19,800 results on '"Numerical Analysis"'
Search Results
2. Round-Off Error Suppression by Statistical Averaging
- Author
-
Andrej Liptaj
- Subjects
statistical averaging ,round-off errors ,numerical analysis ,accuracy ,Mathematics ,QA1-939 - Abstract
Regarding round-off errors as random is often a necessary simplification to describe their behavior. Assuming, in addition, the symmetry of their distributions, we show that one can, in unstable (ill-conditioned) computer calculations, suppress their effect by statistical averaging. For this, one slightly perturbs the argument of fx0 many times and averages the resulting function values. In this text, we forward arguments to support the assumed properties of round-off errors and critically evaluate the validity of the averaging approach in several numerical experiments.
- Published
- 2024
- Full Text
- View/download PDF
3. Enhancing Medical Image Quality Using Fractional Order Denoising Integrated with Transfer Learning
- Author
-
Abirami Annadurai, Vidhushavarshini Sureshkumar, Dhayanithi Jaganathan, and Seshathiri Dhanasekaran
- Subjects
transfer learning ,fractional order ,numerical analysis ,image denoising ,medical imaging ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
In medical imaging, noise can significantly obscure critical details, complicating diagnosis and treatment. Traditional denoising techniques often struggle to maintain a balance between noise reduction and detail preservation. To address this challenge, we propose an “Efficient Transfer-Learning-Based Fractional Order Image Denoising Approach in Medical Image Analysis (ETLFOD)” method. Our approach uniquely integrates transfer learning with fractional order techniques, leveraging pre-trained models such as DenseNet121 to adapt to the specific needs of medical image denoising. This method enhances denoising performance while preserving essential image details. The ETLFOD model has demonstrated superior performance compared to state-of-the-art (SOTA) techniques. For instance, our DenseNet121 model achieved an accuracy of 98.01%, precision of 98%, and recall of 98%, significantly outperforming traditional denoising methods. Specific results include a 95% accuracy, 98% precision, 99% recall, and 96% F1-score for MRI brain datasets, and an 88% accuracy, 91% precision, 95% recall, and 88% F1-score for COVID-19 lung data. X-ray pneumonia results in the lung CT dataset showed a 92% accuracy, 97% precision, 98% recall, and 93% F1-score. It is important to note that while we report performance metrics in this paper, the primary evaluation of our approach is based on the comparison of original noisy images with the denoised outputs, ensuring a focus on image quality enhancement rather than classification performance.
- Published
- 2024
- Full Text
- View/download PDF
4. Synchronization for discrete coupled fuzzy neural networks with uncertain information via observer-based impulsive control.
- Author
-
Zhou, Weisong, Wang, Kaihe, and Zhu, Wei
- Subjects
SYNCHRONIZATION ,INFORMATION sharing ,COMMUNICATION ,NUMERICAL analysis ,MATHEMATICS - Abstract
This paper discussed the synchronization of impulsive fuzzy neural networks (FNNs) with uncertainty of information exchange. Since the data of neural networks (NNs) cannot be completely measured in reality, we designed an observer-based impulsive controller on the basis of the partial measurement results and achieved the purpose of reducing the communication load and the controller load of FNNs. In terms of the Lyapunov stability theory, an impulsive augmented error system (IAES) was established and two sufficient criteria to guarantee the synchronization of our FNNs system were obtained. Finally, we demonstrated the validity of the results by a numerical example. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Finite-time lag synchronization for two-layer complex networks with impulsive effects.
- Author
-
Chu, Yao, Han, Xiuping, and Rakkiyappan, R.
- Subjects
SYNCHRONIZATION ,MATHEMATICS ,STATISTICS ,INITIAL value problems ,NUMERICAL analysis - Abstract
This paper mainly considered the finite-time lag synchronization for two-layer complex networks with impulsive effects. Different types of controllers were designed to achieve the lag synchronization of two-layer complex networks. Several sufficient conditions on lag synchronization in the sense of finite time were derived. The time for synchronization was also estimated. It is important to note that synchronization time was influenced by the initial value, as well as the impulses and impulse sequence. This implied that different impulse effects result in varying synchronization times. Additionally, desynchronizing impulses can extend the synchronization time, whereas synchronizing impulses have the opposite effect. Finally, a numerical example was presented to showcase the practicality and validity of the proposed theoretical criteria. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. Combining GPT and Colab as learning tools for students to explore the numerical solutions of difference equations.
- Author
-
Seebut, Supot, Wongsason, Patcharee, and Kim, Dojin
- Subjects
CLASSROOMS ,STUDENTS ,NUMERICAL analysis ,MATHEMATICS - Abstract
One of the most important things you can do to improve your mathematical application is to learn how to find numerical solutions. However, it was discovered that classrooms teaching methods that use numerical solutions are largely unable to provide students with the successful experience they should have in finding numerical solutions. Since conceptual and procedural knowledge, as well as the ability to perform computational mathematics, must be understood, simultaneously mastering all three can be difficult for most students. This study investigates combining GPT and Colab as learning tools for students to explore numerical solutions in the context of difference equations. The developed learning process works in tandem with the power of GPT and Colab to provide students with a successful experience in finding numerical solutions to difference equations. The survey results show that students have a high level of self-efficacy in finding numerical solutions to difference equations. This reflects today's power of innovation, which can be applied in classroom to improve student skills so that they can use the tools to solve problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
7. A unified approach to solving parabolic Volterra partial integro-differential equations for a broad category of kernels: Numerical analysis and computing
- Author
-
M. Fakharany, Mahmoud M. El-Borai, and M.A. Abu Ibrahim
- Subjects
Volterra PIDE ,Unit step function ,Implicit finite difference scheme ,Numerical integration of open type ,Numerical analysis ,Mathematics ,QA1-939 - Abstract
This work is concerned with solving parabolic Volterra partial integro-differential equations (PIDE) considering differentiable and singular kernels. The implicit finite difference scheme is implemented to approximate the differential operator, and the nonlocal term is discretized based on an open-type formula with two distinct time step sizes related to the nature of the time level to guarantee to avoid the singular terms at the endpoints and denominators. The properties of the plied scheme are investigated, more precisely, its stability and consistency. Four detailed examples are implemented to demonstrate the efficiency and reliability of the applied finite difference scheme.
- Published
- 2024
- Full Text
- View/download PDF
8. Neural networking study of worms in a wireless sensor model in the sense of fractal fractional
- Author
-
Aziz Khan, Thabet Abdeljawad, and Manar A. Alqudah
- Subjects
fractal-fractional operator ,neural networking ,mittag-leffler kernel ,ulam-hyers stability ,banach contraction ,numerical analysis ,Mathematics ,QA1-939 - Abstract
We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.
