Pub Date : 2023-08-15DOI: 10.1080/00207160.2023.2248304
M. T. Hoang
In this paper, we extend the Mickens' methodology to construct a second-order nonstandard finite difference (NSFD) method, which preserves dynamical properties including positivity, local asymptotic stability and especially, global asymptotic stability of a general single-species model. This NSFD method is based on a novel weighted non-local approximation of the right-hand side function in combination with the renormalization of the denominator function. The weight guarantees the dynamic consistency and the nonstandard denominator function ensures the convergence of order 2 of the NSFD method. The result is that we obtain a second-order and dynamically consistent NSFD method. It is proved that the NSFD method is simple and efficient and can be extended for solving a broad range of mathematical models arising in real-world applications. Also, we combine the constructed second-order NSFD method with Richardson's extrapolation technique to generate high-order numerical approximations. Finally, the theoretical findings are illustrated and supported by numerical experiments.
{"title":"A novel second-order nonstandard finite difference method preserving dynamical properties of a general single-species model","authors":"M. T. Hoang","doi":"10.1080/00207160.2023.2248304","DOIUrl":"https://doi.org/10.1080/00207160.2023.2248304","url":null,"abstract":"In this paper, we extend the Mickens' methodology to construct a second-order nonstandard finite difference (NSFD) method, which preserves dynamical properties including positivity, local asymptotic stability and especially, global asymptotic stability of a general single-species model. This NSFD method is based on a novel weighted non-local approximation of the right-hand side function in combination with the renormalization of the denominator function. The weight guarantees the dynamic consistency and the nonstandard denominator function ensures the convergence of order 2 of the NSFD method. The result is that we obtain a second-order and dynamically consistent NSFD method. It is proved that the NSFD method is simple and efficient and can be extended for solving a broad range of mathematical models arising in real-world applications. Also, we combine the constructed second-order NSFD method with Richardson's extrapolation technique to generate high-order numerical approximations. Finally, the theoretical findings are illustrated and supported by numerical experiments.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"1 1","pages":"2047 - 2062"},"PeriodicalIF":1.8,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83465421","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 : 2023-08-15DOI: 10.1080/00207160.2023.2248286
Z. Sabir, D. Baleanu, F. Mallawi, M. Z. Ullah
The purpose of this work is to construct a reliable stochastic framework for solving the SIRC delay differential epidemic system, i.e. SIRC-DDES that is based on the coronavirus dynamics. The design of radial basis (RB) transfer function with the optimization of Bayesian regularization neural network (RB-BRNN) is presented to solve the SIRC-DDES. The SIRC-DDES is classified into susceptible , infected , recovered and cross-immune . The exactness of the RB-BRNN is performed for three cases of SIRC-DDES by using the performances of the obtained and reference results. The mean square error is reduced by using the training, testing and substantiation performances with the reference solutions. The small values of the absolute error around 10−07 to 10−08 and different statistical operator performances based on the error histogram values, transitions of state investigations, correlation and regression tests also approve the accuracy of the proposed technique.
{"title":"A novel radial basis procedure for the SIRC epidemic delay differential model","authors":"Z. Sabir, D. Baleanu, F. Mallawi, M. Z. Ullah","doi":"10.1080/00207160.2023.2248286","DOIUrl":"https://doi.org/10.1080/00207160.2023.2248286","url":null,"abstract":"The purpose of this work is to construct a reliable stochastic framework for solving the SIRC delay differential epidemic system, i.e. SIRC-DDES that is based on the coronavirus dynamics. The design of radial basis (RB) transfer function with the optimization of Bayesian regularization neural network (RB-BRNN) is presented to solve the SIRC-DDES. The SIRC-DDES is classified into susceptible , infected , recovered and cross-immune . The exactness of the RB-BRNN is performed for three cases of SIRC-DDES by using the performances of the obtained and reference results. The mean square error is reduced by using the training, testing and substantiation performances with the reference solutions. The small values of the absolute error around 10−07 to 10−08 and different statistical operator performances based on the error histogram values, transitions of state investigations, correlation and regression tests also approve the accuracy of the proposed technique.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"9 1","pages":"2014 - 2025"},"PeriodicalIF":1.8,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75143180","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 : 2023-08-03DOI: 10.1080/00207160.2023.2239944
Eleonora Arnone, C. de Falco, L. Formaggia, Giorgio Meretti, L. Sangalli
We investigate some computational aspects of an innovative class of PDE-regularized statistical models: Spatial Regression with Partial Differential Equation regularization (SR-PDE). These physics-informed regression methods can account for the physics of the underlying phenomena and handle data observed over spatial domains with nontrivial shapes, such as domains with concavities and holes or curved domains. The computational bottleneck in SR-PDE estimation is the solution of a computationally demanding linear system involving a low-rank but dense block. We address this aspect by innovatively using Sherman–Morrison–Woodbury identity. We also investigate the efficient selection of the smoothing parameter in SR-PDE estimates. Specifically, we propose ad hoc optimization methods to perform Generalized Cross-Validation, coupling suitable reformulation of key matrices, e.g. those based on Sherman–Morrison–Woodbury formula, with stochastic trace estimation, to approximate the equivalent degrees of freedom of the problem. These solutions permit high computational efficiency also in the context of massive data.
