Pub Date : 2024-11-07DOI: 10.1016/j.matcom.2024.10.041
Daniele Peri
In this paper, a multidisciplinary design optimization algorithm, the Normal Boundary Intersection (NBI) method, is applied to the design of some devices of a sailing yacht. The full Pareto front is identified for two different design problems, and the optimal configurations are compared with standard devices. The great efficiency of the optimization algorithm is demonstrated by the wideness and density of the identified Pareto front.
{"title":"Multi-objective optimization of the appendages of a sailing yacht using the Normal Boundary Intersection method","authors":"Daniele Peri","doi":"10.1016/j.matcom.2024.10.041","DOIUrl":"10.1016/j.matcom.2024.10.041","url":null,"abstract":"<div><div>In this paper, a multidisciplinary design optimization algorithm, the Normal Boundary Intersection (NBI) method, is applied to the design of some devices of a sailing yacht. The full Pareto front is identified for two different design problems, and the optimal configurations are compared with standard devices. The great efficiency of the optimization algorithm is demonstrated by the wideness and density of the identified Pareto front.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"229 ","pages":"Pages 885-895"},"PeriodicalIF":4.4,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142650624","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 : 2024-11-07DOI: 10.1016/j.matcom.2024.10.018
Bingzhen Chen , Wenjuan Zhai
In recent years, distributed statistical models have received increasing attention for large-scale data analysis. On the one hand, data sets come from multiple data sources, and are stored in different locations due to limited bandwidth and storage, or privacy protocols, directly centralizing all data together is impossible. On the other hand, the size of data is so large that it is difficult or inefficient to analyze data together. There are two main research aspects to using distributed statistical models to analyze large-scale data. The first one is to study the statistical convergence rate under some mild assumptions. The second one is to establish fast and efficient optimization algorithms considering the property of the loss function. There is a lot of research on the first aspect, but relatively little research on the second one. Motivated by this, we consider the construction of unified algorithms for distributed linear regression with different losses and regularizers. As a result, we designed two type methods, proximal alternating direction method of multipliers (pADMM) and distributed accelerated proximal gradient method with line-search (DAPGL). In order to demonstrate the efficiency of the proposed algorithms, we perform numerical experiments on the distributed Huber-Lasso model and Huber-Group-Lasso model. In view of the numerical results, we can observe that these two algorithms are more competitive than some of state-of-art algorithms. In particular, DAPGL algorithm performs better than pADMM in most cases.
{"title":"Unified algorithms for distributed regularized linear regression model","authors":"Bingzhen Chen , Wenjuan Zhai","doi":"10.1016/j.matcom.2024.10.018","DOIUrl":"10.1016/j.matcom.2024.10.018","url":null,"abstract":"<div><div>In recent years, distributed statistical models have received increasing attention for large-scale data analysis. On the one hand, data sets come from multiple data sources, and are stored in different locations due to limited bandwidth and storage, or privacy protocols, directly centralizing all data together is impossible. On the other hand, the size of data is so large that it is difficult or inefficient to analyze data together. There are two main research aspects to using distributed statistical models to analyze large-scale data. The first one is to study the statistical convergence rate under some mild assumptions. The second one is to establish fast and efficient optimization algorithms considering the property of the loss function. There is a lot of research on the first aspect, but relatively little research on the second one. Motivated by this, we consider the construction of unified algorithms for distributed linear regression with different losses and regularizers. As a result, we designed two type methods, proximal alternating direction method of multipliers (pADMM) and distributed accelerated proximal gradient method with line-search (DAPGL). In order to demonstrate the efficiency of the proposed algorithms, we perform numerical experiments on the distributed Huber-Lasso model and Huber-Group-Lasso model. In view of the numerical results, we can observe that these two algorithms are more competitive than some of state-of-art algorithms. In particular, DAPGL algorithm performs better than pADMM in most cases.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"229 ","pages":"Pages 867-884"},"PeriodicalIF":4.4,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659627","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 : 2024-11-07DOI: 10.1016/j.matcom.2024.10.042
Ruqi Li , Yurong Song , Min Li , Hongbo Qu , Guo-Ping Jiang
To analyze and predict the evolution of contagion dynamics, fractional derivative modeling has emerged as an important technique. However, inferring the dynamical structure of fractional-order models with high degrees of freedom poses a challenge. In this paper, to elucidate the spreading mechanism and non-local properties of disease evolution, we propose a novel fractional-order SEIHDR epidemiological model with variable parameters, incorporating fractional derivatives in the Caputo sense. We compute the basic reproduction number by the next-generation matrix and establish local and global stability conditions based on this reproduction number. By using the fractional Adams–Bashforth method, we validate dynamical behaviors at different equilibrium points in both autonomous and non-autonomous scenarios, while qualitatively analyze the effects of fractional order on the dynamics. To effectively address the inverse problem of the proposed fractional SEIHDR model, we construct a fractional Physics-Informed Neural Network framework to simultaneously infer time-dependent parameters, fractional orders, and state components. Graphical results based on the COVID-19 pandemic data from Canada demonstrate the effectiveness of the proposed framework.