- Published
- 2023
- Full Text
- View/download PDF
9. The Estimation of Different Kinds of Integral Inequalities for a Generalized Class of Convex Mapping and a Harmonic Set via Fuzzy Inclusion Relations and Their Applications in Quadrature Theory
- Author
-
Ali Althobaiti, Saad Althobaiti, and Miguel Vivas Cortez
- Subjects
fuzzy up and down harmonically ℏ-convexity ,numerical analysis ,fuzzy Aumann integral inequalities ,Mathematics ,QA1-939 - Abstract
The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings (F-N-V-Ms), as fuzzy theory relies on the unit interval, which is crucial to resolving problems with interval analysis and fuzzy number theory. In this paper, a new harmonic convexities class of fuzzy numbers has been introduced via up and down relation. We show several Hermite–Hadamard (H⋅H) and Fejér-type inequalities by the implementation of fuzzy Aumann integrals using the newly defined class of convexities. Some nontrivial examples are also presented to validate the main outcomes.
- Published
- 2024
- Full Text
- View/download PDF
10. Analyzing a Dynamical System with Harmonic Mean Incidence Rate Using Volterra–Lyapunov Matrices and Fractal-Fractional Operators
- Author
-
Muhammad Riaz, Faez A. Alqarni, Khaled Aldwoah, Fathea M. Osman Birkea, and Manel Hleili
- Subjects
local and global stability ,Volterra–Lyapunov (V-L) matrices ,Lyapunov function ,fractional calculus ,Ulam–Hyers (UH) stability approach ,numerical analysis ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics.
- Published
- 2024
- Full Text
- View/download PDF
11. Analysis of Tensor Approximation Schemes for Continuous Functions
- Author
-
Griebel, Michael and Harbrecht, Helmut
- Subjects
Numerical analysis ,Mathematics - Abstract
In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights., Author(s): Michael Griebel [sup.1] [sup.2], Helmut Harbrecht [sup.3] Author Affiliations: (1) grid.10388.32, 0000 0001 2240 3300, Institut für Numerische Simulation, Universität Bonn, , Friedrich-Hirzebruch-Allee 7, 53115, Bonn, Germany (2) grid.418688.b, [...]
- Published
- 2023
- Full Text
- View/download PDF
12. Numerical Analysis of the Diversity of Keyboard Instrument Playing Fingerings
- Author
-
Wang Huaijin
- Subjects
numerical analysis ,a keyboard instrument ,playing fingering ,characteristics of diversity ,15a60 ,Mathematics ,QA1-939 - Abstract
Fingering is the foundation of keyboard instrument performance and an important part of keyboard music, but few people pay attention to its development. The evolution from the early variety of playing fingerings to the thumb-centered five-fingerings is even less mentioned. At present, steganographic analysis mainly focuses on the diversity and high dimensionality of features. Faced with the emerging new steganography, it is difficult for a single feature to cover and express the influence of steganography process on the multi-dimensional distribution of images. Therefore, it is necessary to combine various features through certain methods to analyze the changes of image properties before and after steganography embedding in a larger range and more types. The experimental results show that the experimental comparison diagram of Comb-RichModel based on diversity features and other steganographic analysis algorithms can be seen by replacing filtering and adding SPAM features. Compared with numerical analysis, the detection accuracy of the Comb-RichModel with diversified features has been improved to some extent, and its detection performance has improved stably from low embedding rate to high embedding rate. It is proved that the numerical analysis can effectively analyze the diversity characteristics of keyboard instrument playing fingerings.
- Published
- 2023
- Full Text
- View/download PDF
13. A mathematical model of a diphtheria outbreak in Rohingya settlement in Bangladesh
- Author
-
Asma Akter Akhi, Farah Tasnim, Saima Akter, and Md. Kamrujjaman
- Subjects
slir model ,diphtheria ,stability analysis ,model validation ,numerical analysis ,Mathematics ,QA1-939 - Abstract
In this paper, we study the dynamics of the diphtheria outbreak among the immunocompromised group of people, the Rohingya ethnic group. Approximately 800,000 Rohingya refugees are living in the Balukhali refugee camp in Cox’s Bazar. The camp is densely populated with the scarcity of proper food, healthcare, and sanitation. Subsequently, in November 2017 a diphtheria epidemic occurred in this camp. To keep up with the pace of the disease spread, medical demands, and disaster planning, we set out to predict diphtheria outbreaks among Bangladeshi Rohingya immigrants. We adopted a modified Susceptible-Latent-Infectious-Recovered (SLIR) transmission model to forecast the possible implications of the diphtheria outbreak in the Rohingya camps of Bangladesh. We discussed two distinct situations: the daily confirmed cases and cumulative data with unique consequences of diphtheria. Data for statistical and numerical simulations were obtained from \cite{Matsuyama}. We used the fourth-order Runge-Kutta method to obtain numerical simulations for varying parameters of the model which would demonstrate conclusive estimates. Daily and cumulative data predictions were explored for alternative values of the parameters i.e., disease transmission rate $(\beta)$ and recovery rate $(\gamma)$. Additionally, the average basic reproduction number for the parameters $\beta$ and $\gamma$ was calculated and displayed graphically. Our analysis demonstrated that the diphtheria outbreak would be under control if the maintenance could perform properly. The results of this research can be utilized by the Bangladeshi government and other humanitarian organizations to forecast disease outbreaks. Furthermore, it might help them to make detailed and practical planning to avoid the worst scenario.
- Published
- 2023
- Full Text
- View/download PDF
14. Numerical investigation of non-transient comparative heat transport mechanism in ternary nanofluid under various physical constraints
- Author
-
Adnan, Waseem Abbas, Sayed M. Eldin, and Mutasem Z. Bani-Fwaz
- Subjects
ternary nanofluid ,heat transfer ,magnetic field ,wedge ,numerical analysis ,Mathematics ,QA1-939 - Abstract
Significance: The study of non-transient heat transport mechanism in mono nano as well as ternary nanofluids attracts the researchers because of their promising heat transport characteristics. Applications of these fluids spread in industrial and various engineering disciplines more specifically in chemical and applied thermal engineering. Due of huge significance of nanofluids, the study is organized for latest class termed as ternary nanofluids along with induced magnetic field. Methodology: The model development done via similarity equations and the properties of ternary nanoparticles, resulting in a nonlinear mathematical model. To analyze the physical results with parametric values performed via RKF-45 scheme. Study findings: The physical results of the model reveal that the velocity $ F{'}\left(\eta \right) $ increased with increasing $ m = 0.1, 0.2, 0.3 $ and $ {\lambda }_{1} = 1.0, 1.2, 1.3 $. However, velocity decreased with increasing $ {\delta }_{1} $. Tangential velocity $ G{'}\left(\eta \right) $ reduces rapidly near the wedge surface and increased with increasing $ {M}_{1} = 1.0, 1.2, 1.3 $. Further, the heat transport in ternary nanofluid was greater than in the hybrid and mono nanofluids. Shear drag and the local thermal gradient increased with increasing $ {\lambda }_{1} $ and these quantities were greatest in the ternary nanofluid.
