Pub Date : 2025-02-18DOI: 10.1016/j.apnum.2025.02.010
Josefa Caballero , Hanna Okrasińska-Płociniczak , Łukasz Płociniczak , Kishin Sadarangani
We consider a generalization of a functional equation that models the learning process in various animal species. The equation can be considered nonlocal, as it is built with a convex combination of the unknown function evaluated at mixed arguments. This makes the equation contain two terms with vanishing delays. We prove the existence and uniqueness of the solution in the Hölder space which is a natural function space to consider. In the second part of the paper, we devise an efficient numerical collocation method used to find an approximation to the main problem. We prove the convergence of the scheme and, in passing, several properties of the linear interpolation operator acting on the Hölder space. Numerical simulations verify that the order of convergence of the method (measured in the supremum norm) is equal to the order of Hölder continuity.
{"title":"Functional equation arising in behavioral sciences: solvability and collocation scheme in Hölder spaces","authors":"Josefa Caballero , Hanna Okrasińska-Płociniczak , Łukasz Płociniczak , Kishin Sadarangani","doi":"10.1016/j.apnum.2025.02.010","DOIUrl":"10.1016/j.apnum.2025.02.010","url":null,"abstract":"<div><div>We consider a generalization of a functional equation that models the learning process in various animal species. The equation can be considered nonlocal, as it is built with a convex combination of the unknown function evaluated at mixed arguments. This makes the equation contain two terms with vanishing delays. We prove the existence and uniqueness of the solution in the Hölder space which is a natural function space to consider. In the second part of the paper, we devise an efficient numerical collocation method used to find an approximation to the main problem. We prove the convergence of the scheme and, in passing, several properties of the linear interpolation operator acting on the Hölder space. Numerical simulations verify that the order of convergence of the method (measured in the supremum norm) is equal to the order of Hölder continuity.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 268-282"},"PeriodicalIF":2.2,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446103","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}
Pub Date : 2025-02-17DOI: 10.1016/j.apnum.2025.02.008
Miloud Sadkane , Roger B. Sidje
Numerical methods are proposed to quantify the magnitude of the growth reachable by solutions of systems of delayed linear difference and differential equations that are assumed to be asymptotically stable. A foundation based on an alternating maximization algorithm is established to address the discrete-time case. Following that, it is shown how to reuse this foundation for the continuous-time case, by converting to the discrete-time case through an approximation scheme that uses a backward differentiation formula (BDF) to produce a discretization in time. This indirect conversion approach raises new theoretical questions that are examined thoroughly. The proposed methods apply to systems with constant or variable coefficients. Numerical experiments are included to demonstrate their performance and reliability on several examples.
{"title":"Estimating the growth of solutions of linear delayed difference and differential equations by alternating maximization","authors":"Miloud Sadkane , Roger B. Sidje","doi":"10.1016/j.apnum.2025.02.008","DOIUrl":"10.1016/j.apnum.2025.02.008","url":null,"abstract":"<div><div>Numerical methods are proposed to quantify the magnitude of the growth reachable by solutions of systems of delayed linear difference and differential equations that are assumed to be asymptotically stable. A foundation based on an alternating maximization algorithm is established to address the discrete-time case. Following that, it is shown how to reuse this foundation for the continuous-time case, by converting to the discrete-time case through an approximation scheme that uses a backward differentiation formula (BDF) to produce a discretization in time. This indirect conversion approach raises new theoretical questions that are examined thoroughly. The proposed methods apply to systems with constant or variable coefficients. Numerical experiments are included to demonstrate their performance and reliability on several examples.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 254-267"},"PeriodicalIF":2.2,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427575","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}
Pub Date : 2025-02-13DOI: 10.1016/j.apnum.2025.02.007
H. Ould Sidi , A.S. Hendy , M.M. Babatin , L. Qiao , M.A. Zaky
In this work, we determine the unknown Robin coefficient in a degenerate parabolic equation. In inverse analysis, the problem under consideration is nonlinear with an ill-formulated operator and nonlocal. For the stable identification of the unknown Robin coefficient, the inverse problem is formulated into a regularised optimization problem. We discuss a variety of practical challenges associated with the problem. The finite element approximation is used to discretize the continuous optimization problem. The convergence and stability analyses are also discussed. Morozov's discrepancy principle is used with the conjugate gradient procedure to construct an iterative scheme. Finally, experiment results are reported to demonstrate the efficiency of the proposed schemes.
