Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.10.017
Guo Jiang, Yuanqin Chen, Jiayi Ying
This paper presents an efficient numerical method for solving nonlinear stochastic delay integro-differential equations based on block pulse functions. Firstly, the equation is transformed into an algebraic system by the integral delay operator matrixes of block pulse functions. Then, error analysis is conducted on the method. Finally, some numerical examples are provided to validate the method. This work provides numerical solutions for the stochastic delay integro-differential equations by global approximation method. This method has the advantages of simple calculation and higher error accuracy.
{"title":"Numerical solutions of stochastic delay integro-differential equations by block pulse functions","authors":"Guo Jiang, Yuanqin Chen, Jiayi Ying","doi":"10.1016/j.apnum.2024.10.017","DOIUrl":"10.1016/j.apnum.2024.10.017","url":null,"abstract":"<div><div>This paper presents an efficient numerical method for solving nonlinear stochastic delay integro-differential equations based on block pulse functions. Firstly, the equation is transformed into an algebraic system by the integral delay operator matrixes of block pulse functions. Then, error analysis is conducted on the method. Finally, some numerical examples are provided to validate the method. This work provides numerical solutions for the stochastic delay integro-differential equations by global approximation method. This method has the advantages of simple calculation and higher error accuracy.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 214-230"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141228","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}
For a linear time-invariant system , the Kronecker canonical form (KCF) of the matrix pencil provides the controllability indices, also called column minimal indices, of the system and their sum corresponds to the dimension of the controllable subspace. In this paper we introduce a fast numerical algorithm for computing the sets of column/row minimal indices of a singular pencil using a rank-updating technique and the properties of piecewise arithmetic progression sequences defined by the size of the null spaces of appropriate Toeplitz matrices. The method is demonstrated and tested on various data sets.
{"title":"A rank-updating technique for the Kronecker canonical form of singular pencils","authors":"Dimitrios Christou , Marilena Mitrouli , Dimitrios Triantafyllou","doi":"10.1016/j.apnum.2024.01.015","DOIUrl":"10.1016/j.apnum.2024.01.015","url":null,"abstract":"<div><div>For a linear time-invariant system <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>A</mi><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>B</mi><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span><span>, the Kronecker canonical form<span> (KCF) of the matrix pencil </span></span><span><math><mo>(</mo><mi>s</mi><mi>I</mi><mo>−</mo><mi>A</mi><mspace></mspace><mo>|</mo><mspace></mspace><mi>B</mi><mo>)</mo></math></span><span> provides the controllability indices, also called column minimal indices, of the system and their sum corresponds to the dimension of the controllable subspace. In this paper we introduce a fast numerical algorithm for computing the sets of column/row minimal indices of a singular pencil </span><span><math><mi>s</mi><mi>F</mi><mo>−</mo><mi>G</mi></math></span><span> using a rank-updating technique and the properties of piecewise arithmetic progression sequences defined by the size of the null spaces of appropriate Toeplitz matrices. The method is demonstrated and tested on various data sets.</span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 135-145"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557855","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.2024.04.002
S. Elgharbi , M. Essaouini , B. Abouzaid , H. Safouhi
Over the last four decades, Sinc methods have occupied an important place in numerical analysis due to their simplicity and great performance. An incorporation of the Sinc collocation method with double exponential transformation is used to solve the two-dimensional time dependent Schrödinger equation. Numerical comparison between the double exponential and single exponential approaches is made to illustrate the superiority of the double exponential Sinc method.
{"title":"Solving the two-dimensional time-dependent Schrödinger equation using the Sinc collocation method and double exponential transformations","authors":"S. Elgharbi , M. Essaouini , B. Abouzaid , H. Safouhi","doi":"10.1016/j.apnum.2024.04.002","DOIUrl":"10.1016/j.apnum.2024.04.002","url":null,"abstract":"<div><div>Over the last four decades, Sinc methods have occupied an important place in numerical analysis due to their simplicity and great performance. An incorporation of the Sinc collocation method with double exponential transformation is used to solve the two-dimensional time dependent Schrödinger equation. Numerical comparison between the double exponential and single exponential approaches is made to illustrate the superiority of the double exponential Sinc method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 222-231"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140772854","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.2024.01.001
Alessandro Buccini , Fei Chen , Omar De la Cruz Cabrera , Lothar Reichel
This paper discusses and develops new methods for fitting trigonometric curves, such as circles, ellipses, and dumbbells, to data points in the plane. Available methods for fitting circles or ellipses are very sensitive to outliers in the data, and are time consuming when the number of data points is large. The present paper focuses on curve fitting methods that are attractive to use when the number of data points is large. We propose a direct method for fitting circles, and two iterative methods for fitting ellipses and dumbbell curves based on trigonometric polynomials. These methods efficiently minimize the sum of the squared geometric distances between the given data points and the fitted curves. In particular, we are interested in detecting the general shape of an object such as a galaxy or a nebula. Certain nebulae, for instance, the one shown in the experiment section, have a dumbbell shape. Methods for fitting dumbbell curves have not been discussed in the literature. The methods developed are not very sensitive to errors in the data points. The use of random subsampling of the data points to speed up the computations also is discussed. The techniques developed in this paper can be applied to fitting other kinds of curves as well.
