graphical behavior of many parameters on the temperature, concentration, and velocity regions. The numerical findings of skin friction and Nusselt number are also presented here, and the outcomes are compared with the results of the BVP5c and Milne's (Predictor-Corrector) methods. The obtained results indicated that the motion (velocity) of Jeffery fluid is restricted by the enhancement in Deborah number, retardation to relaxation parameter, Hartmann number, and viscosity parameter, whereas augmentation happens in fluid velocity due to melting parameter and stretching ratio parameter. The temperature region can be increased by the source of the viscosity parameter, Hartmann number, heat source/sink parameters, and Eckert number. The decline in the concentration distribution is noted due to activation energy and the reaction rate parameter, while an upsurge in concentration occurs due to the Soret number.

Integrated optimization of ply number, layer thickness, and fiber angle for variable-stiffness composites using dynamic multi-fidelity surrogate model

To fully exploit the efficiency of variable-stiffness composite laminates with spatially varied fiber orientation angles, this paper aims at presenting a novel optimization framework for integrated design of ply number, layer thickness, and fiber angle. The optimization problem is innovatively formulated based on the definition of a ground laminate with redundant layers. The basic optimization idea is to seek both unnecessary and necessary layers in this ground laminate. For unnecessary layers, they can be removed and assigned with small-value ply thicknesses, while necessary layers are retained in the ground laminate and corresponding ply thicknesses and fiber angles are optimally determined using discrete and continuous variables, respectively. Since variable-stiffness composite laminates always require high-fidelity analysis models to accurately capture the spatial characteristics of varying fibers, this results in a time-consuming process. To alleviate this problem, a multi-fidelity surrogate model with an exponent-based comprehensive correction is originally proposed based on Gaussian process regression, generating an approximate problem to replace the original one. The genetic algorithm and sequential quadratic programming method are sequentially employed to solve this approximate problem with mixed design variables. The solution from this procedure is dynamically added to the sampling dataset to update the constructed surrogate model. Numerical benchmark problems and cases studies of a composite plate and a solar wing structure are addressed, demonstrating the efficacy of the newly proposed optimization strategy.

Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set

In this paper, a steepest descent method is developed for the robust counterpart of uncertain multiobjective optimization problems under finite uncertainty. The robust counterpart under consideration is the minimum of objective wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. To solve this robust counterpart with the help of the steepest descent method, a subproblem is constructed and solved to find a descent direction. An Armijo-type inexact line search technique is employed to find a suitable step length. With the help of the descent direction and step length, a sequence is generated. The convergence of the proposed method is justified under some common assumptions. Finally, the algorithm is verified and compared with one existing method with the help of some numerical problems.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

MHD Casson flow across a stretched surface in a porous material: a numerical study

In this study, we examine the nature of magnetohydrodynamic (MHD) Casson flow of fluid across a stretched surface in a porous material. It studies how the behaviour of Casson fluids is affected by a number of variables, including thermal radiation, chemical processes, Joule heating, and viscosity dissipation. The Keller box strategy, based on the finite difference method (FDM), is used to tackle the complex numerical problem. Graphical representations are used to show the effects of different system parts. Comprehensive tables displaying surface transfer of mass, heat, and drag rates are given for your convenience. The study focuses on how particle motion transforms kinetic energy into heat. Increased Brownian motion leads to a higher temperature profile and a reduced concentration profile. Thicker concentration profiles are created by increased Lewis number (Le\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Le$$\\end{document}) values and rates of chemical reactions, resulting in changes in mass transfer across fluids. This in-depth investigation focuses on the complicated interactions between various variables and how they influence the Casson fluid's behaviour in the system under study.

Generalized Laplace Transform with Adomian Decomposition Method for Solving Fractional Differential Equations Involving ψ-Caputo Derivative

In this study, we introduced the ψ-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, ψ, in the Caputo sense. To validate the effectiveness of this method, we applied the derived recurrent scheme of the ψ-Laplace Adomian decomposition on several test numerical problems, including a real-life scenario in pharmacokinetics that models the movement of drug concentration in human blood. The solutions obtained closely matched the known solutions for the test problems. Additionally, in the pharmacokinetics case, the results were consistent with the available physical data. Consequently, this method simplifies the verification of numerous related aspects and proves advantageous in solving various ψ-fractional differential equations.

Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization

This is Part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev families of node points, be it Lobatto, Radau, or Gauss, then the resulting collection of trajectory optimization methods satisfies the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an O(1) computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin’s- and covector-mapping principles, where the latter was developed in Part I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the totality of necessary conditions that result from an application of Pontryagin’s principle. Numerical problems are solved to illustrate all these ideas in addition to demonstrating near-exponential rates of convergence. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: 1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, 2) discontinuous and even Dirac delta covector trajectories can be well approximated, and 3) extremal solutions over dense grids can be computed in a stable and efficient manner.

Neural Network-Based Nonconservative Predefined-Time Backstepping Control for Uncertain Strict-Feedback Nonlinear Systems.

