Computational Mathematics and Mathematical Modelling

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 11020

Special Issue Editors

Physical Processes Modeling Laboratory, Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, Kamchatskiy Kray, 684034 Paratunka, Russia
Interests: fractional calculus; fractional oscillators; fractional dynamics; numerical methods; mathematical modeling
Special Issues, Collections and Topics in MDPI journals
Chief of the Informatics and Computer Graphics Department, Tashkent State Transport University, 1, Odilkhodjaev Str., Tashkent 100167, Uzbekistan
Interests: interpolation; approximation; numerical integration; numerical methods for differential and integral equations
Head of Laboratory of Computational Mathematucs, V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b University Street, Tashkent 100174, Uzbekistan
Interests: interpolation; approximation; numerical integration; numerical methods for differential and integral equations

Special Issue Information

Dear Colleagues,

It is known that computational mathematics deals with mathematical research in all areas of science and technology where computation plays an important role.

Computational mathematics includes:

  • The development of accurate and optimal numerical methods for solving models of natural processes;
  • The construction and analysis of numerical approximation algorithms to solve differential and integral equations;
  • The development of computational formulas for the numerical integration, interpolation, and approximation of a set of given data;
  • In addition, it is desirable to have algorithms which will be developed fulfilling accuracy, efficiency, robustness, and stability requirements.

In this Special Issue, we focus on the development of new computational tools to better understand data and structure. Topics that may be featured in the Special Issue may include (but are not limited to):

  • Interpolation and polynomial approximation;
  • Numerical differentiation and integration;
  • Numerical solution to ordinary differential equations;
  • Numerical solution to partial differential equations;
  • Approximation;
  • Numerical solution of nonlinear systems of equations;
  • Finite element methods.

Dr. Roman Parovik
Prof. Dr. Kholmat Mahkambaevich Shadimetov
Prof. Dr. Abdullo Rakhmonovich Hayotov
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • interpolation
  • approximation
  • numerical integration
  • numerical methods for differential equations
  • mathematical modeling using differential equations
  • optimization
  • discretization
  • iterative methods
  • stability theory

Published Papers (12 papers)

