Pub Date : 2023-06-01DOI: 10.1080/00207160.2023.2214643
Sayed Arsalan Sajjadi, H. Najafi, H. Aminikhah
In this paper, we introduce a new fractional basis function based on Lagrange polynomials. We define the new interpolation formula for approximation of the solutions of the second kind weakly singular Volterra integral equations. The product integration method is used for the numerical solution of these equations based on Jacobi polynomials. It is known that the weakly singular Volterra integral equations typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. We use the suitable transformations to overcome this non-smooth behaviour. An upper error bound of the proposed method is determined and the convergence analysis is discussed. Finally, some numerical examples with non-smooth solutions are prepared to test the efficiency and accuracy of the method.
{"title":"Convergence analysis of a novel fractional product integration method for solving the second kind weakly singular Volterra integral equations with non-smooth solutions based on Jacobi polynomials","authors":"Sayed Arsalan Sajjadi, H. Najafi, H. Aminikhah","doi":"10.1080/00207160.2023.2214643","DOIUrl":"https://doi.org/10.1080/00207160.2023.2214643","url":null,"abstract":"In this paper, we introduce a new fractional basis function based on Lagrange polynomials. We define the new interpolation formula for approximation of the solutions of the second kind weakly singular Volterra integral equations. The product integration method is used for the numerical solution of these equations based on Jacobi polynomials. It is known that the weakly singular Volterra integral equations typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. We use the suitable transformations to overcome this non-smooth behaviour. An upper error bound of the proposed method is determined and the convergence analysis is discussed. Finally, some numerical examples with non-smooth solutions are prepared to test the efficiency and accuracy of the method.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"1 1","pages":"1794 - 1808"},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86464010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-27DOI: 10.1080/00207160.2023.2217303
Kanyanee Saechou, A. Kangtunyakarn
The purpose of this research is to study the generalized modification of the system of equilibrium problems (GMSEP) and a lemma is established to show the property of this problem. Then, we prove a strong convergence theorem for finding a common element of the set of the solutions of the fixed points problem and the set of the solutions of the GMSEP under some suitable conditions, in which , where are coefficients in the main iteration. Moreover, we prove strong convergence theorems for finding solutions to the generalized equilibrium problem, the system of equilibrium problems, the variational inequality problem, the general system of variational inequality problems, and the minimization problem. Finally, we give two numerical examples, one of which shows the rate of convergence of the main iteration while the other shows the rate of convergence of the main iteration but the sum of coefficients equals 1.
{"title":"Approximating solutions of the generalized modification of the system of equilibrium problems and fixed point problem of a nonexpansive mapping","authors":"Kanyanee Saechou, A. Kangtunyakarn","doi":"10.1080/00207160.2023.2217303","DOIUrl":"https://doi.org/10.1080/00207160.2023.2217303","url":null,"abstract":"The purpose of this research is to study the generalized modification of the system of equilibrium problems (GMSEP) and a lemma is established to show the property of this problem. Then, we prove a strong convergence theorem for finding a common element of the set of the solutions of the fixed points problem and the set of the solutions of the GMSEP under some suitable conditions, in which , where are coefficients in the main iteration. Moreover, we prove strong convergence theorems for finding solutions to the generalized equilibrium problem, the system of equilibrium problems, the variational inequality problem, the general system of variational inequality problems, and the minimization problem. Finally, we give two numerical examples, one of which shows the rate of convergence of the main iteration while the other shows the rate of convergence of the main iteration but the sum of coefficients equals 1.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"10 29 1","pages":"1821 - 1838"},"PeriodicalIF":1.8,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88731623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-26DOI: 10.1080/00207160.2023.2219349
Chen J. Wang, Houping Dai, Meng-jun Li, Yinghua Feng
This paper investigates a class of (2 + 1)-dimensional coupled nonlinear evolution equation with time-dependent coefficients in an inhomogeneous medium via the Hirota bilinear method. Combining the long wave limit method and complex conjugate transform, the higher-order breather and lump solutions are initially constructed. Furthermore, hybrid solutions among N-soliton, lump and breather solutions are derived by linear constraints on the parameters. Meanwhile, the dynamic evolution behaviour of some special concrete solutions under different time-dependent coefficients is presented visually in the form of images.