- Published
- 2023
- Full Text
- View/download PDF
15. NUMERICAL SOLUTION OF FRACTIONAL VOLTERRA INTEGRAL EQUATIONS BASED ON RATIONAL CHEBYSHEV APPROXIMATION.
- Author
-
DENIZ, S., ÖZGER, F., ÖZGER, Z. O., MOHIUDDINE, S. A., and ERSOY, M. T.
- Subjects
- *
NUMERICAL analysis , *MATHEMATICS , *FIXED point theory , *CHEBYSHEV polynomials , *ORTHOGONAL polynomials - Abstract
We aim to give the numerical method for solving the fractional Volterra integral equations of first and second kinds. We here use the techniques based upon rational Chebyshev functions and Riemann-Liouville fractional integrals. Some illustrative experiments with a view of estimating error and graphics are given in order to show the validity and applicability of the technique. Our experiments show that the new technique has high accuracy and is very efficient when compare to the other approaches existing in literature. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
16. Approximate message passing for compressed sensing magnetic resonance imaging
- Author
-
Millard, Charles, Tanner, Jared, Hess, Aaron, and Mailhe, Boris
- Subjects
Magnetic resonance imaging ,Signal processing ,Numerical analysis ,Image reconstruction ,Mathematics - Abstract
Magnetic Resonance Imaging (MRI) is a non-invasive, non-ionising imaging modality with unrivalled soft tissue contrast. A key consideration for MRI is data acquisition time, which is limited by inherent technological and physiological constraints. Compressed sensing is a relatively recent framework that can reduce the MRI acquisition time by undersampling randomly and exploiting presumed redundancies in the data. The Approximate Message Passing (AMP) algorithm is an iterative compressed sensing method that efficiently reconstructs signals that have been sampled with i.i.d. sub-Gaussian sensing matrices. However, when Fourier coefficients of a signal with non-uniform spectral density are sampled, such as in MRI, AMP performs poorly in practice. In response, this thesis proposes the Variable Density Approximate Message Passing (VDAMP) algorithm for undersampled MRI data. We present three versions of VDAMP: single-coil VDAMP, where receiver coil sensitivities are ignored, Parallel-VDAMP (P-VDAMP), which includes coil sensitivities, and Denoising-P-VDAMP (D-P-VDAMP), which incorporates the statistical modelling capabilities of neural networks. Central to VDAMP is a property that we term "coloured state evolution", where the difference between the intermediate image estimate at a given iteration and the ground truth is distributed according to a zero-mean Gaussian with known covariance. We demonstrate that coloured state evolution can be leveraged to yield an algorithm that converges rapidly, and to a competitive reconstruction quality, without the need to hand-tune model parameters.
- Published
- 2021
17. Computing multiple solutions of topology optimization problems
- Author
-
Papadopoulos, Ioannis, Farrell, Patrick, and Süli, Endre
- Subjects
Mathematics ,Finite element method ,Constrained optimization ,Differential equations, Partial ,Nonconvex programming ,Multigrid methods (Numerical analysis) ,Numerical analysis - Abstract
Topology optimization finds the optimal material distribution of a continuum in a domain, subject to PDE and volume constraints. Density-based models often result in a PDE, volume and inequality constrained, nonconvex, infinite-dimensional optimization problem. These problems can exhibit many local minima. In practice, heuristics are used to aid the search for better minima, but these can fail even in the simplest of cases. In this thesis we address two core issues related to the nonconvexity of topology optimization problems: the convergence of the discretization and the computation of the solutions. First, we consider the convergence of a finite element discretization of a fluid topology optimization problem. Results available in literature show that there exists a sequence of finite element solutions that weakly(-*) converges to a solution of the infinite-dimensional problem. We improve on these classical results. In particular, by fixing any isolated minimizer, we show that there exists a sequence of finite element solutions that \emph{strongly} converges to that minimizer. Moreover, these results hold for both traditional conforming finite element methods and more sophisticated divergence-free discontinuous Galerkin finite element methods. We then focus on developing a solver that can systematically compute multiple minimizers of a general density-based topology optimization problem. This leads to the successful computation of 42 distinct solutions of a two-dimensional fluid topology optimization problem. Finally, by developing preconditioners for the linear systems that arise during the optimization process, we are able to apply the solver to three-dimensional fluid topology optimization problems. This culminates in an example where we compute 11 distinct three-dimensional solutions.
- Published
- 2021
18. On low-rank plus sparse matrix sensing
- Author
-
Vary, Simon and Tanner, Jared
- Subjects
510 ,Statistics ,Mathematics ,Numerical analysis - Abstract
Expressing a matrix as the sum of a low-rank matrix plus a sparse matrix is a flexible model capturing global and local features in data, and is the foundation of robust principal component analysis (Candès et al., 2011). This thesis is concerned with low-rank plus sparse matrix sensing - the problem of recovering a matrix that is formed as the sum of a low-rank and a sparse matrix, and the two components, from a number of measurements far smaller than the dimensionality of the matrix. It is well-known, that inverse problems over low-rank matrices, such as robust principal component analysis and matrix completion, require a low coherence between the low-rank matrix and the canonical basis. However, in this thesis, we demonstrate that the well-posedness issue is even more fundamental; in some cases, both robust principal component analysis and matrix completion can fail to have any solutions due to the fact that the set of low-rank plus sparse matrices is not closed. As a consequence, the lower restricted isometry constants (RICs) cannot be upper bounded for some low-rank plus sparse matrices unless further restrictions are imposed on the constituents. We close the set of low-rank plus sparse matrices by posing an additional bound on the Frobenius norm of the low-rank component, and ensure the optimisation is well-posed and that the RICs can be bounded. We show that constraining the incoherence of the low-rank component also closes the set provided μ < √mn/ (r √s) and satisfies a certain additivity property necessary for the analysis of recovery algorithms. Compressed sensing, matrix completion, and their variants have established that data satisfying low complexity models can be efficiently measured and recovered from a number of measurements proportional to the model complexity rather than the ambient dimension (Foucart and Rauhut, 2013). This thesis develops similar guarantees showing that m × n matrices that can be expressed as the sum of a rank-r matrix and a s-sparse matrix can be recovered by computationally tractable methods.