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Research

14 pages, 3166 KiB  
Article
Mathematical Modeling of Multi-Phenomena Anisotropic Systems: Ejection of Primary Aerosols during the Fast Pyrolysis of Biomass
by Mario A. Sánchez, Juan C. Maya, Farid Chejne, Brennan Pecha and Adriana M. Quinchía-Figueroa
Mathematics 2024, 12(6), 925; https://doi.org/10.3390/math12060925 - 21 Mar 2024
Viewed by 338
Abstract
This study introduces a novel particle model for biomass fast pyrolysis, incorporating an anisotropic cylindrical particle to address mass and energy transport coupled with aerosol ejection, which previous models have overlooked. The main contribution lies in developing a model that considers aerosol generation [...] Read more.
This study introduces a novel particle model for biomass fast pyrolysis, incorporating an anisotropic cylindrical particle to address mass and energy transport coupled with aerosol ejection, which previous models have overlooked. The main contribution lies in developing a model that considers aerosol generation in anisotropic cylindrical particles for the first time, addressing bubbling dynamics and bursting within the liquid phase. The population balance equation describes bubble dynamics and aerosol formation, capturing phenomena like nucleation, growth, coalescence, and bursting. The model employs the method of moments with bubble volume as an internal variable, substantially reducing computational costs by eliminating dependence on this variable. Results highlight the significant impact of anisotropy and particle size on aerosol ejection: smaller, less elongated particles experience faster heating, quicker conversion, and the increased accumulation of the liquid intermediate phase. Specifically, 1 mm diameter particles yield higher concentrations of metaplast and bio-oil aerosols, exceeding 15%, compared to concentrations below 11% for 3 mm particles. This model provides insights into aerosol structure (volume, surface area), aiding in understanding aerosol reactivity at the reactor scale. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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21 pages, 449 KiB  
Article
Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order
by Dmitrii Tverdyi and Roman Parovik
Mathematics 2023, 11(15), 3358; https://doi.org/10.3390/math11153358 - 31 Jul 2023
Cited by 1 | Viewed by 745
Abstract
The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by [...] Read more.
The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by the dependence of the current value of the solution function on previous values in the time interval. In terms of mathematics, the heredity here is described using a fractional differentiation operator in the Gerasimov–Caputo sense of variable order. As an example, we consider the Cauchy problem for the non-linear fractional Riccati equation with non-constant coefficients. An efficient parallel implementation algorithm has been proposed for the known sequential non-local explicit finite-difference numerical solution scheme. This implementation of the algorithm is a hybrid one, since it uses both GPU and CPU computational nodes. The program code of the parallel implementation of the algorithm is described in C and CUDA C languages, and is developed using OpenMP and CUDA hardware, as well as software architectures. This paper presents a study on the computational efficiency of the proposed parallel algorithm based on data from a series of computational experiments that were obtained using a computing server NVIDIA DGX STATION. The average computation time is analyzed in terms of the following: running time, acceleration, efficiency, and the cost of the algorithm. As a result, it is shown on test examples that the hybrid version of the numerical algorithm can give a significant performance increase of 3–5 times in comparison with both the sequential version of the algorithm and OpenMP implementation. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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22 pages, 1262 KiB  
Article
On a Fast Hough/Radon Transform as a Compact Summation Scheme over Digital Straight Line Segments
by Dmitry Nikolaev, Egor Ershov, Alexey Kroshnin, Elena Limonova, Arseniy Mukovozov and Igor Faradzhev
Mathematics 2023, 11(15), 3336; https://doi.org/10.3390/math11153336 - 29 Jul 2023
Cited by 1 | Viewed by 1034
Abstract
The Hough transform, interpreted as the discretization of the Radon transform, is a widely used tool in image processing and machine vision. The primary way to speed it up is to employ the Brady–Yong algorithm. However, the accuracy of the straight line discretization [...] Read more.
The Hough transform, interpreted as the discretization of the Radon transform, is a widely used tool in image processing and machine vision. The primary way to speed it up is to employ the Brady–Yong algorithm. However, the accuracy of the straight line discretization utilized in this algorithm is limited. In this study, we propose a novel algorithm called ASD2 that offers fast computation of the Hough transform for images of arbitrary sizes. Our approach adopts a computation scheme similar to the Brady–Yong algorithm but incorporates the best possible line discretization for improved accuracy. By employing the Method of Four Russians, we demonstrate that for an image of size n×n where n=8q and qN, the computational complexity of the ASD2 algorithm is O(n8/3) when summing over O(n2) digital straight line segments. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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20 pages, 373 KiB  
Article
Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator
by Kholmat Shadimetov, Aziz Boltaev and Roman Parovik
Mathematics 2023, 11(14), 3114; https://doi.org/10.3390/math11143114 - 14 Jul 2023
Cited by 1 | Viewed by 677
