Computers & Mathematics with Applications最新文献_第10页

An optimization-based positivity-preserving limiter in semi-implicit discontinuous Galerkin schemes solving Fokker–Planck equations 求解Fokker-Planck方程的半隐式不连续Galerkin格式中基于优化的正保持极限器

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-19 DOI: 10.1016/j.camwa.2025.05.008

Chen Liu , Jingwei Hu , William T. Taitano , Xiangxiong Zhang

引用次数: 0

Modal analysis of biological structures based on the smoothed finite element methods 基于光滑有限元法的生物结构模态分析

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-17 DOI: 10.1016/j.camwa.2025.05.006

Jingui Zhao , Guirong Liu , Jinhui Zhao , Gang Wang , Zhonghu Wang , Zirui Li

{"title":"Modal analysis of biological structures based on the smoothed finite element methods","authors":"Jingui Zhao , Guirong Liu , Jinhui Zhao , Gang Wang , Zhonghu Wang , Zirui Li","doi":"10.1016/j.camwa.2025.05.006","DOIUrl":"10.1016/j.camwa.2025.05.006","url":null,"abstract":"<div><div>The smoothed finite element model exhibits a \"softening effect,\" resulting in reduced stiffness compared to the standard finite element model. This study employs the smoothed finite element methods (S-FEMs) with automatically generated tetrahedral meshes to perform modal analysis of biological structures subjected to arbitrary dynamic forces. Various S-FEM models are developed, including Edge-based, Face-based, and Node-based cross-element smoothing domains within the tetrahedral mesh framework, referred to as ES/FS/NS-FEM-T4. Using the gradient smoothing technique, the process of obtaining the strain-displacement matrix requires only the value of the shape function, not the inverse of the shape function, and no mapping is required. Additionally, by incorporating a Taylor expansion term for the strain gradient within the node-based smoothing domain framework, we introduce a stable node-based smoothed finite element method (SNS-FEM). Furthermore, the Lanczos algorithm and the modal superposition technique are integrated into our S-FEM models to compute the transient response of bone structures within the human body. The results obtained from S-FEMs are evaluated against the standard finite element method with respect to accuracy, convergence, and computational efficiency.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"191 ","pages":"Pages 188-227"},"PeriodicalIF":2.9,"publicationDate":"2025-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144070929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

High-order implicit Runge-Kutta Fourier pseudospectral methods for wave equations 波动方程的高阶隐式龙格-库塔傅立叶伪谱方法

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-16 DOI: 10.1016/j.camwa.2025.05.007

Ian T. Morgan , Youzuo Lin , Songting Luo

{"title":"High-order implicit Runge-Kutta Fourier pseudospectral methods for wave equations","authors":"Ian T. Morgan , Youzuo Lin , Songting Luo","doi":"10.1016/j.camwa.2025.05.007","DOIUrl":"10.1016/j.camwa.2025.05.007","url":null,"abstract":"<div><div>The dispersion error, also known as the pollution effect, is one of the main difficulties in numerical solutions to the wave propagation problem at high wavenumbers. The pollution effect, especially in mesh-based methods, can potentially be controlled by using either finer meshes or higher-order discretizations. Using finer meshes often leads to large systems that are computationally expensive to solve, especially for medium to high wavenumbers. Therefore, higher-order approximations are preferred to achieve good accuracy with manageable complexity. In this work, we will present high-order methods with implicit Runge-Kutta time integration and Fourier pseudospectral spatial approximations for the wave equation in a domain of interest surrounded by a sponge layer. At each time step, applying an appropriate A-stable implicit Runge-Kutta time-stepping method results in a modified Helmholtz equation that needs to be solved, for which an efficient iterative functional evaluation method with Fourier pseudospectral approximations will be proposed. The functional evaluation method transforms the equation into a functional iteration problem associated with an exponential operator that can be solved iteratively with guaranteed efficient convergence, where the exponential operator is evaluated by high-order operator splitting techniques and Fourier pseudospectral approximations. Numerical experiments are performed to demonstrate the effectiveness of the proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"192 ","pages":"Pages 1-13"},"PeriodicalIF":2.9,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144067223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Physics informed neural network framework for unsteady discretized reduced order system 非定常离散化降阶系统的物理信息神经网络框架

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-13 DOI: 10.1016/j.camwa.2025.05.004