- Published
- 2021
19. Nested multilevel Monte Carlo methods and a modified Euler-Maruyama scheme utilising approximate Gaussian random variables suitable for vectorised hardware and low-precisions
- Author
-
Sheridan-Methven, Oliver, Giles, Michael, and Goodyer, Christopher
- Subjects
518 ,Mathematics ,Numerical Analysis ,Computer Science - Abstract
We present a modified Euler-Maruyama scheme using approximate random variables, produced by the inverse transform method, using cheap approximations to the inverse Gaussian cumulative distribution function. We analyse the error for two approximations: a piecewise constant approximation on equally spaced intervals, and a piecewise linear approximation using geometric intervals dense at the singularities. High speed implementations faster than Intel's MKL are provided, suitable for modern vector hardware. The error between the approximations from the exact and modified Euler-Maruyama schemes is bounded by the error from the approximate random variables. We incorporate this scheme into a multilevel Monte Carlo framework producing a nested scheme, and show that the discretisation error couples to the random variables' approximation error. The result directly extends to Lipschitz and differentiable payoff functions. For Lipschitz and non-differentiable payoff functions simulated using a time step delta, there is a transition from a variance decay of O(delta) to order(delta
1/2 ) as the discretisation error becomes dominant. These variance bounds are demonstrated numerically for geometric Brownian motion and a variety of payoff functions of varying smoothness. For approximate random variables computed in low-precision, a model for the accumulated rounding error is developed and assessed. Half-precision is viable for a range of coarse path simulations, and can be extended further by incorporating a Kahan compensated summation. We empirically demonstrate these ideas are transferable to the Milstein scheme, and the more difficult Cox-Ingersoll-Ross process and its non-central chi-squared distribution. We estimate that under the Black-Scholes model, options can be priced using path simulations with approximate Gaussian random variables, obtaining a five-times or more speed improvement without losing accuracy.- Published
- 2021
20. Cubic B-Spline method for the solution of the quadratic Riccati differential equation
- Author
-
Osama Ala'yed, Belal Batiha, Diala Alghazo, and Firas Ghanim
- Subjects
cubic b-spline method ,nonlinear equations ,numerical analysis ,riccati differential equation ,first-order differential equations ,Mathematics ,QA1-939 - Abstract
The quadratic Riccati equations are first-order nonlinear differential equations with numerous applications in various applied science and engineering areas. Therefore, several numerical approaches have been derived to find their numerical solutions. This paper provided the approximate solution of the quadratic Riccati equation via the cubic b-spline method. The convergence analysis of the method is discussed. The efficiency and applicability of the proposed approach are verified through three numerical test problems. The obtained results are in good settlement with the exact solutions. Moreover, the numerical results indicate that the proposed cubic b-spline method attains a superior performance compared with some existing methods.
- Published
- 2023
- Full Text
- View/download PDF
21. Preconditioning for thermal reservoir simulation
- Author
-
Roy, Thomas, Jönsthövel, Tom, Wathen, Andy, and Lemon, Christopher
- Subjects
620.1 ,Mathematics ,Fluid mechanics ,Applied mathematics ,Oil reservoir engineering--Simulation methods ,Numerical analysis - Abstract
Multiphase flow through porous media can be modelled as a complex system of partial differential equations. Such models can be used to optimize the recovery of oil and gas from subsurface reservoirs. In the case of highly viscous oils, thermal recovery techniques are typically used to enhance their extraction. To simulate this, models describing the flow of fluids (typically oil, water, and gas) are coupled with a model for heat flow. Thermal reservoir simulation entails solving these highly coupled systems. Their complexity and the computational effort needed to solve them motivate the need for highly efficient solvers. In reservoir simulation, most of the computational time is spent on solving linearized systems with a preconditioned Krylov subspace iterative method. Industry-standard preconditioning techniques are based on the approach introduced by Wallis in 1983, the Constrained Pressure Residual method (CPR). This preconditioner is a two-stage process involving the solution of a restricted pressure system. While initially designed for isothermal reservoir simulation, CPR is also the standard for thermal cases. However, its treatment of the conservation of energy equation does not incorporate heat diffusion, which is often dominant in thermal cases. We are interested in preconditioners specifically designed for thermal reservoir simulation. In this thesis, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted pressure-temperature system is solved in the first stage. To study the effects of both pressure and temperature on fluid and heat flow, we first consider a model of non-isothermal single-phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Then, we extend this method for multiphase flow as a solver for the first stage of CPTR. We present a comparison of the algorithmic performance of the different preconditioning approaches under mesh refinement and parallelization.
- Published
- 2020
22. Simpson’s Variational Integrator for Systems with Quadratic Lagrangians
- Author
-
Juan Antonio Rojas-Quintero, François Dubois, and José Guadalupe Cabrera-Díaz
- Subjects
ordinary differential equations ,oscillator ,numerical analysis ,symplectic scheme ,Mathematics ,QA1-939 - Abstract
This contribution proposes a variational symplectic integrator aimed at linear systems issued from the least action principle. An internal quadratic finite-element interpolation of the state is performed at each time step. Then, the action is approximated by Simpson’s quadrature formula. The implemented scheme is implicit, symplectic, and conditionally stable. It is applied to the time integration of systems with quadratic Lagrangians. The example of the linearized double pendulum is treated. Our method is compared with Newmark’s variational integrator. The exact solution of the linearized double pendulum example is used for benchmarking. Simulation results illustrate the precision and convergence of the proposed integrator.
- Published
- 2024
- Full Text
- View/download PDF
23. The interpolated variational iteration method for solving a class of nonlinear optimal control problems.
- Author
-
Shirazian, Mohammad
- Subjects
NUMERICAL analysis ,BOUNDARY value problems ,PONTRYAGIN spaces ,MATHEMATICS ,POLYNOMIALS - Abstract
Despite the variety of methods available to solve nonlinear optimal control problems, numerical methods are still evolving to solve these problems. This paper deals with the numerical solution of nonlinear optimal control affine problems by the interpolated variational iteration method, which was introduced in 2016 to improve the variational iteration method. For this purpose, the optimality conditions are first derived as a two-point boundary value problem and then converted to an initial value problem with the unknown initial values for costates. The speed and convergence of the method are compared with the existing methods in the form of three examples, and the initial values of the costates are obtained by an efficient technique in each iteration. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
24. Number Line Estimation Patterns and Their Relationship With Mathematical Performance.
- Author
-
Ruiz, Carola, Kohnen, Saskia, and Bull, Rebecca
- Subjects
MATHEMATICS ,MATHEMATICS education ,LOGARITHMS ,NUMERICAL analysis ,COGNITIVE ability - Abstract
There is ongoing debate regarding what performance on the number line estimation task represents and its role in mathematics learning. The patterns followed by children’s estimates on the number line task could provide insight into this. This study investigates children’s estimation patterns on the number line task and assesses whether mathematics achievement is associated with these estimation patterns. Singaporean children (n = 324, Age M = 6.2 years, Age SD = 0.3 years) in their second year of kindergarten were assessed on the number line task (0-100) and their mathematical performance (Numerical Operations and Mathematical Reasoning subtests from WIAT II). The results show that most children’s number line estimation patterns can be explained by at least one mathematical model (i.e., linear, logarithmic, unbounded power model, one-cycle power model, two-cycle power model). But the findings also highlight the high percentage of participants for which more than one model shows similar support. Children’s mathematical achievement differed based on the models that best explained children’s estimation patterns. Children whose estimation patterns corresponded to a more advanced model tended to show higher mathematical achievement. Limitations of drawing conclusions regarding what performance on the number line task represents based on models that best explain the estimation patterns are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