Abstract
It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be [...] Read more.
It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be constructed. In this paper, we construct a discrete analogue Dm(hβ) of the differential operator d2mdx2m+2dmdxm+1 in the Hilbert space W2(m,0). We develop an algorithm for constructing optimal quadrature formulas exact on exponential-trigonometric functions using a discrete operator. Based on this algorithm, in m=2, we give an optimal quadrature formula exact for trigonometric functions. Finally, we present the rate of convergence of the optimal quadrature formula in the Hilbert space W2(2,0) for the case m=2. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
17 pages, 3128 KiB  
Article
Oscillatory Behavior of Heat Transfer and Magnetic Flux of Electrically Conductive Fluid Flow along Magnetized Cylinder with Variable Surface Temperature
by Zia Ullah, Nifeen H. Altaweel, Musaad S. Aldhabani, Kaouther Ghachem, Muapper Alhadri and Lioua Kolsi
Mathematics 2023, 11(14), 3045; https://doi.org/10.3390/math11143045 - 10 Jul 2023
Viewed by 516
Abstract
The present study deals with electrically conductive fluid flow across a heated circular cylinder to examine the oscillatory magnetic flux and heat transfer in the presence of variable surface temperature. The proposed mathematical formulation is time-dependent, which is the source of the amplitude [...] Read more.
The present study deals with electrically conductive fluid flow across a heated circular cylinder to examine the oscillatory magnetic flux and heat transfer in the presence of variable surface temperature. The proposed mathematical formulation is time-dependent, which is the source of the amplitude and fluctuation in this analysis. The designed fluctuating nonlinear computational model is associated with the differential equations under specific boundary conditions. The governing equations are converted into dimensionless form by using adequate dimensionless variables. To simplify the resolution of the set of governing equations, it is further reduced. The effects of surface temperature parameter β, magnetic force number ξ, buoyancy parameter λ, Prandtl number Pr, and magnetic Prandtl parameter γ are investigated. The main finding of the current study is related to the determination of the temperature distribution for each inclination angle. It is seen that a higher amplitude of the heat transfer rate occurs as the surface temperature increases. It is also noticed that the oscillatory magnetic flux becomes more important as the magnetic Prandtl number increases at each position. The present magneto-thermal analysis is significantly important in practical applications such as power plants, thermally insulated engines, and nuclear reactor cooling. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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15 pages, 1191 KiB  
Article
A Proposed Analytical and Numerical Treatment for the Nonlinear SIR Model via a Hybrid Approach
by Abdulrahman B. Albidah
Mathematics 2023, 11(12), 2749; https://doi.org/10.3390/math11122749 - 17 Jun 2023
Cited by 1 | Viewed by 636
Abstract
This paper re-analyzes the nonlinear Susceptible–Infected–Recovered (SIR) model using a hybrid approach based on the Laplace–Padé technique. The proposed approach is successfully applied to extract several analytic approximations for the infected and recovered individuals. The domains of applicability of such analytic approximations are [...] Read more.
This paper re-analyzes the nonlinear Susceptible–Infected–Recovered (SIR) model using a hybrid approach based on the Laplace–Padé technique. The proposed approach is successfully applied to extract several analytic approximations for the infected and recovered individuals. The domains of applicability of such analytic approximations are addressed. In addition, the present results are validated through various comparisons with the Runge–Kutta numerical method. The obtained analytical results agree with the numerical ones for a wide range of numbers of contacts featured in the studied model. The efficiency of the present analysis reveals that it can be implemented to deal with other systems describing real-life phenomena. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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29 pages, 3675 KiB  
Article
An Analysis of a Fractional-Order Model of Colorectal Cancer and the Chemo-Immunotherapeutic Treatments with Monoclonal Antibody
by Ali Alhajraf, Ali Yousef and Fatma Bozkurt
Mathematics 2023, 11(10), 2374; https://doi.org/10.3390/math11102374 - 19 May 2023
Cited by 1 | Viewed by 864
Abstract
The growth of colorectal cancer tumors and their reactions to chemo-immunotherapeutic treatment with monoclonal antibodies (mAb) are discussed in this paper using a system of fractional order differential equations (FDEs). mAb medications are still at the research stage; however, this research takes into [...] Read more.
The growth of colorectal cancer tumors and their reactions to chemo-immunotherapeutic treatment with monoclonal antibodies (mAb) are discussed in this paper using a system of fractional order differential equations (FDEs). mAb medications are still at the research stage; however, this research takes into account the mAbs that are already in use. The major goal is to demonstrate the effectiveness of the mAb medication Cetuximab and the significance of IL-2 levels in immune system support. The created model is broken down into four sub-systems: cell populations, irinotecan (CPT11) concentration for treatment, IL-2 concentration for immune system support, and monoclonal antibody Cetuximab. We show the existence and uniqueness of the initial value problem (IVP). After that, we analyze the stability of the equilibrium points (disease-free and co-existing) using the Routh–Hurwitz criteria. In addition, in applying the discretization process, we demonstrate the global stability of the constructed system around the equilibrium points based on specific conditions. In the end, simulation results were carried out to support the theory of the manuscript. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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21 pages, 4632 KiB  