{"title":"Dynamic analysis and data-driven inference of a fractional-order SEIHDR epidemic model with variable parameters","authors":"Ruqi Li , Yurong Song , Min Li , Hongbo Qu , Guo-Ping Jiang","doi":"10.1016/j.matcom.2024.10.042","DOIUrl":"10.1016/j.matcom.2024.10.042","url":null,"abstract":"<div><div>To analyze and predict the evolution of contagion dynamics, fractional derivative modeling has emerged as an important technique. However, inferring the dynamical structure of fractional-order models with high degrees of freedom poses a challenge. In this paper, to elucidate the spreading mechanism and non-local properties of disease evolution, we propose a novel fractional-order SEIHDR epidemiological model with variable parameters, incorporating fractional derivatives in the Caputo sense. We compute the basic reproduction number by the next-generation matrix and establish local and global stability conditions based on this reproduction number. By using the fractional Adams–Bashforth method, we validate dynamical behaviors at different equilibrium points in both autonomous and non-autonomous scenarios, while qualitatively analyze the effects of fractional order on the dynamics. To effectively address the inverse problem of the proposed fractional SEIHDR model, we construct a fractional Physics-Informed Neural Network framework to simultaneously infer time-dependent parameters, fractional orders, and state components. Graphical results based on the COVID-19 pandemic data from Canada demonstrate the effectiveness of the proposed framework.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 1-19"},"PeriodicalIF":4.4,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659321","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 : 2024-11-06DOI: 10.1016/j.matcom.2024.10.043
Carmine Valentino , Giovanni Pagano , Dajana Conte , Beatrice Paternoster , Francesco Colace , Mario Casillo
The use of Artificial Neural Networks (ANNs) has spread massively in several research fields. Among the various applications, ANNs have been exploited for the solution of Partial Differential Equations (PDEs). In this context, the so-called Physics-Informed Neural Networks (PINNs) are considered, i.e. neural networks generally constructed in such a way as to compute a continuous approximation in time and space of the exact solution of a PDE.
In this manuscript, we propose a new step-by-step approach that allows to define PINNs capable of providing numerical solutions of PDEs that are discrete in time and continuous in space. This is done by establishing connections between the network outputs and the numerical approximations computed by a classical one-stage method for stiff Initial Value Problems (IVPs). Links are also highlighted between the step-by-step PINNs derived here, and the time discrete models based on Runge–Kutta (RK) methods proposed so far in literature. To evaluate the efficiency of the new approach, we build such PINNs to solve a nonlinear diffusion–reaction PDE model describing the process of production of renewable energy through dye-sensitized solar cells. The numerical experiments show that not only the new step-by-step PINNs are able to well reproduce the model solution, but also highlight that the proposed approach can constitute an improvement over existing continuous and time discrete models.
