Pub Date : 2023-11-10DOI: 10.1080/00207160.2023.2279511
M. Yousuf, Shahzad Sarwar
AbstractA computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.Keywords: Inhomogeneous parabolic PDEsReal pole rational approximationComputationally fastfractional distributed order PDEsRiesz derivativeDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.
{"title":"A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations","authors":"M. Yousuf, Shahzad Sarwar","doi":"10.1080/00207160.2023.2279511","DOIUrl":"https://doi.org/10.1080/00207160.2023.2279511","url":null,"abstract":"AbstractA computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.Keywords: Inhomogeneous parabolic PDEsReal pole rational approximationComputationally fastfractional distributed order PDEsRiesz derivativeDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"121 31","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136450","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 : 2023-11-01DOI: 10.1080/00207160.2023.2279006
Shuaikang Wang, Yongbin Ge, Tingfu Ma
AbstractFirst, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.Keywords: Three-dimensional nonlinear wave equationNonlinear difference schemeSixth-order accuracyLinearized difference schemeRichardson extrapolationDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is partially supported by National Natural Science Foundation of China (12161067), Natural Science Foundation of Ningxia (2022AAC02023, 2022AAC03313), the Key Research and Development Program of Ningxia (2021YCZX0036, 2021BEB04053), the Scientific Research Program in Higher Institution of Ningxia (NGY2020110), National Youth Top-notch Talent Support Program of Ningxia.Data AvailabilityThe data used to support the findings of this study are available from the corresponding author upon request. Conflicts of InterestThe authors declare no conflict of interest.
{"title":"Sixth-order Finite Difference Schemes for Nonlinear Wave Equations with Variable Coefficients in Three Dimensions","authors":"Shuaikang Wang, Yongbin Ge, Tingfu Ma","doi":"10.1080/00207160.2023.2279006","DOIUrl":"https://doi.org/10.1080/00207160.2023.2279006","url":null,"abstract":"AbstractFirst, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.Keywords: Three-dimensional nonlinear wave equationNonlinear difference schemeSixth-order accuracyLinearized difference schemeRichardson extrapolationDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is partially supported by National Natural Science Foundation of China (12161067), Natural Science Foundation of Ningxia (2022AAC02023, 2022AAC03313), the Key Research and Development Program of Ningxia (2021YCZX0036, 2021BEB04053), the Scientific Research Program in Higher Institution of Ningxia (NGY2020110), National Youth Top-notch Talent Support Program of Ningxia.Data AvailabilityThe data used to support the findings of this study are available from the corresponding author upon request. Conflicts of InterestThe authors declare no conflict of interest.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"143 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135371205","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 : 2023-10-23DOI: 10.1080/00207160.2023.2274278
Hongling Shi, Minghui Song, Mingzhu Liu
AbstractThis paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.Keywords: Modified partially truncated EM methodstochastic differential equations with piecewise continuous argumentsstrong convergenceconvergence ratemean square exponential stabilityDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is supported by the NSF of PR China (No. 12071101 and No. 11671113).
{"title":"Convergence and stability of modified partially truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments","authors":"Hongling Shi, Minghui Song, Mingzhu Liu","doi":"10.1080/00207160.2023.2274278","DOIUrl":"https://doi.org/10.1080/00207160.2023.2274278","url":null,"abstract":"AbstractThis paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.Keywords: Modified partially truncated EM methodstochastic differential equations with piecewise continuous argumentsstrong convergenceconvergence ratemean square exponential stabilityDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is supported by the NSF of PR China (No. 12071101 and No. 11671113).","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"11 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135368481","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 : 2023-10-22DOI: 10.1080/00207160.2023.2274277
Zhaolin Yan, Jianfang Gao
AbstractThe purpose of this paper is to study oscillation and non-oscillation of Runge-Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge-Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function is discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge-Kutta method. To confirm the theoretical results, the numerical examples are given.Keywords: oscillationnumerical solutionRunge-Kutta methodsimpulsive delay differential equationspiecewise constant argumentsDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.