25. Does Spontaneous Attention to Relations Predict Conceptual Knowledge of Negative Numbers?
- Author
-
Prather, Richard
- Subjects
MATHEMATICS ,NEGATIVE numbers ,ARITHMETIC ,NUMERICAL analysis ,CALCULATORS - Abstract
Mastery of mathematics depends on the people’s ability to manipulate and abstract values such as negative numbers. Knowledge of arithmetic principles does not necessarily generalize from positive number arithmetic to arithmetic involving negative numbers (Prather & Alibali, 2008, https://doi.org/10.1080/03640210701864147). In this study, we evaluate the relationship between participant’s knowledge of the Relation to Operands arithmetic principle in both positive and negative numbers and their spontaneous on numerical relations. Additionally, we tested if the feedback that directs attention to relations affects participants’ attention to relation and their arithmetic principle knowledge. This study contributes to our understanding of the specific skills and cognitive processes that are associated with understanding high-level mathematics. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
26. Interplay of harvesting and the growth rate for spatially diversified populations and the testing of a decoupled scheme
- Author
-
Md. Mashih Ibn Yasin Adan, Md. Kamrujjaman, Md. Mamun Molla, Muhammad Mohebujjaman, and Clarisa Buenrostro
- Subjects
harvesting ,diffusion ,global analysis ,competition ,numerical analysis ,Biotechnology ,TP248.13-248.65 ,Mathematics ,QA1-939 - Abstract
The loss and degradation of habitat, Allee effects, climate change, deforestation, hunting-overfishing and human disturbances are alarming and significant threats to the extinction of many species in ecology. When populations compete for natural resources, food supply and habitat, survival to extinction and various other issues are visible. This paper investigates the competition of two species in a heterogeneous environment that are subject to the effect of harvesting. The most realistic harvesting case is connected with the intrinsic growth rate, and the harvesting functions are developed based on this clause instead of random choice. We prove the existence and uniqueness of the solution to the model. Theoretically, we state that, when species coexist, one may drive the other to die out, so both species become extinct, considering all possible rational values of parameters. These results highlight a worthy-of attention study between two populations based on harvesting coefficients. Finally, we solve the model for two spatial dimensions by using a backward Euler, decoupled and linearized time-stepping fully discrete algorithm in a series of examples and observe a match between the theoretical and numerical findings.
- Published
- 2023
- Full Text
- View/download PDF
27. An iterative technique for solving path planning in identified environments by using a skewed block accelerated algorithm
- Author
-
A'qilah Ahmad Dahalan and Azali Saudi
- Subjects
rotated iterative scheme ,numerical analysis ,laplace's equation ,accelerated method ,path finding ,obstacle avoidance ,Mathematics ,QA1-939 - Abstract
Currently, designing path-planning concepts for autonomous robot systems remains a topic of high interest. This work applies computational analysis through a numerical approach to deal with the path-planning problem with obstacle avoidance over a robot simulation. Based on the potential field produced by Laplace's equation, the formation of a potential function throughout the simulation configuration regions is obtained. This potential field is typically employed as a guide in the global approach of robot path-planning. An extended variant of the over-relaxation technique, namely the skewed block two-parameter over relaxation (SBTOR), otherwise known as the explicit decoupled group two-parameter over relaxation method, is presented to obtain the potential field that will be used for solving the path-planning problem. Experimental results with a robot simulator are presented to demonstrate the performance of the proposed approach on computing the harmonic potential for solving the path-planning problem. In addition to successfully validating pathways generated from various locations, it is also demonstrated that SBTOR outperforms existing over-relaxation algorithms in terms of the number of iterations, as well as the execution time.
- Published
- 2023
- Full Text
- View/download PDF
28. Analytical and numerical investigation of the Hindmarsh-Rose model neuronal activity
- Author
-
Abdon Atangana and Ilknur Koca
- Subjects
hindmarsh model ,chaotic number ,nonlocality ,numerical analysis ,Biotechnology ,TP248.13-248.65 ,Mathematics ,QA1-939 - Abstract
In this work, a set of nonlinear equations capable of describing the transit of the membrane potential's spiking-bursting process which is shown in experiments with a single neuron was taken into consideration. It is well known that this system, which is built on dynamical dimensionless variables, can reproduce chaos. We arrived at the chaotic number after first deriving the equilibrium point. We added different nonlocal operators to the classical model's foundation. We gave some helpful existence and uniqueness requirements for each scenario using well-known theorems like Lipchitz and linear growth. Before using the numerical solution on the model, we analyzed a general Cauchy issue for several situations, solved it numerically and then demonstrated the numerical solution's convergence. The results of numerical simulations are given.
- Published
- 2023
- Full Text
- View/download PDF
29. Shape optimisation and robust solvers for incompressible flow
- Author
-
Wechsung, Florian and Farrell, Patrick
- Subjects
Mathematics ,Numerical Analysis - Abstract
When designing a new car or a plane, engineers need to solve the Navier-Stokes equations to understand how air flows around the object. Based on experience and intuition, they modify the design slightly, then solve the equations again, and inspect the changes. This process is iterated many times until a final design that minimises or maximises some quantity of interest, such as drag or lift, is found. The goal of shape optimisation is to automate this type of process. In this thesis we address several issues related to shape optimisation. Focussing on the case when the shape is discretised using a mesh and when PDE constraints are solved using the finite element method, we describe a reformulation of the shape derivative as the derivative of the pushforward from the reference element. This viewpoint allows for automated calculation of shape derivatives in finite element software. When shape optimisation is performed by deforming an initial mesh, the choice of deformation is important. We propose a new Hilbert space structure on the space of deformations that results in high mesh quality of the deformed domains. We then focus on the solution of a particular PDE constraint given by the steady, incompressible Navier-Stokes equations that govern laminar flow. The solution of these equations becomes challenging for large Reynolds number. We develop augmented Lagrangian based preconditioners that exhibit robust performance as the Reynolds number is increased. The effectiveness and scalability of the developed solvers is demonstrated for a range of test problems.