{"title":"Step-by-step time discrete Physics-Informed Neural Networks with application to a sustainability PDE model","authors":"Carmine Valentino , Giovanni Pagano , Dajana Conte , Beatrice Paternoster , Francesco Colace , Mario Casillo","doi":"10.1016/j.matcom.2024.10.043","DOIUrl":"10.1016/j.matcom.2024.10.043","url":null,"abstract":"<div><div>The use of Artificial Neural Networks (ANNs) has spread massively in several research fields. Among the various applications, ANNs have been exploited for the solution of Partial Differential Equations (PDEs). In this context, the so-called Physics-Informed Neural Networks (PINNs) are considered, i.e. neural networks generally constructed in such a way as to compute a continuous approximation in time and space of the exact solution of a PDE.</div><div>In this manuscript, we propose a new step-by-step approach that allows to define PINNs capable of providing numerical solutions of PDEs that are discrete in time and continuous in space. This is done by establishing connections between the network outputs and the numerical approximations computed by a classical one-stage method for stiff Initial Value Problems (IVPs). Links are also highlighted between the step-by-step PINNs derived here, and the time discrete models based on Runge–Kutta (RK) methods proposed so far in literature. To evaluate the efficiency of the new approach, we build such PINNs to solve a nonlinear diffusion–reaction PDE model describing the process of production of renewable energy through dye-sensitized solar cells. The numerical experiments show that not only the new step-by-step PINNs are able to well reproduce the model solution, but also highlight that the proposed approach can constitute an improvement over existing continuous and time discrete models.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 541-558"},"PeriodicalIF":4.4,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129346","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 : 2024-11-06DOI: 10.1016/j.matcom.2024.10.037
Ismael Chirino Aguinaga , Nicolas Patin , Patrice Gomez , Vincent Lanfranchi , Jeanne-Marie Dalbavie
This paper focuses on the control of an n-port Multi-Active Bridge, providing insights into switching sequences, a time-domain model, and an optimization algorithm concerning conduction and switching losses. Central to this analysis is the introduction of a novel concept known as winding voltage sequences, which encapsulate the switching sequences employed by the converter bridges. The algorithm proposed in this article aims to derive analytical expressions for the currents flowing through all transformer windings (and switches) for each switching sequence. This control methodology is characterized by a set of parameters—inter-leg and inter-bridge phase shifts—whose diverse values are systematically explored to attain an optimal solution.
{"title":"Algorithm for sequences exploration and optimization of a Multi-Active-Bridge","authors":"Ismael Chirino Aguinaga , Nicolas Patin , Patrice Gomez , Vincent Lanfranchi , Jeanne-Marie Dalbavie","doi":"10.1016/j.matcom.2024.10.037","DOIUrl":"10.1016/j.matcom.2024.10.037","url":null,"abstract":"<div><div>This paper focuses on the control of an n-port <em>Multi-Active Bridge</em>, providing insights into switching sequences, a time-domain model, and an optimization algorithm concerning conduction and switching losses. Central to this analysis is the introduction of a novel concept known as winding voltage sequences, which encapsulate the switching sequences employed by the converter bridges. The algorithm proposed in this article aims to derive analytical expressions for the currents flowing through all transformer windings (and switches) for each switching sequence. This control methodology is characterized by a set of parameters—inter-leg and inter-bridge phase shifts—whose diverse values are systematically explored to attain an optimal solution.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 278-288"},"PeriodicalIF":4.4,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759025","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 : 2024-11-03DOI: 10.1016/j.matcom.2024.10.039
Vincenzo Schiano Di Cola , Vittorio Bauduin , Marco Berardi , Filippo Notarnicola , Salvatore Cuomo
Physics-Informed Neural Networks (PINNs) are considered a powerful tool for solving partial differential equations (PDEs), particularly for the groundwater flow (GF) problem. In this paper, we investigate how the deep learning (DL) architecture, within the PINN framework, is connected to the ability to compute a more or less accurate numerical GF solution, so the link ‘PINN architecture - numerical performance’ is explored. Specifically, this paper explores the effect of various DL components, such as different activation functions and neural network structures, on the computational framework. Through numerical results and on the basis of some theoretical foundations of PINNs, this research aims to improve the explicability of PINNs to resolve, in this case, the one-dimensional GF equation. Moreover, our problem involves source terms described by a Dirac delta function, providing insights into the role of DL architecture in solving complex PDEs.