{"title":"Computationally efficient techniques for spatial regression with differential regularization","authors":"Eleonora Arnone, C. de Falco, L. Formaggia, Giorgio Meretti, L. Sangalli","doi":"10.1080/00207160.2023.2239944","DOIUrl":"https://doi.org/10.1080/00207160.2023.2239944","url":null,"abstract":"We investigate some computational aspects of an innovative class of PDE-regularized statistical models: Spatial Regression with Partial Differential Equation regularization (SR-PDE). These physics-informed regression methods can account for the physics of the underlying phenomena and handle data observed over spatial domains with nontrivial shapes, such as domains with concavities and holes or curved domains. The computational bottleneck in SR-PDE estimation is the solution of a computationally demanding linear system involving a low-rank but dense block. We address this aspect by innovatively using Sherman–Morrison–Woodbury identity. We also investigate the efficient selection of the smoothing parameter in SR-PDE estimates. Specifically, we propose ad hoc optimization methods to perform Generalized Cross-Validation, coupling suitable reformulation of key matrices, e.g. those based on Sherman–Morrison–Woodbury formula, with stochastic trace estimation, to approximate the equivalent degrees of freedom of the problem. These solutions permit high computational efficiency also in the context of massive data.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"93 1","pages":"1971 - 1991"},"PeriodicalIF":1.8,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79585054","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 : 2023-07-23DOI: 10.1080/00207160.2023.2239947
A. Abubakar, P. Kumam, Jin-kui Liu, Hassan Mohammad, C. Tammer
This work presents a new three-term projection algorithm for solving nonlinear monotone equations. The paper is aimed at constructing an efficient and competitive algorithm for finding approximate solutions of nonlinear monotone equations. This is based on a new choice of the conjugate gradient direction which satisfies the sufficient descent condition. The convergence of the algorithm is shown under Lipschitz continuity and monotonicity of the involved operator. Numerical experiments presented in the paper show that the algorithm needs a less number of iterations in comparison with existing algorithms. Furthermore, the proposed algorithm is applied to solve signal recovery problems.
{"title":"New three-term conjugate gradient algorithm for solving monotone nonlinear equations and signal recovery problems","authors":"A. Abubakar, P. Kumam, Jin-kui Liu, Hassan Mohammad, C. Tammer","doi":"10.1080/00207160.2023.2239947","DOIUrl":"https://doi.org/10.1080/00207160.2023.2239947","url":null,"abstract":"This work presents a new three-term projection algorithm for solving nonlinear monotone equations. The paper is aimed at constructing an efficient and competitive algorithm for finding approximate solutions of nonlinear monotone equations. This is based on a new choice of the conjugate gradient direction which satisfies the sufficient descent condition. The convergence of the algorithm is shown under Lipschitz continuity and monotonicity of the involved operator. Numerical experiments presented in the paper show that the algorithm needs a less number of iterations in comparison with existing algorithms. Furthermore, the proposed algorithm is applied to solve signal recovery problems.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"42 1","pages":"1992 - 2013"},"PeriodicalIF":1.8,"publicationDate":"2023-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74766572","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 : 2023-07-22DOI: 10.1080/00207160.2023.2239953
S. Husain, Mohd Asad
ABSTRACT To address the split best proximity point and monotone variational inclusion problems in real Hilbert spaces, we present and investigate projection and viscosity approximation methods. Under a few reasonable assumptions, we prove some weak and strong convergence theorems for the aforementioned methods. The efficiency of the proposed method is demonstrated by some numerical examples. Some well-known recent results in this area have been improved, generalized, and extended as an outcome of this paper.
{"title":"Viscosity approximation method for split best proximity point and monotone variational inclusion problem","authors":"S. Husain, Mohd Asad","doi":"10.1080/00207160.2023.2239953","DOIUrl":"https://doi.org/10.1080/00207160.2023.2239953","url":null,"abstract":"ABSTRACT To address the split best proximity point and monotone variational inclusion problems in real Hilbert spaces, we present and investigate projection and viscosity approximation methods. Under a few reasonable assumptions, we prove some weak and strong convergence theorems for the aforementioned methods. The efficiency of the proposed method is demonstrated by some numerical examples. Some well-known recent results in this area have been improved, generalized, and extended as an outcome of this paper.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"94 1","pages":"1941 - 1954"},"PeriodicalIF":1.8,"publicationDate":"2023-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85423970","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 : 2023-07-21DOI: 10.1080/00207160.2023.2239948
Ashisha Kumar, P. Goswami
This paper deals with the numerical solution of the general Emden–Fowler equation using the Haar wavelet collocation method. This method transforms the differential equation into a system of nonlinear equations. These equations are further solved by Newton's method to obtain the Haar coefficients, and finally the solution to the problem is acquired using these coefficients. We have taken many examples of fifth- and sixth-order equations and implemented our method on those examples. The graphs show the efficiency of the solution for resolution L = 3 and the maximum absolute error of our approach. The error tables give a good picture of the accuracy of this approach.