- Published
- 2019
30. Numerical analysis of fractional-order nonlinear Gardner and Cahn-Hilliard equations.
- Author
-
Iqbal, Naveed, Alshammari, Mohammad, and Wajaree Weera
- Subjects
NUMERICAL analysis ,MATHEMATICS ,TIME series analysis ,LAPLACE distribution ,POWER series - Abstract
In this study, the suggested residual power series transform method is used to compute the numerical solution of the fractional-order nonlinear Gardner and Cahn-Hilliard equations and the result is discovered in a fast convergent series. The leverage and effcacy of the suggested technique are demonstrated by the test examples provided. The achieved results are proved graphically. The current method handles the series solution in a sizable admissible domain in a powerful way. It provides a simple means of modifying the solution's convergence zone. Results with graphs expressly demonstrate the effectiveness and abilities of the suggested method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
31. An iterative technique for solving path planning in identified environments by using a skewed block accelerated algorithm.
- Author
-
Dahalan, A'qilah Ahmad and Saudi, Azali
- Subjects
AUTONOMOUS robots ,LAPLACE distribution ,MATHEMATICS ,NUMERICAL analysis ,SKEWNESS (Probability theory) - Abstract
Currently, designing path-planning concepts for autonomous robot systems remains a topic of high interest. This work applies computational analysis through a numerical approach to deal with the path-planning problem with obstacle avoidance over a robot simulation. Based on the potential field produced by Laplace's equation, the formation of a potential function throughout the simulation configuration regions is obtained. This potential field is typically employed as a guide in the global approach of robot path-planning. An extended variant of the over-relaxation technique, namely the skewed block two-parameter over relaxation (SBTOR), otherwise known as the explicit decoupled group two-parameter over relaxation method, is presented to obtain the potential field that will be used for solving the path-planning problem. Experimental results with a robot simulator are presented to demonstrate the performance of the proposed approach on computing the harmonic potential for solving the path-planning problem. In addition to successfully validating pathways generated from various locations, it is also demonstrated that SBTOR outperforms existing over-relaxation algorithms in terms of the number of iterations, as well as the execution time. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. Convergence and error estimates for pseudo-polyharmonic div-curl and elastic interpolation on a bounded domain
- Author
-
Mohammed-Najib Benbourhim, Abderrahman Bouhamidi, and Pedro Gonzalez-Casanova
- Subjects
Approximation theory ,interpolation and approximation ,convergence and error estimates ,numerical analysis ,functional analysis ,Mathematics ,QA1-939 - Abstract
This paper establishes convergence rates and error estimates for the pseudo-polyharmonic div-curl and elastic interpolation. This type of interpolation is based on a combination of the divergence and the curl of a multivariate vector field and minimizing an appropriate functional energy related to the divergence and curl. Convergence rates and error estimates are established when the interpolated vector field is assumed to be in the classical fractional vectorial Sobolev space on an open bounded set with a Lipschitz-continuous boundary. The error estimates introduced in this work are sharp and the rate of convergence depends algebraically on the fill distance of the scattered data nodes. More precisely, the order of convergence depends, essentially, on the smoothness of the target vector field, on the dimension of the Euclidean space and on the null space of corresponding Sobolev semi-norm.
- Published
- 2023
- Full Text
- View/download PDF
33. Theoretical and numerical analysis of a chaotic model with nonlocal and stochastic differential operators
- Author
-
İlknur Koca and Abdon Atangana
- Subjects
Nonlinear model ,Chaotic number ,Stochastic effect ,Numerical analysis ,Applied mathematics. Quantitative methods ,T57-57.97 ,Mathematics ,QA1-939 - Abstract
A set of nonlinear ordinary differential equations has been considered in this paper. The work tries to establish some theoretical and analytical insights when the usual time-deferential operator is replaced with the Caputo fractional derivative. Using the Caratheodory principle and other additional conditions, we established that the system has a unique system of solutions. A variety of well-known approaches were used to investigate the system. The stochastic version of this system was solved using a numerical approach based on Lagrange interpolation, and numerical simulation results were produced.
- Published
- 2023
- Full Text
- View/download PDF
34. Mathematics and Modeling in Finance
- Subjects
finance ,mathematics ,modeling ,numerical analysis ,stochastic ,machine learning ,Finance ,HG1-9999 ,Mathematics ,QA1-939 - Published
- 2023
35. Chaotic Steady States of the Reinartz Oscillator: Mathematical Evidence and Experimental Confirmation
- Author
-
Jiri Petrzela
- Subjects
Reinartz oscillator ,generalized transistor ,two-port admittance parameters ,numerical analysis ,hyperchaos ,chaos ,Mathematics ,QA1-939 - Abstract
This paper contributes to the problem of chaos and hyperchaos localization in the fundamental structure of analog building blocks dedicated to single-tone harmonic signal generation. This time, the known Reinartz sinusoidal oscillator is addressed, considering its conventional topology, both via numerical analysis and experiments using a flow-equivalent lumped electronic circuit. It is shown that physically reasonable values of circuit parameters can result in robust dynamical behavior characterized by a pair of positive Lyapunov exponents. Mandatory numerical results prove that discovered strange attractors exhibit all necessary fingerprints of structurally stable chaos. The new “chaotic” parameters are closely related to the standard operation of the investigated analog functional block. A few interestingly shaped, strange attractors have been captured as oscilloscope screenshots.
- Published
- 2023
- Full Text
- View/download PDF
36. Analysis and Optimal Control Measures of a Typhoid Fever Mathematical Model for Two Socio-Economic Populations
- Author
-
Stephen Ekwueme Aniaku, Obiora Cornelius Collins, and Ifeanyi Sunday Onah
- Subjects
typhoid fever ,reproduction number ,stability analysis ,optimal control ,numerical analysis ,Mathematics ,QA1-939 - Abstract
Typhoid fever is an infectious disease that affects humanity worldwide; it is particularly dangerous in areas with communities of a lower socio-economic status, where many individuals are exposed to a dirty environment and unclean food. A mathematical model is formulated to analyze the impact of control measures such as vaccination of susceptible humans, treatment of infected humans and sanitation in different socio-economic communities. The model assumed that the population comprises of two socio-economic classes. The essential dynamical system analysis of our model was appropriately carried out. The impact of the control measures was analyzed, and the optimal control theory was applied on the control model to explore the impact of the different control measures. Numerical simulation of the models and the optimal controls were carried out and the obtained results indicate that the overall combination of the control measures eradicates typhoid fever in the population, but the controls are more optimal in higher socio-economic status communities.
- Published
- 2023
- Full Text
- View/download PDF
37. Does freelancing have a future? Mathematical analysis and modeling
- Author
-
Fareeha Sami Khan, M. Khalid, Ali Hasan Ali, Omar Bazighifan, Taher A. Nofal, and Kamsing Nonlaopon
- Subjects
freelancing ,social science ,mathematical modeling ,differential equation ,numerical analysis ,simulation ,Biotechnology ,TP248.13-248.65 ,Mathematics ,QA1-939 - Abstract
During the past few years, freelancing has grown exponentially due to the pandemic and subsequent economical changes in the world. In fact, in the last ten years, a drastic increase in freelancing has been observed; people quit their jobs to be their own boss. There are various reasons for this: downsizing of employees, not having fun in their jobs, unemployment, part time work to earn more, etc. Observing this vast change, many individuals on Facebook/YouTube, NGOs, and government departments started teaching freelancing as a course; to date, thousands of youngsters have been trained to start their careers as freelancers. It has been observed that the ratio of informed freelancers is more successful than those who start their careers independently. We construct a compartmental model to explore the influence of information on the expansion of freelancing in this article, which was motivated by this surge in freelancing. Following that, the model is subjected to dynamical analysis utilizing dynamical systems and differential equation theory. To validate our analytical conclusions, we used numerical simulation.