Article
Combined Micro-Structural Effects of Linearly Increasing Reynolds Number and Mean Inflow Velocity on Flow Fields with Mesh Independence Analysis in Non-Classical Framework
by Nazim Hussain Hajano, Muhammad Sabeel Khan, Lisheng Liu, Mumtaz Ali Kaloi and Hai Mei
Mathematics 2023, 11(9), 2074; https://doi.org/10.3390/math11092074 - 27 Apr 2023
Viewed by 759
Abstract
The monolithic Eulerian formulation has widely been employed for solving numerically fluid–structure interaction (FSI) problems of finite structural displacement using the same mathematical variational formulation for fluid and structural dynamics. Recently, different physical features of fluid flow have been analyzed using this approach [...] Read more.
The monolithic Eulerian formulation has widely been employed for solving numerically fluid–structure interaction (FSI) problems of finite structural displacement using the same mathematical variational formulation for fluid and structural dynamics. Recently, different physical features of fluid flow have been analyzed using this approach to such coupling problems by computing the classical benchmark solutions in a non-classical framework. Despite producing decent results, the analysis of micro-structural characteristics of fluid flow by applying the classical benchmark solutions still needs to be enhanced and extended further for such coupling problems. In this paper, the classical benchmark solutions have been enhanced and extended further for analyzing the combined micro-structural effects of linearly increasing Reynolds number Re and mean inflow velocity U¯ on flow fields with mesh independence analysis by employing a monolithic Eulerian formulation in a non-classical framework. To this aim, the Cosserat fluids theory is taken into account for the present coupling problem by considering three micro-rotational degrees of freedom (dof) of fluid particles. The model equations of the proposed Cosserat fluid–structure interaction (CFSI) problem are derived using underlying laws of continuum mechanics. A numerical section presents the implementation of the benchmark problem with test examples followed by a detailed evaluation of the obtained results. The results obtained indicate that a uniform linear increase in Reynolds number Re and mean inflow velocity U¯ produce the significant combined micro-structural effects on the micro-rotation velocity field ω, and this effect is found increasing on the increase of both parameters. This combined effect of increasing Re and U¯ on the velocity field u is also observed to be very significant in a sense that horizontal and vertical flow velocity profiles experience great variation by maintaining the same pattern on each increasing value of Re and U¯ at any particular location in the computational domain. Further, the mesh independence analysis is employed to verify the convergence of obtained results. The study concludes that the linear increase in Reynolds number and mean inflow velocity affect micro-rotational velocity field significantly at the micro-structural level with mesh independence analysis. Finally, some future recommendations to enhance and extend the study with some of its limitations are presented. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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17 pages, 4797 KiB  
Article
Hydrothermal Mixed Convection in a Split-Lid-Driven Triangular Cavity Suspended by NEPCM
by Obai Younis, Sameh E. Ahmed, Aissa Abderrahmane, Abdulaziz Alenazi and Ahmed M. Hassan
Mathematics 2023, 11(6), 1323; https://doi.org/10.3390/math11061323 - 09 Mar 2023
Cited by 1 | Viewed by 1090
Abstract
A numerical investigation of the magnetohydrodynamics of a mixed convection of nano-enhanced phase change material (NEPCM) within a triangular chamber containing an elliptical heat source is presented in this article. The forced convection has resulted from the movement of the upper cavity, while [...] Read more.
A numerical investigation of the magnetohydrodynamics of a mixed convection of nano-enhanced phase change material (NEPCM) within a triangular chamber containing an elliptical heat source is presented in this article. The forced convection has resulted from the movement of the upper cavity, while the free convection is due to the temperature difference between the heat source and cold inclined sidewalls. Four cases are considered based on the directions of the moving of the upper wall parts, namely, Case 1, where the left part is moving in the positive direction of the X-axis and the right part moves in the opposite direction (1(+−)), Case 2, where the two parts move in the positive direction of the X-axis (2(++)), Case 3, where the two parts move in the negative direction of the X-axis (3(− −)), and Case 4, where the left part moves in the negative direction of the X-axis and the right part moves in the negative direction (4(−+)). The Galerkin finite element method (GFEM) is employed for addressing the governing equations of the system under study. The impacts of the Reynolds number (1Re100), the inclination angle of the elliptic heat source (0γ90), the nanoparticles volume fraction ϕ (0%ϕ8%) and the movement directions of the parts of the upper wall (four cases) are presented and discussed. The results suggested that increasing Re enhanced the heat transfer rate, while increasing Ha reduced it. The vertical positions of the elliptical heat source resulted in the maximum heat transmission rate. At the highest Re, changing the location of the heat source from horizontal (γ=0) to vertical (γ=90) enhanced the average Nusselt number by 60%, while choosing Case 1 for upper wall movement increased the average Nusselt number by 300% compared to Cases 2 and 3. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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16 pages, 5084 KiB  