Rahul Halder , Giovanni Stabile , Gianluigi Rozza

{"title":"Physics informed neural network framework for unsteady discretized reduced order system","authors":"Rahul Halder , Giovanni Stabile , Gianluigi Rozza","doi":"10.1016/j.camwa.2025.05.004","DOIUrl":"10.1016/j.camwa.2025.05.004","url":null,"abstract":"<div><div>This work addresses the development of a physics-informed neural network (PINN) with a loss term derived from a discretized time-dependent full-order and reduced-order system. In this work, first, the governing equations are discretized using a finite difference scheme (whereas any other discretization technique can be adopted), then projected on a reduced or latent space using the Proper Orthogonal Decomposition (POD)-Galerkin approach, and next, the residual arising from discretized reduced order equation is considered as an additional loss penalty term alongside the data-driven loss term using different variants of deep learning method such as Artificial neural network (ANN), Long Short-Term Memory based neural network (LSTM). The LSTM neural network has been proven to be very effective for time-dependent problems in a purely data-driven environment. The current work demonstrates the LSTM network's potential over ANN networks in PINN as well. The major difficulties in coupling PINN with external forward solvers often arise from the inability to access the discretized forms of the governing equation directly through the PINN solver and also to include those forms in the computational graph of the network. This poses a significant challenge, especially when a gradient-based optimization approach is considered in the neural network. Therefore, we propose an additional step in the PINN algorithm to overcome these difficulties. The proposed methods are applied to a pitch-plunge airfoil motion governed by rigid-body dynamics and a one-dimensional viscous Burgers' equation. The potential of using discretized governing equations instead of a continuous form lies in the flexibility of input to the PINN. The current work also demonstrates the prediction capability of various discretized-physics-informed neural networks outside the domain where the data is available or where the governing equation-based residuals are minimized.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"191 ","pages":"Pages 167-187"},"PeriodicalIF":2.9,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Numerical simulation for pulmonary airway reopening in alveolar duct by lattice Boltzmann method 点阵玻尔兹曼方法对肺泡管内气道重开的数值模拟

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-13 DOI: 10.1016/j.camwa.2025.05.005

Qianyu Lv , Bing He , Chunyan Qin , Binghai Wen

{"title":"Numerical simulation for pulmonary airway reopening in alveolar duct by lattice Boltzmann method","authors":"Qianyu Lv , Bing He , Chunyan Qin , Binghai Wen","doi":"10.1016/j.camwa.2025.05.005","DOIUrl":"10.1016/j.camwa.2025.05.005","url":null,"abstract":"<div><div>Aerosols, which are generated by the rupture of the liquid plug in the pulmonary respiratory tract, are important carriers of the viruses of infectious respiratory diseases, such as flu, tuberculosis, COVID-19, and Measles. In this study, we investigate liquid plug rupture and aerosol generation in the low respiratory tract with the alveolar structures by the chemical-potential multiphase lattice Boltzmann method. In a single alveolus duct, the opening expedites a unilateral break of the liquid plug due to a portion of the liquid flowing into the alveolus, and a microdroplet is yielded in the rupture. Aerosol would be deflected and reintegrated into the liquid film when the force is not great enough, which generates greater shear stresses to the inner wall where the microdroplet falls. In two alveoli duct, the rupture times of the upper and lower neck of the liquid plug depend on the radius ratio of the upper and lower alveolar. After the rupture of the liquid plug, the movement trajectory of the droplet is influenced by the alveoli structure to move forward or upward deflection. Interestingly, with the increase of radius ratio of the upper and lower alveolar, the mass of the fluid inflow into the alveoli decreases, while the mass of the aerosol generated by the rupture increase. This work contributes to understanding complex flow properties in the pulmonary airways, and the model can be extended to study the transport of liquid plugs and the generation of aerosol particles in more complex respiratory tract structures.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 206-218"},"PeriodicalIF":2.9,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A new numerical strategy for the drift-diffusion equations based on bridging the hybrid mixed and exponential fitted methods 基于桥接混合法和指数拟合法的漂移扩散方程的一种新的数值策略

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-09 DOI: 10.1016/j.camwa.2025.04.028