{"title":"An inverse problem of Robin coefficient identification in parabolic equations with interior degeneracy from terminal observation data","authors":"H. Ould Sidi , A.S. Hendy , M.M. Babatin , L. Qiao , M.A. Zaky","doi":"10.1016/j.apnum.2025.02.007","DOIUrl":"10.1016/j.apnum.2025.02.007","url":null,"abstract":"<div><div>In this work, we determine the unknown Robin coefficient in a degenerate parabolic equation. In inverse analysis, the problem under consideration is nonlinear with an ill-formulated operator and nonlocal. For the stable identification of the unknown Robin coefficient, the inverse problem is formulated into a regularised optimization problem. We discuss a variety of practical challenges associated with the problem. The finite element approximation is used to discretize the continuous optimization problem. The convergence and stability analyses are also discussed. Morozov's discrepancy principle is used with the conjugate gradient procedure to construct an iterative scheme. Finally, experiment results are reported to demonstrate the efficiency of the proposed schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 242-253"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422494","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}
Pub Date : 2025-02-12DOI: 10.1016/j.apnum.2025.02.005
Xin Zhang , Yuanfeng Jin
In this paper, the generalized Korteweg-de Vries–Benjamin Bona Mahony (GKdV-BBM) equation is investigated by two compact finite difference methods. One is a two-level-nonlinear difference scheme and another is a three-level-linearized difference scheme. Both of the schemes provide second and fourth-order accuracy in time and space, respectively. It is important that they preserve certain properties of the original equation, such as conservative properties. The solvability of the proposed numerical schemes is proved by Brouwer's fixed point theorem and mathematical induction, respectively. The unconditional convergence of the proposed difference schemes are also established through the discrete energy method, without imposing any restrictions on the grid ratios. Finally, numerical results are presented to confirm the theoretical findings, and they also demonstrate the efficiency and reliability of the proposed compact approaches.
{"title":"Two fourth-order conservative compact difference schemes for the generalized Korteweg–de Vries–Benjamin Bona Mahony equation","authors":"Xin Zhang , Yuanfeng Jin","doi":"10.1016/j.apnum.2025.02.005","DOIUrl":"10.1016/j.apnum.2025.02.005","url":null,"abstract":"<div><div>In this paper, the generalized Korteweg-de Vries–Benjamin Bona Mahony (GKdV-BBM) equation is investigated by two compact finite difference methods. One is a two-level-nonlinear difference scheme and another is a three-level-linearized difference scheme. Both of the schemes provide second and fourth-order accuracy in time and space, respectively. It is important that they preserve certain properties of the original equation, such as conservative properties. The solvability of the proposed numerical schemes is proved by Brouwer's fixed point theorem and mathematical induction, respectively. The unconditional convergence of the proposed difference schemes are also established through the discrete energy method, without imposing any restrictions on the grid ratios. Finally, numerical results are presented to confirm the theoretical findings, and they also demonstrate the efficiency and reliability of the proposed compact approaches.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 223-241"},"PeriodicalIF":2.2,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422493","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}
Pub Date : 2025-02-11DOI: 10.1016/j.apnum.2025.02.004
Chen-Xiao Gao, Fang Chen
The partially randomized extended Kaczmarz method with residual is effective for solving large sparse linear systems. In this paper, an improved variant of this method is proposed and its expected exponential convergence rate is proved. In addition, numerical results show that this method can preform better than the partially randomized extended Kaczmarz method with residual.
{"title":"Modified partially randomized extended Kaczmarz method with residual for solving large sparse linear systems","authors":"Chen-Xiao Gao, Fang Chen","doi":"10.1016/j.apnum.2025.02.004","DOIUrl":"10.1016/j.apnum.2025.02.004","url":null,"abstract":"<div><div>The partially randomized extended Kaczmarz method with residual is effective for solving large sparse linear systems. In this paper, an improved variant of this method is proposed and its expected exponential convergence rate is proved. In addition, numerical results show that this method can preform better than the partially randomized extended Kaczmarz method with residual.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 215-222"},"PeriodicalIF":2.2,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422492","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}
Pub Date : 2025-02-10DOI: 10.1016/j.apnum.2025.02.003
Jie Li, Lifang Pei
An accurate and efficient parametric finite element method (PFEM) is proposed to simulate numerically the evolution of closed curves under a nonlocal perimeter-conserved generalized curvature flow. We firstly present a variational formulation and show that it preserves two fundamental geometric structures of the flow, i.e., enclosed area increase and perimeter conservation. Then the semi-discrete parametric finite element scheme is proposed and its geometric structure preserving property is rigorously proved. On this basis, an implicit fully discrete scheme is established, which preserves the area-increasing property at the discretized level and enjoys asymptotic equal mesh distribution property. At last, extensive numerical results confirm the good performance of the proposed PFEM, including second-order accuracy in space, area-increasing and the excellent mesh quality.