{"title":"Numerical solution of general Emden–Fowler equation using Haar wavelet collocation method","authors":"Ashisha Kumar, P. Goswami","doi":"10.1080/00207160.2023.2239948","DOIUrl":"https://doi.org/10.1080/00207160.2023.2239948","url":null,"abstract":"This paper deals with the numerical solution of the general Emden–Fowler equation using the Haar wavelet collocation method. This method transforms the differential equation into a system of nonlinear equations. These equations are further solved by Newton's method to obtain the Haar coefficients, and finally the solution to the problem is acquired using these coefficients. We have taken many examples of fifth- and sixth-order equations and implemented our method on those examples. The graphs show the efficiency of the solution for resolution L = 3 and the maximum absolute error of our approach. The error tables give a good picture of the accuracy of this approach.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"36 1","pages":"1918 - 1940"},"PeriodicalIF":1.8,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77203312","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 : 2023-07-21DOI: 10.1080/00207160.2023.2239943
Haide Gou, Y. Jia
In this paper, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of multi-term time-fractional measure differential equations with initial conditions in Banach spaces. Firstly, we look for a suitable concept of S-asymptotically ω-periodic mild solution to our concerned problem, by means of the Laplace transform and -resolvent family . Secondly, the existence of S-asymptotically ω-periodic mild solutions for the mentioned system is obtained by utilizing regulated functions and fixed point theorem. Finally, as the application of abstract results, an example is given to illustrate our main results.
{"title":"A study on mild solutions for multi-term time fractional measure differential equations","authors":"Haide Gou, Y. Jia","doi":"10.1080/00207160.2023.2239943","DOIUrl":"https://doi.org/10.1080/00207160.2023.2239943","url":null,"abstract":"In this paper, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of multi-term time-fractional measure differential equations with initial conditions in Banach spaces. Firstly, we look for a suitable concept of S-asymptotically ω-periodic mild solution to our concerned problem, by means of the Laplace transform and -resolvent family . Secondly, the existence of S-asymptotically ω-periodic mild solutions for the mentioned system is obtained by utilizing regulated functions and fixed point theorem. Finally, as the application of abstract results, an example is given to illustrate our main results.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"69 1","pages":"1896 - 1917"},"PeriodicalIF":1.8,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89976289","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 : 2023-06-30DOI: 10.1080/00207160.2023.2226254
Zihao Tian, Yanhua Cao, Xiaozhong Yang
The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE.
{"title":"A new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation","authors":"Zihao Tian, Yanhua Cao, Xiaozhong Yang","doi":"10.1080/00207160.2023.2226254","DOIUrl":"https://doi.org/10.1080/00207160.2023.2226254","url":null,"abstract":"The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"25 1","pages":"1877 - 1895"},"PeriodicalIF":1.8,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77696109","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 : 2023-06-19DOI: 10.1080/00207160.2023.2214254
F. Asadi-Mehregan, P. Assari, M. Dehghan
In this research paper, we introduce a numerical approach to solve a particular type of nonlinear integro-differential equations derived from Volterra's population model. This model characterizes the growth of a biological species in a closed system and includes an integral term to consider the influence of toxin accumulation on the species, along with the conventional terms found in the logistic equation. The proposed technique estimates the solution of integro-differential equations utilizing the discrete Galerkin scheme using the moving least squares (MLS) algorithm. The locally weighted least squares polynomial fitting, known as the MLS method, is a valuable approach for approximating unknown functions. Since the offered scheme does not require any cell structures, it can be known as a meshless local discrete Galerkin method. Moreover, we obtain the error estimate of the proposed approach. The validity and efficiency of the newly developed technique are assessed over several nonlinear integro-differential equations.
{"title":"On the numerical solution of a population growth model of a species living in a closed system based on the moving least squares scheme","authors":"F. Asadi-Mehregan, P. Assari, M. Dehghan","doi":"10.1080/00207160.2023.2214254","DOIUrl":"https://doi.org/10.1080/00207160.2023.2214254","url":null,"abstract":"In this research paper, we introduce a numerical approach to solve a particular type of nonlinear integro-differential equations derived from Volterra's population model. This model characterizes the growth of a biological species in a closed system and includes an integral term to consider the influence of toxin accumulation on the species, along with the conventional terms found in the logistic equation. The proposed technique estimates the solution of integro-differential equations utilizing the discrete Galerkin scheme using the moving least squares (MLS) algorithm. The locally weighted least squares polynomial fitting, known as the MLS method, is a valuable approach for approximating unknown functions. Since the offered scheme does not require any cell structures, it can be known as a meshless local discrete Galerkin method. Moreover, we obtain the error estimate of the proposed approach. The validity and efficiency of the newly developed technique are assessed over several nonlinear integro-differential equations.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":"72 1","pages":"1757 - 1778"},"PeriodicalIF":1.8,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79123669","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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