Abstract The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.
{"title":"Richardson extrapolation method for solving the Riesz space fractional diffusion problem","authors":"Ren‐jun Qi, Zhi‐zhong Sun","doi":"10.1002/num.23076","DOIUrl":"https://doi.org/10.1002/num.23076","url":null,"abstract":"Abstract The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.
{"title":"A semi‐Lagrangian ε$$ varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate","authors":"Yaowen Lu, Duy‐Minh Dang","doi":"10.1002/num.23075","DOIUrl":"https://doi.org/10.1002/num.23075","url":null,"abstract":"Abstract We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135729697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract An efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.
{"title":"Implicit Runge‐Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian","authors":"Yanming Zhang, Yu Li, Yuexin Yu, Wansheng Wang","doi":"10.1002/num.23074","DOIUrl":"https://doi.org/10.1002/num.23074","url":null,"abstract":"Abstract An efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.
{"title":"Two‐grid finite element method on grade meshes for time‐fractional nonlinear Schrödinger equation","authors":"Hanzhang Hu, Yanping Chen, Jianwei Zhou","doi":"10.1002/num.23073","DOIUrl":"https://doi.org/10.1002/num.23073","url":null,"abstract":"Abstract A two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A spectral‐Galerkin method based on Legendre‐Fourier approximation for fourth‐order problems in cylindrical regions is studied in this paper. By the cylindrical coordinate transformation, a three‐dimensional fourth‐order problem in a cylindrical region is transformed into a sequence of decoupled fourth‐order problems with two dimensions and the corresponding pole conditions are also derived. With appropriately constructed weighted Sobolev space, a weak form is established. Based on this weak form, a spectral‐Galerkin discretization scheme is proposed and its error is rigorously analyzed by defining a new class of projection operators. Then, a set of efficient basis functions are used to write the discrete scheme as the linear systems with a sparse matrix based on tensor product. Numerical examples are presented to show the efficiency and high‐accuracy of the developed method. Finally, an application of the proposed method to the fourth‐order Steklov problem and the corresponding numerical experiments once again confirm the efficiency and spectral accuracy of the method.
{"title":"A Legendre spectral‐Galerkin method for fourth‐order problems in cylindrical regions","authors":"Jihui Zheng, Jing An","doi":"10.1002/num.23071","DOIUrl":"https://doi.org/10.1002/num.23071","url":null,"abstract":"Abstract A spectral‐Galerkin method based on Legendre‐Fourier approximation for fourth‐order problems in cylindrical regions is studied in this paper. By the cylindrical coordinate transformation, a three‐dimensional fourth‐order problem in a cylindrical region is transformed into a sequence of decoupled fourth‐order problems with two dimensions and the corresponding pole conditions are also derived. With appropriately constructed weighted Sobolev space, a weak form is established. Based on this weak form, a spectral‐Galerkin discretization scheme is proposed and its error is rigorously analyzed by defining a new class of projection operators. Then, a set of efficient basis functions are used to write the discrete scheme as the linear systems with a sparse matrix based on tensor product. Numerical examples are presented to show the efficiency and high‐accuracy of the developed method. Finally, an application of the proposed method to the fourth‐order Steklov problem and the corresponding numerical experiments once again confirm the efficiency and spectral accuracy of the method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we analyze two classes of fully discrete spectral volume schemes (SV) for solving the one‐dimensional scalar hyperbolic equation. These two schemes are constructed by using the forward Euler (EU) method or the second‐order Runge–Kutta (RK2) method in time‐discretization, and by letting a piecewise k th degree( is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with Gauss–Legendre points (LSV) or right‐Radau points (RRSV). We prove that for the EU‐SV schemes, the weak (2) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. While for the RK2‐SV schemes, the weak (4) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. Here and are, respectively, the spacial and temporal mesh sizes and the constant is independent of and . Our theoretical findings have been justified by several numerical experiments.