{"title":"Numerical oscillation and non-oscillation analysis of the mixed type impulsive differential equation with piecewise constant arguments","authors":"Zhaolin Yan, Jianfang Gao","doi":"10.1080/00207160.2023.2274277","DOIUrl":"https://doi.org/10.1080/00207160.2023.2274277","url":null,"abstract":"AbstractThe purpose of this paper is to study oscillation and non-oscillation of Runge-Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge-Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function is discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge-Kutta method. To confirm the theoretical results, the numerical examples are given.Keywords: oscillationnumerical solutionRunge-Kutta methodsimpulsive delay differential equationspiecewise constant argumentsDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"229 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135461885","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 : 2023-10-18DOI: 10.1080/00207160.2023.2272589
Fariba Bakrani Balani, Masoud Hajarian
AbstractWe introduce a new block preconditioner for the solution of weighted Toeplitz regularized least-squares problems written in augmented system form. The proposed preconditioner is obtained based on the new splitting of coefficient matrix which results in an unconditionally convergent stationary iterative method. Spectral analysis of the preconditioned matrix is investigated. In particular, we show that the preconditioned matrix has a very nice eigenvalue distribution which can lead to fast convergence of the preconditioned Krylov subspace methods such as GMRES. Numerical experiments are reported to demonstrate the performance of preconditioner used with (flexible) GMRES method in the solution of augmented system form of weighted Toeplitz regularized least-squares problems.Keywords: PreconditioningSplittingLeast-squares problemsWeighted Toeplitz matricesAMS classification 2010:: 65F1065F50DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe authors express their thanks to the referees for the comments and constructive suggestions, which were valuable in improving the quality of the manuscript.
{"title":"A new block preconditioner for weighted Toeplitz regularized least-squares problems","authors":"Fariba Bakrani Balani, Masoud Hajarian","doi":"10.1080/00207160.2023.2272589","DOIUrl":"https://doi.org/10.1080/00207160.2023.2272589","url":null,"abstract":"AbstractWe introduce a new block preconditioner for the solution of weighted Toeplitz regularized least-squares problems written in augmented system form. The proposed preconditioner is obtained based on the new splitting of coefficient matrix which results in an unconditionally convergent stationary iterative method. Spectral analysis of the preconditioned matrix is investigated. In particular, we show that the preconditioned matrix has a very nice eigenvalue distribution which can lead to fast convergence of the preconditioned Krylov subspace methods such as GMRES. Numerical experiments are reported to demonstrate the performance of preconditioner used with (flexible) GMRES method in the solution of augmented system form of weighted Toeplitz regularized least-squares problems.Keywords: PreconditioningSplittingLeast-squares problemsWeighted Toeplitz matricesAMS classification 2010:: 65F1065F50DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe authors express their thanks to the referees for the comments and constructive suggestions, which were valuable in improving the quality of the manuscript.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135889289","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 : 2023-10-10DOI: 10.1080/00207160.2023.2269430
Peng Wang, Detong Zhu
AbstractIn this paper, we propose a single timescale stochastic quasi-Newton method for solving the stochastic optimization problems. The objective function of the problem is a composition of two smooth functions and their derivatives are not available. The algorithm sets to approximate sequences to estimate the gradient of the composite objective function and the inner function. The matrix correction parameters are given in BFGS update form for avoiding the assumption that Hessian matrix of objective is positive definite. We show the global convergence of the algorithm. The algorithm achieves the complexity O(ϵ−1) to find an ϵ−approximate stationary point and ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance ϵ. The numerical results of nonconvex binary classification problem using the support vector machine and a multicall classification problem using neural networks are reported to show the effectiveness of the algorithm.Keywords: stochastic optimizationquasi-Newton methodBFGS update techniquemachine learning2010: 49M3765K0590C3090C56DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe author thanks the support of National Natural Science Foundation (11371253) and Hainan Natural Science Foundation (120MS029).
{"title":"A single timescale stochastic quasi-Newton method for stochastic optimization","authors":"Peng Wang, Detong Zhu","doi":"10.1080/00207160.2023.2269430","DOIUrl":"https://doi.org/10.1080/00207160.2023.2269430","url":null,"abstract":"AbstractIn this paper, we propose a single timescale stochastic quasi-Newton method for solving the stochastic optimization problems. The objective function of the problem is a composition of two smooth functions and their derivatives are not available. The algorithm sets to approximate sequences to estimate the gradient of the composite objective function and the inner function. The matrix correction parameters are given in BFGS update form for avoiding the assumption that Hessian matrix of objective is positive definite. We show the global convergence of the algorithm. The algorithm achieves the complexity O(ϵ−1) to find an ϵ−approximate stationary point and ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance ϵ. The numerical results of nonconvex binary classification problem using the support vector machine and a multicall classification problem using neural networks are reported to show the effectiveness of the algorithm.Keywords: stochastic optimizationquasi-Newton methodBFGS update techniquemachine learning2010: 49M3765K0590C3090C56DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe author thanks the support of National Natural Science Foundation (11371253) and Hainan Natural Science Foundation (120MS029).","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293592","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 : 2023-10-10DOI: 10.1080/00207160.2023.2269438
None Garima, Kapil K Sharma
AbstractThis article focuses on the investigation of two-dimensional elliptic singularly perturbed problems that incorporate positive and negative shifts, the solution of this class of problems may demonstrate regular/parabolic/degenerate or interior boundary layers. The goal of this article is to establish the development of numerical techniques for two-dimensional elliptic singularly perturbed problems with positive and negative shifts having regular boundary layers. The three numerical schemes are proposed to estimate the solution of this class of problems based on the fitted operator and fitted mesh finite-difference methods. The fitted operator finite difference method is analyzed for convergence. The effect of shift terms on the solution behavior is demonstrated through numerical experiments. The paper concludes by providing several numerical results that demonstrate the performance of these three numerical schemes.Keywords: Singularly perturbed problemDifferential-difference equationsUpwind SchemeHybrid SchemeFitted operator finite-difference methodDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe first author acknowledges the financial support received from the Council of Scientific and Industrial Research (File No.- 09/1112(0006)/2018-EMR-I) in the form of Senior Research Fellowship.Conflict of interestThe authors declare that they have no conflict of interest.