{"title":"Parametric finite element method for a nonlocal curvature flow","authors":"Jie Li, Lifang Pei","doi":"10.1016/j.apnum.2025.02.003","DOIUrl":"10.1016/j.apnum.2025.02.003","url":null,"abstract":"<div><div>An accurate and efficient parametric finite element method (PFEM) is proposed to simulate numerically the evolution of closed curves under a nonlocal perimeter-conserved generalized curvature flow. We firstly present a variational formulation and show that it preserves two fundamental geometric structures of the flow, i.e., enclosed area increase and perimeter conservation. Then the semi-discrete parametric finite element scheme is proposed and its geometric structure preserving property is rigorously proved. On this basis, an implicit fully discrete scheme is established, which preserves the area-increasing property at the discretized level and enjoys asymptotic equal mesh distribution property. At last, extensive numerical results confirm the good performance of the proposed PFEM, including second-order accuracy in space, area-increasing and the excellent mesh quality.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 197-214"},"PeriodicalIF":2.2,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387497","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}
Pub Date : 2025-02-05DOI: 10.1016/j.apnum.2025.02.002
Xindan Zhang , Jianping Zhao , Yanren Hou
In this paper, we develop a priori error estimates for the finite element approximations of parabolic optimal control problems with time delay and pointwise control constraints. At first, we derive the first-order optimality systems for the control problems and the corresponding regularity results. Then, to approximate the problem we use the piecewise linear and continuous finite elements for the space discretization of the state, while the piecewise constant discontinuous Galerkin method is used for the time discretization. For the control discretization, we consider variational discretization. We show order of convergence rate for the control in the norm, which is new to the best of our knowledge. Finally, some numerical examples are provided to confirm our theoretical results.
{"title":"Finite element error estimation for parabolic optimal control problems with time delay","authors":"Xindan Zhang , Jianping Zhao , Yanren Hou","doi":"10.1016/j.apnum.2025.02.002","DOIUrl":"10.1016/j.apnum.2025.02.002","url":null,"abstract":"<div><div>In this paper, we develop a priori error estimates for the finite element approximations of parabolic optimal control problems with time delay and pointwise control constraints. At first, we derive the first-order optimality systems for the control problems and the corresponding regularity results. Then, to approximate the problem we use the piecewise linear and continuous finite elements for the space discretization of the state, while the piecewise constant discontinuous Galerkin method is used for the time discretization. For the control discretization, we consider variational discretization. We show <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> order of convergence rate for the control in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm, which is new to the best of our knowledge. Finally, some numerical examples are provided to confirm our theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 176-196"},"PeriodicalIF":2.2,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378941","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}
Pub Date : 2025-02-05DOI: 10.1016/j.apnum.2025.02.001
Dana Černá
Multi-stage expansion and contraction options are real options enabling an investment project to be scaled up or down in response to market conditions at predetermined future dates. We examine an investment project focused on producing a specific commodity, with the project value dependent on the market price of this commodity. We then study the value of options to either increase or decrease production at specific future dates based on predetermined factors and costs. Under the assumption that the commodity price follows a geometric Brownian motion and the volatility is stochastic, multiple partial differential equations represent the valuation model for these options. This paper aims to establish two new pricing models for multi-stage expansion and contraction options: one where variance follows a geometric Brownian motion and another governed by the Cox–Ingersoll–Ross process. Another aim is to propose and analyze an efficient wavelet-based numerical method for these models. The method employs the Galerkin method with a recently constructed orthogonal cubic spline wavelet basis and the Crank-Nicolson scheme enhanced by Richardson extrapolation. We establish the existence and uniqueness of the solution, provide error estimates for the proposed method, and derive bounds for condition numbers of the resulting matrices arising from discretization. The method is applied to options related to iron-ore mining investment projects to verify the relevance of the method and show its benefits, which are a high-order convergence rate, well-conditioned discretization matrices, and an efficient solution of the resulting system of equations using a small number of iterations.