Aline C. da Rocha

{"title":"A new numerical strategy for the drift-diffusion equations based on bridging the hybrid mixed and exponential fitted methods","authors":"Aline C. da Rocha","doi":"10.1016/j.camwa.2025.04.028","DOIUrl":"10.1016/j.camwa.2025.04.028","url":null,"abstract":"<div><div>We present a new discretization scheme to solve the stationary drift-diffusion equations based on the hybrid mixed finite element method. A convenient change of variables is adopted and the partial differential equations of the system are decoupled and linearized through Gummel's map. This gives rise to three equations that need to be solved in a staggered fashion: one of reaction-diffusion type (Poisson) and two exhibiting a diffusion-reaction character (continuity equations). The Poisson's equation is solved by the classical hybrid mixed finite element method, while the continuity equations are discretized by a new version of the hybrid mixed exponential fitted method. The novelty here lies on the bridging terms between Poisson and each continuity equation, pursued by exploring direct relations between the Lagrange multipliers, thereby avoiding the use of a projection operator. The static condensation technique is adopted to reduce the number of degrees of freedom. Moreover, the finite dimensional functional spaces characterizing the hybrid mixed methods are chosen to ensure that the discrete problems satisfy the discrete maximum principle when a mesh of rectangular elements is used. Numerical experiments simulating semiconductor devices are presented, showing that the proposed methodology is capable of producing solutions free from spurious oscillations and accurate fluxes without the need of highly refined or complex meshes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 185-205"},"PeriodicalIF":2.9,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143924078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A new energy dissipation-preserving Crank-Nicolson type nonconforming FEM for damped wave equation with cubic nonlinearity 具有三次非线性的阻尼波动方程的一种新的保持能量耗散的Crank-Nicolson型非协调有限元方法

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-08 DOI: 10.1016/j.camwa.2025.04.029

Dongyang Shi , Xuemiao Xu

{"title":"A new energy dissipation-preserving Crank-Nicolson type nonconforming FEM for damped wave equation with cubic nonlinearity","authors":"Dongyang Shi , Xuemiao Xu","doi":"10.1016/j.camwa.2025.04.029","DOIUrl":"10.1016/j.camwa.2025.04.029","url":null,"abstract":"<div><div>In this article, an energy dissipative Crank-Nicolson (C-N) type fully discrete nonconforming finite element method (FEM) is developed for the damped wave equation with cubic nonlinearity, and its unconditional superconvergence behavior is rigorously analyzed for the nonconforming <span><math><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></math></span> element. By introducing an auxiliary variable <span><math><mi>p</mi><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, the problem is converted to a proper parabolic system, a new C-N type fully discrete scheme is presented, which preserves the energy dissipation property so as to ensure the boundedness of the numerical solutions in the broken <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm. Then, the existence and uniqueness of the numerical solutions are proved strictly. Also, thanks to the dissipation property mentioned above, along with specific characteristics of <span><math><mi>E</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>r</mi><mi>o</mi><mi>t</mi></mrow></msubsup></math></span> element and interpolation post-processing technique, the unconditional superclose and superconvergence estimates of order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> for original variable <em>u</em> and auxiliary variable <em>p</em> in the broken <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm are derived, respectively, without any restriction between the mesh size <em>h</em> and the time step <em>τ</em> that is usually required in the previous literature. Finally, the theoretical findings and good performance of the proposed method are confirmed by some numerical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 170-184"},"PeriodicalIF":2.9,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143924077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

High order direct discontinuous Galerkin method for elliptic interface problem on arbitrary polygon fitted meshes 任意多边形拟合网格上椭圆界面问题的高阶直接不连续伽辽金法

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-07 DOI: 10.1016/j.camwa.2025.04.027

Hanlin Guo , Li Yin , Xia Cui

{"title":"High order direct discontinuous Galerkin method for elliptic interface problem on arbitrary polygon fitted meshes","authors":"Hanlin Guo , Li Yin , Xia Cui","doi":"10.1016/j.camwa.2025.04.027","DOIUrl":"10.1016/j.camwa.2025.04.027","url":null,"abstract":"<div><div>In this article, we aim to develop a high order direct discontinuous Galerkin (DDG) method solving elliptic interface problem on arbitrary polygon fitted meshes. Elliptic interface problem with the homogeneous or non-homogeneous interface conditions can be solved in the uniform discrete DDG formulation. Numerical analysis results show that high order DDG method for polygonal elliptic interface problem on arbitrary polygon fitted meshes can reach the optimal <em>k</em>th order convergence in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm and the optimal <span><math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>th order convergence in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm. By refining meshes nearby the curved interface, errors coming from polygonal approximating interface can be reduced. A sequence of numerical experiments are carried out to verify the optimal convergence of DDG method with high order <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> approximations to deal with several different interface situations. It means that DDG method is capable of handing interface problems with complicated geometries meshes and interface situations.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"191 ","pages":"Pages 144-166"},"PeriodicalIF":2.9,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143912082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A new numerical approach using the VOF method to model the two-layered Herschel-Bulkley blood flow in microvessels 一种利用VOF方法模拟微血管双层Herschel-Bulkley血流的新方法