{"title":"The Robust Numerical Schemes for Two-Dimensional Elliptical Singularly Perturbed Problems with Space Shifts","authors":"None Garima, Kapil K Sharma","doi":"10.1080/00207160.2023.2269438","DOIUrl":"https://doi.org/10.1080/00207160.2023.2269438","url":null,"abstract":"AbstractThis article focuses on the investigation of two-dimensional elliptic singularly perturbed problems that incorporate positive and negative shifts, the solution of this class of problems may demonstrate regular/parabolic/degenerate or interior boundary layers. The goal of this article is to establish the development of numerical techniques for two-dimensional elliptic singularly perturbed problems with positive and negative shifts having regular boundary layers. The three numerical schemes are proposed to estimate the solution of this class of problems based on the fitted operator and fitted mesh finite-difference methods. The fitted operator finite difference method is analyzed for convergence. The effect of shift terms on the solution behavior is demonstrated through numerical experiments. The paper concludes by providing several numerical results that demonstrate the performance of these three numerical schemes.Keywords: Singularly perturbed problemDifferential-difference equationsUpwind SchemeHybrid SchemeFitted operator finite-difference methodDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe first author acknowledges the financial support received from the Council of Scientific and Industrial Research (File No.- 09/1112(0006)/2018-EMR-I) in the form of Senior Research Fellowship.Conflict of interestThe authors declare that they have no conflict of interest.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293741","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 : 2023-10-05DOI: 10.1080/00207160.2023.2265500
F. Afiatdoust, M. H. Heydari, M. M. Hosseini
AbstractIn this paper, a block-by-block scheme is proposed for a class of nonlinear fractional integro-differential equations. This method is based on the Gauss-Lobatto numerical integration method, which shows the high accuracy at all time intervals. Also, the method convergence for this type of equations is proved and it is shown that the order of convergence is at least eight. Finally, the high accuracy, fast calculations and good performance of the method are investigated by solving some numerical examples.Keywords: Nonlinear fractional integro-differentia equationsGauss-Lobatto quadrature ruleBlock-by-block methodDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.
{"title":"A block-by-block approach for nonlinear fractional integro-differential equations","authors":"F. Afiatdoust, M. H. Heydari, M. M. Hosseini","doi":"10.1080/00207160.2023.2265500","DOIUrl":"https://doi.org/10.1080/00207160.2023.2265500","url":null,"abstract":"AbstractIn this paper, a block-by-block scheme is proposed for a class of nonlinear fractional integro-differential equations. This method is based on the Gauss-Lobatto numerical integration method, which shows the high accuracy at all time intervals. Also, the method convergence for this type of equations is proved and it is shown that the order of convergence is at least eight. Finally, the high accuracy, fast calculations and good performance of the method are investigated by solving some numerical examples.Keywords: Nonlinear fractional integro-differentia equationsGauss-Lobatto quadrature ruleBlock-by-block methodDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134946922","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 : 2023-10-03DOI: 10.1080/00207160.2023.2266051
Lech A. Grzelak, Juliusz Jablecki, Dariusz Gatarek
AbstractThis paper studies equity basket options – i.e. multi-dimensional derivatives whose payoffs depend on the value of a weighted sum of the underlying stocks – and develops a new and innovative approach to ensure consistency between options on individual stocks and on the index comprising them. Specifically, we show how to resolve a well-known problem that when individual constituent distributions of an equity index are inferred from the single-stock option markets and combined in a multi-dimensional local/stochastic volatility model, the resulting basket option prices will not generate a skew matching that of the options on the equity index corresponding to the basket. To address this “insufficient skewness”, we proceed in two steps. First, we propose an “effective” local volatility model by mapping the general multi-dimensional basket onto a collection of marginal distributions. Second, we build a multivariate dependence structure between all the marginal distributions assuming a jump-diffusion model for the effective projection parameters, and show how to calibrate the basket to the index smile. Numerical tests and calibration exercises demonstrate an excellent fit for a basket of as many as 30 stocks with fast calculation time.Keywords: Basket OptionsIndex SkewMonte CarloLocal VolatilityStochastic VolatilityCollocation MethodsDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. Notes1 Cf. the prospectus availible in the online records of the U.S. Securities and Exchange Commission at: https://www.sec.gov/Archives/edgar/data/19617/000089109221003578/e13291-424b2.htm2 For example, in the Bloomberg basket options pricing template correlations are, by default, estimated over a 5 year period, whereby to eliminate noise, a given percentile of rolling 6-month cross-correlation estimates is chosen in the parameterization of the full correlation matrix.3 We define the skew here loosely as the difference in implied volatilities between the 85-120% ATM levels.4 As an alternative to [27] one could consider Kou's jump-diffusion model [18] which has the additional benefit of separating the upside and downside skew. However, in this case, we opt for the simplicity and parsimony of Merton's approach5 Without loss of generality, we shall henceforth think of the underlying assets as stocks, however the method developed below is obviously general and, mutatis mutandis, applies to other instruments as well.6 The proposed framework can also be extended with a stochastic volatility process. Such an extension is trivial and will, for simplicity, be omitted.7 The respective dynami