Article
Effect of Buoyancy Force on an Unsteady Thin Film Flow of Al2O3/Water Nanofluid over an Inclined Stretching Sheet
by Sumayyah Alabdulhadi, Sakhinah Abu Bakar, Anuar Ishak, Iskandar Waini and Sameh E. Ahmed
Mathematics 2023, 11(3), 739; https://doi.org/10.3390/math11030739 - 01 Feb 2023
Cited by 6 | Viewed by 1104
Abstract
The present study looks at the heat transfer and the unsteady thin film flow of Al2O3 water nanofluid past an inclined stretching sheet having a buoyancy force effect. The boundary value problem solver (bvp4c) package in Matlab is utilized in [...] Read more.
The present study looks at the heat transfer and the unsteady thin film flow of Al2O3 water nanofluid past an inclined stretching sheet having a buoyancy force effect. The boundary value problem solver (bvp4c) package in Matlab is utilized in solving the converted set of ordinary differential equations (ODEs). The multi-shape Al2O3 nanoparticles’ impact with respect to the flow as well as heat transfer characteristics are studied and visually displayed for certain governing parameter values, which include the mixed convection, inclination angle, magnetic, slip, and Biot number. Thus, the skin friction coefficient and the local Nusselt number are also determined. Here, the platelet shape of Al2O3 nanoparticles possesses a high heat transfer and flow rate based on the outcomes. In addition, increasing the slip and magnetic parameters improves the temperature, whereas increasing the buoyancy and inclination angle parameters has reverse effects. The results also show that increasing the unsteadiness parameter and the magnetic parameter reduces the film thickness. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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17 pages, 1501 KiB  
Article
Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type
by Valentine Aleksandrovich Kim, Roman Ivanovich Parovik and Zafar Ravshanovich Rakhmonov
Mathematics 2023, 11(3), 558; https://doi.org/10.3390/math11030558 - 20 Jan 2023
Cited by 2 | Viewed by 959
Abstract
The article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann–Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate the [...] Read more.
The article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann–Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate the theoretical results. Using the Runge rule, the results of the implicit scheme are compared with the results of the explicit scheme. Phase trajectories and oscillograms for a Duffing oscillator with a fractional derivative of variable order of the Riemann–Liouville type are constructed, chaotic modes are detected using the spectrum of maximum Lyapunov exponents and Poincare sections. Q-factor surfaces, amplitude-frequency and phase-frequency characteristics are constructed for the study of forced oscillations. The results of the study showed that the implicit finite-difference scheme shows more accurate results than the explicit one. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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16 pages, 2708 KiB  
Article
Mixed-Integer Conic Formulation of Unit Commitment with Stochastic Wind Power
by Haiyan Zheng, Liying Huang and Ran Quan
Mathematics 2023, 11(2), 346; https://doi.org/10.3390/math11020346 - 09 Jan 2023
Viewed by 955
Abstract
Due to the high randomness and volatility of renewable energy sources such as wind energy, the traditional thermal unit commitment (UC) model is no longer applicable. In this paper, in order to reduce the possible negative effects of an inaccurate wind energy forecast, [...] Read more.
Due to the high randomness and volatility of renewable energy sources such as wind energy, the traditional thermal unit commitment (UC) model is no longer applicable. In this paper, in order to reduce the possible negative effects of an inaccurate wind energy forecast, the chance-constrained programming (CCP) method is used to study the UC problem with uncertainty wind power generation, and chance constraints such as power balance and spinning reserve are satisfied with a predetermined probability. In order to effectively solve the CCP problem, first, we used the sample average approximation (SAA) method to transform the chance constraints into deterministic constraints and to obtain a mixed-integer quadratic programming (MIQP) model. Then, the quadratic terms were incorporated into the constraints by introducing some auxiliary variables, and some second-order cone constraints were formed by combining them with the output characteristics of thermal unit; therefore, a tighter mixed-integer second-order cone programming (MISOCP) formulation was obtained. Finally, we applied this method to some systems including 10 to 100 thermal units and 1 to 2 wind units, and we invoked MOSEK in MATLAB to solve the MISOCP formulation. The numerical results obtained within 24 h confirm that not only is the MISOCP formulation a successful reformulation that can achieve better suboptimal solutions, but it is also a suitable method for solving the large-scale uncertain UC problem. In addition, for systems of up to 40 units within 24 h that do not consider wind power and pollution emissions, the numerical results were compared with those of previously published methods, showing that the MISOCP formulation is very promising, given its excellent performance. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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