{"title":"Higher-order breather, lump and hybrid solutions of (2 + 1)-dimensional coupled nonlinear evolution equations with time-dependent coefficients","authors":"Chen J. Wang, Houping Dai, Meng-jun Li, Yinghua Feng","doi":"10.1080/00207160.2023.2219349","DOIUrl":"https://doi.org/10.1080/00207160.2023.2219349","url":null,"abstract":"This paper investigates a class of (2 + 1)-dimensional coupled nonlinear evolution equation with time-dependent coefficients in an inhomogeneous medium via the Hirota bilinear method. Combining the long wave limit method and complex conjugate transform, the higher-order breather and lump solutions are initially constructed. Furthermore, hybrid solutions among N-soliton, lump and breather solutions are derived by linear constraints on the parameters. Meanwhile, the dynamic evolution behaviour of some special concrete solutions under different time-dependent coefficients is presented visually in the form of images.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"21 1","pages":"1860 - 1876"},"PeriodicalIF":1.8,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81464449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-25DOI: 10.1080/00207160.2023.2217307
Eva G. Villalba, J. L. Hueso, E. Martínez
It is well known that the Steffensen-type methods approximate the derivative appearing in Newton's scheme by means of the first-order divided difference operator. The generalized multistep Steffensen iterative method consists of composing the method with itself m times. Specifically, the divided difference is held constant for every m steps before it is updated. In this work, we introduce a modification to this method, in order to accelerate the convergence order. In the proposed, scheme we compute the divided differences in first and second step and use the divided difference from the second step in the following m−1 steps. We perform an exhaustive study of the computational efficiency of this scheme and also introduce memory to this method to speed up convergence without performing new functional evaluations. Finally, some numerical examples are studied to verify the usefulness of these algorithms.
{"title":"Generalized multistep Steffensen iterative method. Solving the model of a photomultiplier device","authors":"Eva G. Villalba, J. L. Hueso, E. Martínez","doi":"10.1080/00207160.2023.2217307","DOIUrl":"https://doi.org/10.1080/00207160.2023.2217307","url":null,"abstract":"It is well known that the Steffensen-type methods approximate the derivative appearing in Newton's scheme by means of the first-order divided difference operator. The generalized multistep Steffensen iterative method consists of composing the method with itself m times. Specifically, the divided difference is held constant for every m steps before it is updated. In this work, we introduce a modification to this method, in order to accelerate the convergence order. In the proposed, scheme we compute the divided differences in first and second step and use the divided difference from the second step in the following m−1 steps. We perform an exhaustive study of the computational efficiency of this scheme and also introduce memory to this method to speed up convergence without performing new functional evaluations. Finally, some numerical examples are studied to verify the usefulness of these algorithms.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"24 1","pages":"1839 - 1859"},"PeriodicalIF":1.8,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81272053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-22DOI: 10.1080/00207160.2023.2217302
Le Jiang, Cheng-long Xu
This paper presents a unified semi-stochastic kernel regression method for pricing options under general stochastic volatility model. The method combines semi-stochastic sampling for initial asset values with Monte Carlo simulations to construct a least-squares based kernel function regression solution. This approach can not only approximates option prices, but also determines the Greeks of option. The least square problem is augmented with weighted derivative constraints, which enables flexible adjustment of approximate errors for both option prices and Greeks. Numerical results show the efficiency of the proposed method for the Vanilla option and some exotic options: Asian option, Lookback option, discretely monitored Barrier option and the Basket option with several assets under the stochastic volatility model.
{"title":"A new options pricing method: semi-stochastic kernel regression method with constraints","authors":"Le Jiang, Cheng-long Xu","doi":"10.1080/00207160.2023.2217302","DOIUrl":"https://doi.org/10.1080/00207160.2023.2217302","url":null,"abstract":"This paper presents a unified semi-stochastic kernel regression method for pricing options under general stochastic volatility model. The method combines semi-stochastic sampling for initial asset values with Monte Carlo simulations to construct a least-squares based kernel function regression solution. This approach can not only approximates option prices, but also determines the Greeks of option. The least square problem is augmented with weighted derivative constraints, which enables flexible adjustment of approximate errors for both option prices and Greeks. Numerical results show the efficiency of the proposed method for the Vanilla option and some exotic options: Asian option, Lookback option, discretely monitored Barrier option and the Basket option with several assets under the stochastic volatility model.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"16 1","pages":"1809 - 1820"},"PeriodicalIF":1.8,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78614667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-16DOI: 10.1080/00207160.2023.2214641
Xiaoqin Shen, R. Wu, Shengfeng Zhu
ABSTRACT In this paper, we propose an efficient numerical method for the two-dimensional (2D) linearly elastic clamped plate model. Specifically, we choose the linear Lagrange element to discretize the first two components in the vector displacement space, as well as the Morley element to discretize the third component. Moreover, the existence of discrete solution, uniqueness and a-priori error estimate is analysed. Finally, numerical experiments are presented to verify theoretical results.