{"title":"Investigating neural networks with groundwater flow equation loss","authors":"Vincenzo Schiano Di Cola , Vittorio Bauduin , Marco Berardi , Filippo Notarnicola , Salvatore Cuomo","doi":"10.1016/j.matcom.2024.10.039","DOIUrl":"10.1016/j.matcom.2024.10.039","url":null,"abstract":"<div><div>Physics-Informed Neural Networks (PINNs) are considered a powerful tool for solving partial differential equations (PDEs), particularly for the groundwater flow (GF) problem. In this paper, we investigate how the deep learning (DL) architecture, within the PINN framework, is connected to the ability to compute a more or less accurate numerical GF solution, so the link ‘PINN architecture - numerical performance’ is explored. Specifically, this paper explores the effect of various DL components, such as different activation functions and neural network structures, on the computational framework. Through numerical results and on the basis of some theoretical foundations of PINNs, this research aims to improve the explicability of PINNs to resolve, in this case, the one-dimensional GF equation. Moreover, our problem involves source terms described by a Dirac delta function, providing insights into the role of DL architecture in solving complex PDEs.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 80-93"},"PeriodicalIF":4.4,"publicationDate":"2024-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142700822","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 : 2024-11-01DOI: 10.1016/j.matcom.2024.10.031
Jie Jin , Xiaoyang Lei , Chaoyang Chen , Zhijing Li
As an effective method for time-varying problems solving, zeroing neural network (ZNN) has been frequently applied in science and engineering. In order to improve its performances in practical applications, a fuzzy activation function (FAF) is designed by introducing the fuzzy logic technology, and a fuzzy activation function based zeroing neural network (FAF-ZNN) model for fast solving time-varying matrix inversion (TVMI) is proposed. Rigorous mathematical analysis and comparative simulation experiments with other models guarantee its superior convergence and robustness to noises. In addition, based on the proposed FAF-ZNN model, a new dynamic Arnold map image cryptography algorithm is designed. Specifically, in the new dynamic image encryption, a dynamic key matrix is introduced, and the FAF-ZNN model is applied to fast compute the inversion of the dynamic key matrix for the dynamic Arnold map image cryptography decryption process. The effectiveness of the dynamic image encryption algorithm is verified by experiment results, which enhances the security of existing image encryption algorithms.
{"title":"A fuzzy activation function based zeroing neural network for dynamic Arnold map image cryptography","authors":"Jie Jin , Xiaoyang Lei , Chaoyang Chen , Zhijing Li","doi":"10.1016/j.matcom.2024.10.031","DOIUrl":"10.1016/j.matcom.2024.10.031","url":null,"abstract":"<div><div>As an effective method for time-varying problems solving, zeroing neural network (ZNN) has been frequently applied in science and engineering. In order to improve its performances in practical applications, a fuzzy activation function (FAF) is designed by introducing the fuzzy logic technology, and a fuzzy activation function based zeroing neural network (FAF-ZNN) model for fast solving time-varying matrix inversion (TVMI) is proposed. Rigorous mathematical analysis and comparative simulation experiments with other models guarantee its superior convergence and robustness to noises. In addition, based on the proposed FAF-ZNN model, a new dynamic Arnold map image cryptography algorithm is designed. Specifically, in the new dynamic image encryption, a dynamic key matrix is introduced, and the FAF-ZNN model is applied to fast compute the inversion of the dynamic key matrix for the dynamic Arnold map image cryptography decryption process. The effectiveness of the dynamic image encryption algorithm is verified by experiment results, which enhances the security of existing image encryption algorithms.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 456-469"},"PeriodicalIF":4.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129341","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 : 2024-11-01DOI: 10.1016/j.matcom.2024.10.038
Karel J. in ’t Hout
In this note, we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank–Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed.
In this note, for the temporal discretization of the PDCP, we study an interesting family of diagonally implicit Runge–Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter is preferable.