- Published
- 2022
- Full Text
- View/download PDF
38. Analytical results for positivity of discrete fractional operators with approximation of the domain of solutions
- Author
-
Pshtiwan Othman Mohammed, Donal O'Regan, Dumitru Baleanu, Y. S. Hamed, and Ehab E. Elattar
- Subjects
discrete fractional calculus ,caputo-fabrizio fractional difference ,nabla positivity ,numerical analysis ,Biotechnology ,TP248.13-248.65 ,Mathematics ,QA1-939 - Abstract
We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where $ \left({}_{{c_0}}^{C{F_R}}\nabla^{\theta} \mathtt{F}\right)(t) > -\epsilon\, \Lambda(\theta-1)\, \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1) $ such that $ \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1)\geq 0 $ and $ \epsilon > 0 $. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of $ \epsilon $ and $ \theta $.
- Published
- 2022
- Full Text
- View/download PDF
39. How Memory Counts in Mathematical Development.
- Author
-
Coolen, Ilse E. J. I. and Castronovo, Julie
- Subjects
- *
MEMORY , *MATHEMATICS , *CHILD development , *SHORT-term memory , *NUMERICAL analysis - Abstract
Memory has been well-established as a predictor of mathematics achievement in child development. Nevertheless, empirical evidence remains elusive on the unique role of the different forms of memory and their specific mechanisms as predictors of mathematics development. Therefore, in this study, the role of visuospatial short-term memory, visuospatial working memory, verbal short-term memory, and verbal longterm memory was investigated at three key stages of the development of mathematics (5-6 years, 6-7 years, 7-8 years), as well as their interactions across development. The relation between the different memory types and informal and formal mathematics was also studied. The findings of this study provide empirical support for a shift in the relation between different memory types and mathematics achievement over development with: 1) visuospatial short-term memory predicting informal mathematics achievement at the age of 5-6 years; 2) visuospatial working memory predicting informal and formal mathematics achievement at the age of 6-7 years; and 3) verbal short-term memory predicting formal mathematics achievement at the age of 7-8 years. These shifts clearly appear consistent with children's mathematics curriculum content over time and the requirements of mathematics acquisition at specific stages in development. With these findings, the unique role of various forms of memory in the development of mathematics and the timeframe in which they play a crucial part is highlighted, which should be taken into consideration for future research and possible intervention studies in children's mathematics achievement. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
40. Refined Hermite–Hadamard Inequalities and Some Norm Inequalities.
- Author
-
Yanagi, Kenjiro
- Subjects
- *
NUMERICAL analysis , *MATHEMATICS , *INTEGRALS , *GENERALIZATION , *FUNCTIONAL analysis - Abstract
It is well known that the Hermite–Hadamard inequality (called the HH inequality) refines the definition of convexity of function f (x) defined on [ a , b ] by using the integral of f (x) from a to b. There are many generalizations or refinements of HH inequality. Furthermore HH inequality has many applications to several fields of mathematics, including numerical analysis, functional analysis, and operator inequality. Recently, we gave several types of refined HH inequalities and obtained inequalities which were satisfied by weighted logarithmic means. In this article, we give an N-variable Hermite–Hadamard inequality and apply to some norm inequalities under certain conditions. As applications, we obtain several inequalities which are satisfied by means defined by symmetry. Finally, we obtain detailed integral values. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
41. Pathfinding algorithm based on rotated block AOR technique in structured environment
- Author
-
A'qilah Ahmad Dahalan and Azali Saudi
- Subjects
algorithms ,laplace's equation ,half-sweep iterative method ,numerical analysis ,collision free ,optimal path ,Mathematics ,QA1-939 - Abstract
Harmonic potential fields are commonly used as guidance in a global approach for self-directed robot pathfinding. These harmonic potentials are generated using Laplace's equation solutions. The computation of these harmonic potentials often requires the use of immense amounts of computing resources. This study introduces a numerical technique called Rotated Block Accelerated Over-Relaxation (AOR), also known as Explicit Decoupled Group AOR (EDGAOR), to deal with pathfinding problem. Several robot navigation simulations were performed in a static, structured, known indoor environment to validate the efficiency of the suggested approach. The paths generated by the simulations are shown using several different starting and target positions. The performance of the proposed approach in computing harmonic potentials for solving pathfinding problems is also discussed.
- Published
- 2022
- Full Text
- View/download PDF
42. Scalable two-phase flow solvers
- Author
-
Bootland, Niall, Wathen, Andrew, and Kees, Christopher
- Subjects
515 ,Mathematics ,Numerical analysis ,Navier-Stokes equations--Numerical solutions ,Applied mathematics ,Fluid mechanics - Abstract
Two-phase flows arise in many areas of application such as in the study of coastal and hydraulic processes. Often the fluids involved can be modelled by incompressible phases which have disparate physical properties such as density and viscosity. We utilise a two-phase flow model of immiscible Newtonian fluids. A key part of this model is a set of variable coefficient Navier-Stokes equations. This thesis focuses on the numerical solution of the linear systems which arise after linearisation of these equations. Solving these systems often dominates the computation time when running simulations. One of the challenges in solving the model is that the density and viscosity coefficients are discontinuous and can have large jumps between the two phases. In this work we consider preconditioned iterative Krylov methods to solve the large and sparse linear systems and pay particular attention to incorporating the highly varying coefficients into the block preconditioners that we propose. We will see that such considerations can be essential in order to obtain good performance. An important issue is the scalability of the solution methodology. Here, we will study how the convergence of the iterative solver depends on a grid parameter which controls the refinement of the computational mesh. We will see that the novel preconditioners we propose can lead to convergence which is effectively independent of the grid parameter. We also investigate dependence on other model parameters such as the Reynolds number as well as the density and viscosity ratios between the two fluids. Another topic we examine is the use of a multipreconditioned iterative method allowing more than one preconditioner to be used simultaneously. Our results using this approach show some promising features. Finally, we consider an implementation within a more realistic model used in practice for simulating complex air-water flows. In particular, we will provide results for a problem modelling the breaking of a dam.