{"title":"Analysis of two fully discrete spectral volume schemes for hyperbolic equations","authors":"Ping Wei, Qingsong Zou","doi":"10.1002/num.23072","DOIUrl":"https://doi.org/10.1002/num.23072","url":null,"abstract":"Abstract In this paper, we analyze two classes of fully discrete spectral volume schemes (SV) for solving the one‐dimensional scalar hyperbolic equation. These two schemes are constructed by using the forward Euler (EU) method or the second‐order Runge–Kutta (RK2) method in time‐discretization, and by letting a piecewise k th degree( is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with Gauss–Legendre points (LSV) or right‐Radau points (RRSV). We prove that for the EU‐SV schemes, the weak (2) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. While for the RK2‐SV schemes, the weak (4) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. Here and are, respectively, the spacial and temporal mesh sizes and the constant is independent of and . Our theoretical findings have been justified by several numerical experiments.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136308891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Time‐dependent Maxwell's equations govern electromagnetics. Under certain conditions, we can rewrite these equations into a partial differential equation of second order, which in this case is the vectorial wave equation. For the vectorial wave equation, we examine numerical schemes and their challenges. For this purpose, we consider a space‐time variational setting, that is, time is just another spatial dimension. More specifically, we apply integration by parts in time as well as in space, leading to a space‐time variational formulation with different trial and test spaces. Conforming discretizations of tensor‐product type result in a Galerkin–Petrov finite element method that requires a CFL condition for stability which we study. To overcome the CFL condition, we use a Hilbert‐type transformation that leads to a variational formulation with equal trial and test spaces. Conforming space‐time discretizations result in a new Galerkin–Bubnov finite element method that is unconditionally stable. In numerical examples, we demonstrate the effectiveness of this Galerkin–Bubnov finite element method. Furthermore, we investigate different projections of the right‐hand side and their influence on the convergence rates. This paper is the first step toward a more stable computation and a better understanding of vectorial wave equations in a conforming space‐time approach.
{"title":"Numerical study of conforming space‐time methods for Maxwell's equations","authors":"Julia I. M. Hauser, Marco Zank","doi":"10.1002/num.23070","DOIUrl":"https://doi.org/10.1002/num.23070","url":null,"abstract":"Abstract Time‐dependent Maxwell's equations govern electromagnetics. Under certain conditions, we can rewrite these equations into a partial differential equation of second order, which in this case is the vectorial wave equation. For the vectorial wave equation, we examine numerical schemes and their challenges. For this purpose, we consider a space‐time variational setting, that is, time is just another spatial dimension. More specifically, we apply integration by parts in time as well as in space, leading to a space‐time variational formulation with different trial and test spaces. Conforming discretizations of tensor‐product type result in a Galerkin–Petrov finite element method that requires a CFL condition for stability which we study. To overcome the CFL condition, we use a Hilbert‐type transformation that leads to a variational formulation with equal trial and test spaces. Conforming space‐time discretizations result in a new Galerkin–Bubnov finite element method that is unconditionally stable. In numerical examples, we demonstrate the effectiveness of this Galerkin–Bubnov finite element method. Furthermore, we investigate different projections of the right‐hand side and their influence on the convergence rates. This paper is the first step toward a more stable computation and a better understanding of vectorial wave equations in a conforming space‐time approach.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134911201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85]. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
{"title":"Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise","authors":"Ye Hu, Yubin Yan, Shahzad Sarwar","doi":"10.1002/num.23068","DOIUrl":"https://doi.org/10.1002/num.23068","url":null,"abstract":"Recently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85]. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47872358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we develop a local discontinuous Galerkin (LDG) method to simulate the wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proved for both semi‐discrete and full‐discrete LDG schemes. Numerical results are presented to justify the theoretical analysis and demonstrate the interesting wave concentration property by the electromagnetic concentrator.
{"title":"Analysis and application of a local discontinuous Galerkin method for the electromagnetic concentrator model","authors":"Yunqing Huang, Jichun Li, Xin Liu","doi":"10.1002/num.23069","DOIUrl":"https://doi.org/10.1002/num.23069","url":null,"abstract":"In this paper, we develop a local discontinuous Galerkin (LDG) method to simulate the wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proved for both semi‐discrete and full‐discrete LDG schemes. Numerical results are presented to justify the theoretical analysis and demonstrate the interesting wave concentration property by the electromagnetic concentrator.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42714198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we propose a second‐order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical scheme, we add some starting parts. In addition, we prove the stability and error estimation of the algorithm. Finally, we illustrate the effect of the starting parts and the accuracy of the numerical scheme through some numerical tests.
{"title":"Finite element algorithm with a second‐order shifted composite numerical integral formula for a nonlinear time fractional wave equation","authors":"Haoran Ren, Yang Liu, Baoli Yin, Haiyang Li","doi":"10.1002/num.23066","DOIUrl":"https://doi.org/10.1002/num.23066","url":null,"abstract":"In this article, we propose a second‐order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical scheme, we add some starting parts. In addition, we prove the stability and error estimation of the algorithm. Finally, we illustrate the effect of the starting parts and the accuracy of the numerical scheme through some numerical tests.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47824281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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