{"title":"Fast alternating fitting methods for trigonometric curves for large data sets","authors":"Alessandro Buccini , Fei Chen , Omar De la Cruz Cabrera , Lothar Reichel","doi":"10.1016/j.apnum.2024.01.001","DOIUrl":"10.1016/j.apnum.2024.01.001","url":null,"abstract":"<div><div>This paper discusses and develops new methods for fitting trigonometric curves, such as circles, ellipses, and dumbbells, to data points in the plane. Available methods for fitting circles or ellipses are very sensitive to outliers in the data, and are time consuming when the number of data points is large. The present paper focuses on curve fitting methods that are attractive to use when the number of data points is large. We propose a direct method for fitting circles, and two iterative methods for fitting ellipses and dumbbell curves based on trigonometric polynomials. These methods efficiently minimize the sum of the squared geometric distances between the given data points and the fitted curves. In particular, we are interested in detecting the general shape of an object such as a galaxy or a nebula. Certain nebulae, for instance, the one shown in the experiment section, have a dumbbell shape. Methods for fitting dumbbell curves have not been discussed in the literature. The methods developed are not very sensitive to errors in the data points. The use of random subsampling of the data points to speed up the computations also is discussed. The techniques developed in this paper can be applied to fitting other kinds of curves as well.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 104-134"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139464058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.03.003
H. Alqahtani , C.F. Borges , D.Lj. Djukić , R.M. Mutavdžić Djukić , L. Reichel , M.M. Spalević
The evaluation of Gauss-type quadrature rules is an important topic in scientific computing. To determine estimates or bounds for the quadrature error of a Gauss rule often another related quadrature rule is evaluated, such as an associated Gauss-Radau or Gauss-Lobatto rule, an anti-Gauss rule, an averaged rule, an optimal averaged rule, or a Gauss-Kronrod rule when the latter exists. We discuss how pairs of a Gauss rule and a related Gauss-type quadrature rule can be computed efficiently by a divide-and-conquer method.
{"title":"Computation of pairs of related Gauss-type quadrature rules","authors":"H. Alqahtani , C.F. Borges , D.Lj. Djukić , R.M. Mutavdžić Djukić , L. Reichel , M.M. Spalević","doi":"10.1016/j.apnum.2024.03.003","DOIUrl":"10.1016/j.apnum.2024.03.003","url":null,"abstract":"<div><div><span>The evaluation of Gauss-type quadrature rules is an important topic in scientific computing. To determine estimates or bounds for the quadrature error of a </span>Gauss rule often another related quadrature rule is evaluated, such as an associated Gauss-Radau or Gauss-Lobatto rule, an anti-Gauss rule, an averaged rule, an optimal averaged rule, or a Gauss-Kronrod rule when the latter exists. We discuss how pairs of a Gauss rule and a related Gauss-type quadrature rule can be computed efficiently by a divide-and-conquer method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 32-42"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125590","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.2024.10.018
F. Hecht, S.-M. Kaber
New approximations of the matrix φ functions are developed. These approximations are rational functions of a specific form allowing simple and accurate schemes for linear systems. Furthermore, these approximations are fully parallelizable. Several tests show the efficiency of the method and its good parallelization properties.