{"title":"Orthogonal wavelet method for multi-stage expansion and contraction options under stochastic volatility","authors":"Dana Černá","doi":"10.1016/j.apnum.2025.02.001","DOIUrl":"10.1016/j.apnum.2025.02.001","url":null,"abstract":"<div><div>Multi-stage expansion and contraction options are real options enabling an investment project to be scaled up or down in response to market conditions at predetermined future dates. We examine an investment project focused on producing a specific commodity, with the project value dependent on the market price of this commodity. We then study the value of options to either increase or decrease production at specific future dates based on predetermined factors and costs. Under the assumption that the commodity price follows a geometric Brownian motion and the volatility is stochastic, multiple partial differential equations represent the valuation model for these options. This paper aims to establish two new pricing models for multi-stage expansion and contraction options: one where variance follows a geometric Brownian motion and another governed by the Cox–Ingersoll–Ross process. Another aim is to propose and analyze an efficient wavelet-based numerical method for these models. The method employs the Galerkin method with a recently constructed orthogonal cubic spline wavelet basis and the Crank-Nicolson scheme enhanced by Richardson extrapolation. We establish the existence and uniqueness of the solution, provide error estimates for the proposed method, and derive bounds for condition numbers of the resulting matrices arising from discretization. The method is applied to options related to iron-ore mining investment projects to verify the relevance of the method and show its benefits, which are a high-order convergence rate, well-conditioned discretization matrices, and an efficient solution of the resulting system of equations using a small number of iterations.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 155-175"},"PeriodicalIF":2.2,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143377000","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}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.09.023
Felipe Lepe , Gonzalo Rivera , Jesus Vellojin
The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.
{"title":"Finite element analysis for the Navier-Lamé eigenvalue problem","authors":"Felipe Lepe , Gonzalo Rivera , Jesus Vellojin","doi":"10.1016/j.apnum.2024.09.023","DOIUrl":"10.1016/j.apnum.2024.09.023","url":null,"abstract":"<div><div>The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 1-20"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143097174","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}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.09.024
Yujun Cheng , Miaojuan Peng , Yumin Cheng
This study investigates a hybrid interpolating element-free Galerkin (HIEFG) method for solving 3D convection diffusion problems. The HIEFG approach divides a 3D solution domain into a sequence of interconnected 2D sub-domains, and in these 2D sub-domains, the interpolating element-free Galerkin (IEFG) method is applied to form the discretized equations. The improved interpolating moving least-squares (IMLS) method is used to obtain the shape function of the IEFG method for 2D problems. The finite difference method is employed to combine the discretized equations in 2D sub-domains in the splitting direction. Then, the HIEFG method's formulas are derived for steady-state convection diffusion problems in 3D solution domain. Three numerical examples are used to discuss the impacts of the number of nodes, the number of split layers, and the scaling parameters of the influence domain on the computational precision and CPU time of the HIEFG technique. Imposing boundary conditions directly and the dimension splitting technique in this method significantly improves the computational speed greatly.
{"title":"A hybrid interpolating element-free Galerkin method for 3D steady-state convection diffusion problems","authors":"Yujun Cheng , Miaojuan Peng , Yumin Cheng","doi":"10.1016/j.apnum.2024.09.024","DOIUrl":"10.1016/j.apnum.2024.09.024","url":null,"abstract":"<div><div>This study investigates a hybrid interpolating element-free Galerkin (HIEFG) method for solving 3D convection diffusion problems. The HIEFG approach divides a 3D solution domain into a sequence of interconnected 2D sub-domains, and in these 2D sub-domains, the interpolating element-free Galerkin (IEFG) method is applied to form the discretized equations. The improved interpolating moving least-squares (IMLS) method is used to obtain the shape function of the IEFG method for 2D problems. The finite difference method is employed to combine the discretized equations in 2D sub-domains in the splitting direction. Then, the HIEFG method's formulas are derived for steady-state convection diffusion problems in 3D solution domain. Three numerical examples are used to discuss the impacts of the number of nodes, the number of split layers, and the scaling parameters of the influence domain on the computational precision and CPU time of the HIEFG technique. Imposing boundary conditions directly and the dimension splitting technique in this method significantly improves the computational speed greatly.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 21-37"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143097175","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}
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