{"title":"Numerical method for two-dimensional linearly elastic clamped plate model","authors":"Xiaoqin Shen, R. Wu, Shengfeng Zhu","doi":"10.1080/00207160.2023.2214641","DOIUrl":"https://doi.org/10.1080/00207160.2023.2214641","url":null,"abstract":"ABSTRACT In this paper, we propose an efficient numerical method for the two-dimensional (2D) linearly elastic clamped plate model. Specifically, we choose the linear Lagrange element to discretize the first two components in the vector displacement space, as well as the Morley element to discretize the third component. Moreover, the existence of discrete solution, uniqueness and a-priori error estimate is analysed. Finally, numerical experiments are presented to verify theoretical results.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"37 1","pages":"1779 - 1793"},"PeriodicalIF":1.8,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89153075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-14DOI: 10.1080/00207160.2023.2214242
X. Liu, Z. Yang, Y. Zeng
This paper deals with the numerical properties of a reaction-diffusion susceptible infected susceptible (SIS) epidemic model under a linear external source. A numerical scheme is constructed with a finite difference scheme for the space discretization and an Implicit-Explicit (IMEX) method in time integration. A threshold value, numerical basic reproduction number, is proposed in the long-time stability analysis of numerical solutions. Differently from previous works on the same model, the numerical basic reproduction number can preserve the behaviours of the basic reproduction number of the model, towards which it converges when the spatial stepsize vanishes. Moreover, it plays a role for the discrete dynamics similar to the one played by its continuous counterpart. Some numerical experiments are given in the end to confirm the conclusions and detect the conjecture on the stability of endemic equilibrium (EE) in general case.
{"title":"Long-time numerical properties analysis of a diffusive SIS epidemic model under a linear external source","authors":"X. Liu, Z. Yang, Y. Zeng","doi":"10.1080/00207160.2023.2214242","DOIUrl":"https://doi.org/10.1080/00207160.2023.2214242","url":null,"abstract":"This paper deals with the numerical properties of a reaction-diffusion susceptible infected susceptible (SIS) epidemic model under a linear external source. A numerical scheme is constructed with a finite difference scheme for the space discretization and an Implicit-Explicit (IMEX) method in time integration. A threshold value, numerical basic reproduction number, is proposed in the long-time stability analysis of numerical solutions. Differently from previous works on the same model, the numerical basic reproduction number can preserve the behaviours of the basic reproduction number of the model, towards which it converges when the spatial stepsize vanishes. Moreover, it plays a role for the discrete dynamics similar to the one played by its continuous counterpart. Some numerical experiments are given in the end to confirm the conclusions and detect the conjecture on the stability of endemic equilibrium (EE) in general case.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"94 1","pages":"1737 - 1756"},"PeriodicalIF":1.8,"publicationDate":"2023-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88112947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-08DOI: 10.1080/00207160.2023.2212089
Hong Du, X. Yang, Zhong Chen
A new method of solving the best approximate solution for nonlinear fractional equations with smooth and nonsmooth solutions in reproducing kernel space is proposed in the paper. The nonlinear equation outlines some important equations, such as fractional diffusion-wave equation, nonlinear Klein–Gordon equation and time-fractional sine-Gordon equation. By constructing orthonormal bases in reproducing kernel space using Legendre orthonormal polynomials and Jacobi fractional orthonormal polynomials, the best approximate solution is obtained by searching the minimum of residue in the sense of . Numerical experiments verify that the method has higher accuracy.
{"title":"A new method of solving the best approximate solution for a nonlinear fractional equation","authors":"Hong Du, X. Yang, Zhong Chen","doi":"10.1080/00207160.2023.2212089","DOIUrl":"https://doi.org/10.1080/00207160.2023.2212089","url":null,"abstract":"A new method of solving the best approximate solution for nonlinear fractional equations with smooth and nonsmooth solutions in reproducing kernel space is proposed in the paper. The nonlinear equation outlines some important equations, such as fractional diffusion-wave equation, nonlinear Klein–Gordon equation and time-fractional sine-Gordon equation. By constructing orthonormal bases in reproducing kernel space using Legendre orthonormal polynomials and Jacobi fractional orthonormal polynomials, the best approximate solution is obtained by searching the minimum of residue in the sense of . Numerical experiments verify that the method has higher accuracy.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"12 1","pages":"1702 - 1718"},"PeriodicalIF":1.8,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84978136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-08DOI: 10.1080/00207160.2023.2212307
Ziyi Zhou, Haixiang Zhang, Xuehua Yang, Jie Tang
The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.
{"title":"An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions","authors":"Ziyi Zhou, Haixiang Zhang, Xuehua Yang, Jie Tang","doi":"10.1080/00207160.2023.2212307","DOIUrl":"https://doi.org/10.1080/00207160.2023.2212307","url":null,"abstract":"The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"7 1","pages":"1719 - 1736"},"PeriodicalIF":1.8,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87934627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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