{"title":"A parallel exponential integrator scheme for linear differential equations","authors":"F. Hecht, S.-M. Kaber","doi":"10.1016/j.apnum.2024.10.018","DOIUrl":"10.1016/j.apnum.2024.10.018","url":null,"abstract":"<div><div>New approximations of the matrix <em>φ</em> functions are developed. These approximations are rational functions of a specific form allowing simple and accurate schemes for linear systems. Furthermore, these approximations are fully parallelizable. Several tests show the efficiency of the method and its good parallelization properties.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 356-364"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141231","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.2024.10.006
Sohrab Bazm , Pedro Lima , Somayeh Nemati
In this paper, a class of nonlinear two-dimensional (2D) integral equations of Volterra type, i.e. Volterra-Urysohn integral equations, is studied. Following the ideas of [24], and assuming that the kernels of the integral equation are Lipschitz functions with respect to the dependent variable, the existence and uniqueness of a solution to the integral equation is shown by a technique based on the Picard iterative method. Then, the Euler and trapezoidal discretization methods are used to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. It is proved that the solution of the Euler method has first order convergence to the exact solution of the integral equation while the solution of the trapezoidal method has quadratic convergence. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Some numerical examples are given which confirm our theoretical results.
{"title":"Discretization methods and their extrapolations for two-dimensional nonlinear Volterra-Urysohn integral equations","authors":"Sohrab Bazm , Pedro Lima , Somayeh Nemati","doi":"10.1016/j.apnum.2024.10.006","DOIUrl":"10.1016/j.apnum.2024.10.006","url":null,"abstract":"<div><div>In this paper, a class of nonlinear two-dimensional (2D) integral equations of Volterra type, i.e. Volterra-Urysohn integral equations, is studied. Following the ideas of <span><span>[24]</span></span>, and assuming that the kernels of the integral equation are Lipschitz functions with respect to the dependent variable, the existence and uniqueness of a solution to the integral equation is shown by a technique based on the Picard iterative method. Then, the Euler and trapezoidal discretization methods are used to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. It is proved that the solution of the Euler method has first order convergence to the exact solution of the integral equation while the solution of the trapezoidal method has quadratic convergence. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Some numerical examples are given which confirm our theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 323-337"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141229","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.2024.01.022
Rawaa Awada , Jérôme Carrayrou , Carole Rosier
In this paper, we study theoretically and numerically the Anderson acceleration method. First, we extend the convergence results of Anderson's method for a small depth to general nonlinear cases. More precisely, we prove that the Type-I and Type-II Anderson(1) are locally q-linearly convergent if the fixed point map is a contraction with a Lipschitz constant small enough. We then illustrate the effectiveness of the method by applying it to the resolution of chemical equilibria. This test case has been identified as a challenging one because of the high nonlinearity of the chemical system and stiffness of the transport phenomena. The Newton method (usually Newton-Raphson) has been adopted by quite all the equilibrium and reactive transport codes. But the often ill-conditioned Jacobian matrix and the choice of a bad initial data can lead to convergence problems, especially if solute transport produces sharp concentrations profiles. Here we propose to combine the Anderson acceleration method with a particular formulation of the equilibrium system called the method of positive continued fractions (usually used as preconditioning). As shown by the numerical simulations, this approach makes it possible to considerably improve the robustness of the resolution of chemical equilibria algorithms, especially since it is coupled with a strategy to monitor the depth of the Anderson acceleration method in order to control the condition number.
本文从理论和数值上研究了安德森加速法。首先，我们将小深度 Anderson 方法的收敛结果扩展到一般非线性情况。更准确地说，我们证明了如果定点映射是一个具有足够小的 Lipschitz 常量的收缩，那么第一类和第二类 Anderson(1) 是局部 q 线性收敛的。然后，我们将该方法应用于化学平衡的解析，以说明其有效性。由于化学系统的高度非线性和输运现象的刚性，这个测试案例被认为是一个具有挑战性的案例。牛顿法（通常为 Newton-Raphson）已被相当多的平衡和反应输运代码所采用。但是，雅各布矩阵通常条件不佳，初始数据选择不当，会导致收敛问题，尤其是当溶质迁移产生尖锐的浓度曲线时。在此，我们建议将安德森加速法与平衡系统的一种特殊表述相结合，这种表述被称为正续分数法（通常用作预处理）。正如数值模拟所显示的那样，这种方法可以大大提高化学平衡算法分辨率的稳健性，特别是因为它与一种监测安德森加速方法深度的策略相结合，以控制条件数。