To handle the tracking problem of uncertain strict-feedback nonlinear systems with matched and mismatched composite disturbances, this article studies a predefined-time backstepping controller by resorting to a Lyapunov-based predefined-time dynamic paradigm, a regulation function, and neural networks (NNs). Moreover, an adding-absolute-value (ADV) technique is adopted in the design process to remove the control singularity. Theoretical analyses prove the boundedness of all closed-loop system signals and the predefined-time convergence of the tracking error into an arbitrarily small vicinity of the origin. The proposed controller exhibits four advantages: 1) the actual convergence time is precisely predefined by only one design parameter irrespective of the initial conditions, and the control energy is economized; 2) no unbounded terms are adopted for predefining the actual convergence time, thus avoiding numerical overflow problem under limited memory space and gaining strong noise-tolerant ability; 3) the peaking tracking error and control input magnitude can be effectively reduced by appropriately setting parameters of the regulation function; and 4) the controller is continuous and nonsingular everywhere. Finally, a practical example of a single-link manipulator is presented to validate the efficacy and superiority of our predefined-time controller.

Modeling Challenges and Limitation Principles of Reissner’s Mixed Approaches to Laminates

Displacement-based formulations for composite structures are directly linked to the functional called the Principle of Virtual Displacements and present the minimum number of independent modeled fields, which are the displacements. However, these models usually provide an inadequate approximation of the transverse stress fields and suffer from pathological numerical problems. These issues have been addressed with the introduction of mixed variational formulations where independent stress fields are included in the definition of the corresponding functionals. Particularly effective and widely used are Reissner’s mixed variational theorem and Hellinger–Reissner Principle. Both Reissner’s mixed variational theorem and Hellinger–Reissner principle allow the a priori enforcing of the interlaminar equilibrium because of the axiomatic modeling of the transverse stresses. However, these functionals are different and present their own challenges. Under certain circumstances, Reissner’s Mixed Variational Theorem was shown to produce thickness oscillations of the displacements and transverse stresses. It was not clear if Hellinger-Reissner Principle had similar properties. It is shown that both functionals may present numerical oscillations. Moreover, the reconstructed displacement and stress fields are actually identical between Reissner’s Mixed Variational Theorem and Hellinger-Reissner Principle if the orders of the in-plane stresses of the Hellinger–Reissner Prinicple case are sufficiently large. In other words, a Hellinger–Reissner Principle/Reissner’s Mixed Variational Theorem Limitation Principle is shown to exist in addition to the well-studied limitation principle between Hellinger–Reissner principle and principle of virtual displacements.

A New Integral Transform and Applications: HK-Transform

In this paper, we propose a new integral transform called the Transform and employ it to analytically solve the Volterra integral equations (VIE.) of the first kind. To achieve this First we derive the transform of basic algebraic and transcendental mathematical functions. The fundamental characteristics of the transform, which can be used to solve a variety of functional equations, including, we then discuss the fundamental properties, such as ordinary differential equations and integral equations, then we discover the exact solution for general first class V.I.E. To illustrate the applicability of the transform, numerical problems are examined and handled carefully step-by-step. The results show that the proposed Transform new integral transform produces accurate solutions for first-kind VIE. without requiring time-consuming computations.

Towards a Field-Based Bayesian Evidence Inference from Nested Sampling Data.

Nested sampling (NS) is a stochastic method for computing the log-evidence of a Bayesian problem. It relies on stochastic estimates of prior volumes enclosed by likelihood contours, which limits the accuracy of the log-evidence calculation. We propose to transform the prior volume estimation into a Bayesian inference problem, which allows us to incorporate a smoothness assumption for likelihood-prior-volume relations. As a result, we aim to increase the accuracy of the volume estimates and thus improve the overall log-evidence calculation using NS. The method presented works as a post-processing step for NS and provides posterior samples of the likelihood-prior-volume relation, from which the log-evidence can be calculated. We demonstrate an implementation of the algorithm and compare its results with plain NS on two synthetic datasets for which the underlying evidence is known. We find a significant improvement in accuracy for runs with less than one hundred active samples in NS but a proneness for numerical problems beyond this point.

Application of fractional differential transform method and Bell polynomial for solving system of fractional delay differential equations

In this article, a new numerical technique is presented to obtain numerical solution of a system of fractional delay differential equations (FDDE’s) involving proportional and time dependent delay terms. The fractional derivative is used in Caputo sense. The proposed technique is the combination of fractional differential transform and Bell polynomial. The existence and uniqueness results are discussed for FDDE’s. Three numerical problems are discussed to show reliability and efficiency of the method. Numerical results are compared with exact and Matlab DDENSD solution. The main advantage of the present method is handing effectively the nonlinear terms present in the FDDEs by using Bell polynomial. The present method can deal with both linear and nonlinear FDDEs. The convergence result is discussed, and error analysis is presented in detail.