{"title":"Efficient Pricing and Calibration of High-Dimensional Basket Options","authors":"Lech A. Grzelak, Juliusz Jablecki, Dariusz Gatarek","doi":"10.1080/00207160.2023.2266051","DOIUrl":"https://doi.org/10.1080/00207160.2023.2266051","url":null,"abstract":"AbstractThis paper studies equity basket options – i.e. multi-dimensional derivatives whose payoffs depend on the value of a weighted sum of the underlying stocks – and develops a new and innovative approach to ensure consistency between options on individual stocks and on the index comprising them. Specifically, we show how to resolve a well-known problem that when individual constituent distributions of an equity index are inferred from the single-stock option markets and combined in a multi-dimensional local/stochastic volatility model, the resulting basket option prices will not generate a skew matching that of the options on the equity index corresponding to the basket. To address this “insufficient skewness”, we proceed in two steps. First, we propose an “effective” local volatility model by mapping the general multi-dimensional basket onto a collection of marginal distributions. Second, we build a multivariate dependence structure between all the marginal distributions assuming a jump-diffusion model for the effective projection parameters, and show how to calibrate the basket to the index smile. Numerical tests and calibration exercises demonstrate an excellent fit for a basket of as many as 30 stocks with fast calculation time.Keywords: Basket OptionsIndex SkewMonte CarloLocal VolatilityStochastic VolatilityCollocation MethodsDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. Notes1 Cf. the prospectus availible in the online records of the U.S. Securities and Exchange Commission at: https://www.sec.gov/Archives/edgar/data/19617/000089109221003578/e13291-424b2.htm2 For example, in the Bloomberg basket options pricing template correlations are, by default, estimated over a 5 year period, whereby to eliminate noise, a given percentile of rolling 6-month cross-correlation estimates is chosen in the parameterization of the full correlation matrix.3 We define the skew here loosely as the difference in implied volatilities between the 85-120% ATM levels.4 As an alternative to [27] one could consider Kou's jump-diffusion model [18] which has the additional benefit of separating the upside and downside skew. However, in this case, we opt for the simplicity and parsimony of Merton's approach5 Without loss of generality, we shall henceforth think of the underlying assets as stocks, however the method developed below is obviously general and, mutatis mutandis, applies to other instruments as well.6 The proposed framework can also be extended with a stochastic volatility process. Such an extension is trivial and will, for simplicity, be omitted.7 The respective dynami","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135696060","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}
AbstractIn this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal L2 error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings.Keywords: Virtual element methodnonlinearvariable-order fractional equationtwo-gridpolygonal meshesa priori error estimateMathematics subject classifications: 65M6065N3034K3765M1565M55 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work is supported by the State Key Program of National Natural Science Foundation of China [grant number 11931003] and National Natural Science Foundation of China [grant number 41974133], Hunan Provincial Innovation Foundation for Postgraduate, China [grant number XDCX2021B098], Postgraduate Scientific Research Innovation Project of Hunan Province [grant number CX20210597].
{"title":"A two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes","authors":"Qiling Gu, Yanping Chen, Jianwei Zhou, Yunqing Huang","doi":"10.1080/00207160.2023.2263589","DOIUrl":"https://doi.org/10.1080/00207160.2023.2263589","url":null,"abstract":"AbstractIn this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal L2 error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings.Keywords: Virtual element methodnonlinearvariable-order fractional equationtwo-gridpolygonal meshesa priori error estimateMathematics subject classifications: 65M6065N3034K3765M1565M55 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work is supported by the State Key Program of National Natural Science Foundation of China [grant number 11931003] and National Natural Science Foundation of China [grant number 41974133], Hunan Provincial Innovation Foundation for Postgraduate, China [grant number XDCX2021B098], Postgraduate Scientific Research Innovation Project of Hunan Province [grant number CX20210597].","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135744159","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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