{"title":"Anderson acceleration. Convergence analysis and applications to equilibrium chemistry","authors":"Rawaa Awada , Jérôme Carrayrou , Carole Rosier","doi":"10.1016/j.apnum.2024.01.022","DOIUrl":"10.1016/j.apnum.2024.01.022","url":null,"abstract":"<div><div><span><span>In this paper, we study theoretically and numerically the Anderson acceleration method. First, we extend the convergence results of Anderson's method for a small depth to general nonlinear cases. More precisely, we prove that the Type-I and Type-II Anderson(1) are locally q-linearly convergent if the </span>fixed point<span> map is a contraction with a Lipschitz constant small enough. We then illustrate the effectiveness of the method by applying it to the resolution of chemical equilibria. This test case has been identified as a challenging one because of the high nonlinearity of the chemical system and stiffness of the transport phenomena. The Newton method (usually Newton-Raphson) has been adopted by quite all the equilibrium and reactive transport codes. But the often ill-conditioned </span></span>Jacobian matrix<span><span> and the choice of a bad initial data can lead to </span>convergence problems<span><span>, especially if solute transport produces sharp concentrations profiles. Here we propose to combine the Anderson acceleration method with a particular formulation of the equilibrium system called the method of positive continued fractions (usually used as preconditioning). As shown by the numerical simulations, this approach makes it possible to considerably improve the robustness of the resolution of chemical equilibria algorithms, especially since it is coupled with a strategy to monitor the depth of the Anderson acceleration method in order to control the </span>condition number.</span></span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 60-75"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657122","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.2024.03.012
M.A. Rahhali , T. Garcia , P. Spiteri
The present paper deals with the resolution on cloud architecture of synchronous and asynchronous iterative parallel algorithms of stationary or evolution variational inequations formulated by a multivalued model. The performances of synchronous and asynchronous iterative parallel methods are compared with previous ones obtained on cluster or when grid architecture is used. Thanks to the properties of the algebraic systems resulting from problem discretization we are able to analyze the behavior of the iterative algorithm in particular the convergence and the speed of convergence. The implementation of the studied methods on cloud architecture is described. Then we present various applications in particular the solidification of steel in continuous casting, the cavity pressure calculation described by a problem subject to unilateral constraints and finally a financial problem modeled by American option pricing.
{"title":"Parallel cloud solution of large algebraic multivalued systems","authors":"M.A. Rahhali , T. Garcia , P. Spiteri","doi":"10.1016/j.apnum.2024.03.012","DOIUrl":"10.1016/j.apnum.2024.03.012","url":null,"abstract":"<div><div><span>The present paper deals with the resolution on cloud architecture of synchronous and asynchronous iterative parallel algorithms of stationary or evolution variational inequations formulated by a multivalued model. The performances of synchronous and asynchronous iterative parallel methods are compared with previous ones obtained on cluster or when grid architecture is used. Thanks to the properties of the algebraic systems resulting from problem discretization we are able to analyze the behavior of the iterative algorithm in particular the convergence and the speed of convergence. The implementation of the studied methods on cloud architecture is described. Then we present various applications in particular the solidification of steel in </span>continuous casting<span>, the cavity pressure calculation described by a problem subject to unilateral constraints and finally a financial problem modeled by American option pricing.</span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 366-389"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196987","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.2024.09.021
V.J. Bevia, S. Blanes, J.C. Cortés, N. Kopylov, R.J. Villanueva
This work presents and analyzes a numerical approach to efficiently solve the Liouville equation in the context of random ODEs using GPGPUs. Our method combines wavelet compression-based adaptive mesh refinement, Lagrangian particle methods, and radial basis function interpolation to create a versatile algorithm applicable in multiple dimensions. We discuss the advantages and limitations of this algorithm. To demonstrate its effectiveness, we compute the probability density function for various 2D and 3D random ODE systems with applications in physics and epidemiology.
{"title":"A GPU-accelerated Lagrangian method for solving the Liouville equation in random differential equation systems","authors":"V.J. Bevia, S. Blanes, J.C. Cortés, N. Kopylov, R.J. Villanueva","doi":"10.1016/j.apnum.2024.09.021","DOIUrl":"10.1016/j.apnum.2024.09.021","url":null,"abstract":"<div><div>This work presents and analyzes a numerical approach to efficiently solve the Liouville equation in the context of random ODEs using GPGPUs. Our method combines wavelet compression-based adaptive mesh refinement, Lagrangian particle methods, and radial basis function interpolation to create a versatile algorithm applicable in multiple dimensions. We discuss the advantages and limitations of this algorithm. To demonstrate its effectiveness, we compute the probability density function for various 2D and 3D random ODE systems with applications in physics and epidemiology.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 231-255"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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