- Published
- 2018
43. Predictive Modeling and Control Strategies for the Transmission of Middle East Respiratory Syndrome Coronavirus
- Author
-
Bibi Fatima, Mehmet Yavuz, Mati ur Rahman, Ali Althobaiti, and Saad Althobaiti
- Subjects
MERS-CoV model ,basic reproductive number ,analysis of stability ,equilibria points ,optimality control ,numerical analysis ,Applied mathematics. Quantitative methods ,T57-57.97 ,Mathematics ,QA1-939 ,Electronic computers. Computer science ,QA75.5-76.95 - Abstract
The Middle East respiratory syndrome coronavirus (MERS-CoV) is a highly infectious respiratory illness that poses a significant threat to public health. Understanding the transmission dynamics of MERS-CoV is crucial for effective control and prevention strategies. In this study, we develop a precise mathematical model to capture the transmission dynamics of MERS-CoV. We incorporate some novel parameters related to birth and mortality rates, which are essential factors influencing the spread of the virus. We obtain epidemiological data from reliable sources to estimate the model parameters. We compute its basic reproduction number (R0). Stability theory is employed to analyze the local and global properties of the model, providing insights into the system’s equilibrium states and their stability. Sensitivity analysis is conducted to identify the most critical parameter affecting the transmission dynamics. Our findings revealed important insights into the transmission dynamics of MERS-CoV. The stability analysis demonstrated the existence of stable equilibrium points, indicating the long-term behavior of the epidemic. Through the evaluation of optimal control strategies, we identify effective intervention measures to mitigate the spread of MERS-CoV. Our simulations demonstrate the impact of time-dependent control variables, such as supportive care and treatment, in reducing the number of infected individuals and controlling the epidemic. The model can serve as a valuable tool for public health authorities in designing effective control and prevention strategies, ultimately reducing the burden of MERS-CoV on global health.
- Published
- 2023
- Full Text
- View/download PDF
44. Construction a distributed order smoking model and its nonstandard finite difference discretization
- Author
-
Mehmet Kocabiyik and Mevlüde Yakit Ongun
- Subjects
distributed order fractional differential equation ,nonstandard finite difference method ,smoking model ,numerical analysis ,discretization ,Mathematics ,QA1-939 - Abstract
Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.
- Published
- 2022
- Full Text
- View/download PDF
45. Financial accounting measurement model based on numerical analysis of rigid normal differential equation and rigid generalised functional equation
- Author
-
Liu Qiuhong, Dai Bo, Katib Iyad, and Alhamami Mohammed Alaa
- Subjects
rigid differential equation ,rigid general functional subequation ,numerical analysis ,the financial accounting ,97b10 ,Mathematics ,QA1-939 - Abstract
In order to solve the problems of financial accounting measurement quickly and accurately, this paper starts the analysis from the perspective of mathematics and finance and establishes the differential equation and the generalised functional equation for the related numerical analysis through mathematical knowledge. The results show that the limit and integral of rigid differential equation and the rigid generalised functional equation can improve their role and status in the financial accounting measurement environment so that they can be more widely used in the financial accounting measurement environment and promote the development of the financial accounting environment.
- Published
- 2021
- Full Text
- View/download PDF
46. AppliedMath
- Subjects
applied mathematics ,integrable systems ,asymptotics ,numerical analysis ,scientific computing ,mathematics and physics ,Mathematics ,QA1-939 - Published
- 2022
47. Computational Mathematics for Scientific Discovery and Problem-Solving
- Author
-
Mrs. T Jeyappradha
- Subjects
computational mathematics ,numerical analysis ,algorithms ,mathematical modelling ,scientific computing ,Mathematics ,QA1-939 - Abstract
This article explores about computational mathematics, which integrates numerical analysis, computer science, and mathematics to solve challenging scientific problems. To enable precise simulations, data analysis, optimisation, and mathematical modelling, computational mathematics makes use of cutting-edge algorithms, numerical techniques, and high-performance computing. In order to demonstrate the importance of computational mathematics for decision-making, scientific discovery, and engineering design, this study examines its key ideas, applications, and approaches. This article tries to demonstrate the transformative potential of computational mathematics in solving complicated issues and stimulating creativity by looking at real-world instances and developments, the use of algorithms, numerical analysis, mathematical modelling, and scientific computing in computational mathematics.
- Published
- 2022
48. Numerical analysis of fluid flow behaviour in two sided deep lid driven cavity using the finite volume technique.
- Author
-
Patel, Manoj R., Pandya, Jigisha U., and Patel, Vijay K.
- Subjects
NUMERICAL analysis ,FLUID flow ,COMPUTER simulation ,MATHEMATICS ,REYNOLDS number - Abstract
In the present study, numerical simulations of two-dimensional steady-state incompressible Newtonian fluid flow in one-sided square and two-sided deep lid driven cavities under the aspect ratio K = 1, 4, 6 are reported. For the one-sided lid driven cavity, the upper wall is moved to the right with up to 5000 Reynolds numbers under a grid size of up to 501 × 501. This lends support to previous findings in the literature with Ghia et al.’s results. Three cases are used in this article for the two-sided deep lid driven square cavity specifically. In these cases, the top and lower walls are moved to the right, while the left and right walls remain fixed up to at high Reynolds numbers (5000) under the grid size of up to 201×201. All possible flow solutions are studied in the present article, and flow bifurcation diagrams are constructed as velocity profiles and streamline contours for the same Reynolds number using a finite volume SIMPLE technique. The work done in this paper includes flow properties such as the location of primary and secondary vortices, velocity components, and numerical values for benchmarking purposes, and it is in excellent agreement with previous findings in the literature. A PARAM Shavak, high-performance computing (HPC) computer, was used to execute the calculations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
49. Riesz bases of exponentials for convex polytopes with symmetric faces.
- Author
-
Debernardi, Alberto and Lev, Nir
- Subjects
- *
RIESZ spaces , *MATHEMATICS , *POLYTOPES , *TOPOLOGY , *INTERPOLATION , *NUMERICAL analysis - Abstract
We prove that for any convex polytope Ω ⊂ ℝd which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space L²(Ω). The result is new in all dimensions d greater than one. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
50. Step Truncation Methods for Nonlinear Evolution Equations on Tensor Manifolds
- Author
-
Rodgers, Abram Kay
- Subjects
Mathematics ,Computational physics ,Computer science ,Data Compression ,Low Rank Dynamics ,Matrix Manifolds ,Multilinear Algebra ,Numerical Analysis ,Tensor Decomposition - Abstract
We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. In particular, we develop a mathematical framework for the analysis of these algorithms which encompasses both explicit and implicit time stepping. With this framework we prove convergence of a wide range of step-truncation methods, including one-step and multi-step methods. These methods rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a linear advection problem, a clasas of Fokker-Planck equations, the Allen-Cahn equation, the nonlinear Schrodinger, and a Burgers' equation with uncertain initial condition.
- Published
- 2023
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.