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-06 DOI: 10.1016/j.camwa.2025.04.025

Louiza Achab , Farida Iachachene

{"title":"A new numerical approach using the VOF method to model the two-layered Herschel-Bulkley blood flow in microvessels","authors":"Louiza Achab , Farida Iachachene","doi":"10.1016/j.camwa.2025.04.025","DOIUrl":"10.1016/j.camwa.2025.04.025","url":null,"abstract":"<div><div>In this paper, we propose a novel numerical approach to model the complex blood flow in microvessels using a two-layered fluid representation. The model considers blood flow as two layers of homogeneous, immiscible fluid with different viscosities: a core layer, rich in erythrocytes (red blood cells, RBCs), occupying the central region of the vessel, and a peripheral cell-free plasma layer (CFL) near the vessel walls. The Herschel-Bulkley constitutive model governs the core layer as a non-Newtonian viscoplastic fluid, accounting for its yield stress and shear-thinning behavior. We model the plasma layer as a Newtonian fluid with constant viscosity. We numerically solve the governing equations for fluid motion in an axisymmetric tube geometry to account for unsteady, incompressible flow. We employ the Volume of Fluid (VOF) method to accurately model the interaction between two immiscible fluids. Comparisons with the analytical one-dimensional Herschel-Bulkley model for single-layer fluid flow, two-layered fluid flow, and with the experimental data have shown that the two-layer model is valid and that the proposed method can accurately predict the dynamic behavior of blood flow in microvessels. Furthermore, numerical results reveal the presence of a plug flow region at the centerline of the vessel. The rheological properties of the core fluid, particularly the hematocrit level and yield stress values, significantly influence the thickness of the cell-free layer (CFL) and the plug flow radius. As both hematocrit and yield stress increase, the CFL thickness decreases while the plug flow radius expands. We also observe that the Reynolds number has a minimal impact on the characteristics of the CFL and the plug flow region. These results show that the two-layered numerical approach is a good way to accurately predict how blood flow moves in microvessels.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 154-169"},"PeriodicalIF":2.9,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143908312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Three-dimensional geometrically nonlinear analysis of functionally graded microshell structures using corotational finite element method based on modified couple stress theory 基于修正耦合应力理论的微壳结构三维几何非线性分析

IF 2.9 2区数学

Computers & Mathematics with Applications Pub Date : 2025-05-05 DOI: 10.1016/j.camwa.2025.04.024

Songhao Wang , Zhenghua Qian , Yan Shang

{"title":"Three-dimensional geometrically nonlinear analysis of functionally graded microshell structures using corotational finite element method based on modified couple stress theory","authors":"Songhao Wang , Zhenghua Qian , Yan Shang","doi":"10.1016/j.camwa.2025.04.024","DOIUrl":"10.1016/j.camwa.2025.04.024","url":null,"abstract":"<div><div>Shell structures, characterized by their thin walls, are prone to significant displacements and deformations under loading, leading to geometric nonlinearity while local deformations or strains remain small. The corotational (CR) method, where the total motion is separated into rigid body motion and elastic displacement, effectively simplifies such complex large-rotation problems into local small strain-small displacement-small curvature problems and provides a powerful tool for the nonlinear analysis of shell structures. To further investigate the size effect of porous multi-directional functionally graded materials (FGMs) under geometric nonlinearity, the penalty unsymmetric finite element method (FEM) based on modified couple stress theory (MCST) is combined with CR method. In this framework, independent rotation degree of freedoms (DOFs) are utilized to approximate the elastic rotations in the local coordinate, ensuring that the C<sup>1</sup> continuity requirement of MCST is satisfied in a weak form. Additionally, the stress trial functions are continuously updated at each incremental step according to the current configuration, enhancing the accuracy of the geometric nonlinear analysis. Numerical results demonstrate that this element exhibits high numerical accuracy and effectively captures the size effects of porous multi-directional FGMs under geometric nonlinear conditions.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 122-153"},"PeriodicalIF":2.9,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0