{"title":"A note on the numerical approximation of Greeks for American-style options","authors":"Karel J. in ’t Hout","doi":"10.1016/j.matcom.2024.10.038","DOIUrl":"10.1016/j.matcom.2024.10.038","url":null,"abstract":"<div><div>In this note, we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank–Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed.</div><div>In this note, for the temporal discretization of the PDCP, we study an interesting family of diagonally implicit Runge–Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter <span><math><mi>θ</mi></math></span> is preferable.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 501-516"},"PeriodicalIF":4.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129478","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 : 2024-11-01DOI: 10.1016/j.matcom.2024.10.035
Y. Massoun , A.K. Alomari , C. Cesarano
In this paper, we construct a general framework for presenting an approximate analytic solution of the SIR epidemic model that contains a multi-parameter of a fractional derivative in the sense of Caputo using the homotopy analysis method. Basic ideas of both fractional derivatives and the application of the semi-analytical method for this type of system of fractional differential equation are presented. The study presents the effect of the new parameters on the solution behaviors. The new parameters of the fractional derivative give the researchers additional tools to fit the data with appropriate parameters. A particular case for compares with the fourth Runge Kutta method, the Adams Bashforth Moulton predictor correcter scheme, and the Bernstein wavelet method to show and confirm this effectiveness method.
{"title":"Analytic solution for SIR epidemic model with multi-parameter fractional derivative","authors":"Y. Massoun , A.K. Alomari , C. Cesarano","doi":"10.1016/j.matcom.2024.10.035","DOIUrl":"10.1016/j.matcom.2024.10.035","url":null,"abstract":"<div><div>In this paper, we construct a general framework for presenting an approximate analytic solution of the SIR epidemic model that contains a multi-parameter of a fractional derivative <span><math><mrow><msup><mrow></mrow><mrow><mi>C</mi></mrow></msup><msubsup><mrow><mi>D</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow><mrow><mi>α</mi><mo>,</mo><mi>ρ</mi></mrow></msubsup></mrow></math></span> in the sense of Caputo using the homotopy analysis method. Basic ideas of both fractional derivatives and the application of the semi-analytical method for this type of system of fractional differential equation are presented. The study presents the effect of the new parameters on the solution behaviors. The new parameters of the fractional derivative give the researchers additional tools to fit the data with appropriate parameters. A particular case for <span><math><mrow><mi>α</mi><mo>=</mo><mi>ρ</mi><mo>=</mo><mn>1</mn></mrow></math></span> compares with the fourth Runge Kutta method, the Adams Bashforth Moulton predictor correcter scheme, and the Bernstein wavelet method to show and confirm this effectiveness method.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 484-492"},"PeriodicalIF":4.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129351","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 : 2024-10-31DOI: 10.1016/j.matcom.2024.10.030
Lívia Boda , István Faragó
In mathematics there are several problems arise that can be described by differential equations with particular, highly complex structure. Most of the time, we cannot produce the exact (analytical) solution of these problems, therefore we have to approximate them numerically by using some approximating method. The main aim of this paper is to create numerical methods, based on operator splitting, that well approximate the exact solution of the original ODE systems while having low computational complexity. Starting from an example, based on the relationship between the Lie–Trotter (sequential) and Strang–Marchuk splitting methods, we examine the properties of processed integrator methods. Then we generalize these methods and introduce the new extended processed methods. By examining the consistency and stability of these methods, we establish the one order higher convergence. However, these methods have a higher computational complexity, which we aim to reduce by introducing economic extended processed methods. In this case we show the lower computational complexity and prove the second-order convergence. In the end, we test the analyzed methods in three models: a large-scale linear model, a piecewise-linear model of flutter and the heat conduction equation. Runtimes and errors are also compared.
{"title":"New efficient numerical methods for some systems of linear ordinary differential equations","authors":"Lívia Boda , István Faragó","doi":"10.1016/j.matcom.2024.10.030","DOIUrl":"10.1016/j.matcom.2024.10.030","url":null,"abstract":"<div><div>In mathematics there are several problems arise that can be described by differential equations with particular, highly complex structure. Most of the time, we cannot produce the exact (analytical) solution of these problems, therefore we have to approximate them numerically by using some approximating method. The main aim of this paper is to create numerical methods, based on operator splitting, that well approximate the exact solution of the original ODE systems while having low computational complexity. Starting from an example, based on the relationship between the Lie–Trotter (sequential) and Strang–Marchuk splitting methods, we examine the properties of processed integrator methods. Then we generalize these methods and introduce the new extended processed methods. By examining the consistency and stability of these methods, we establish the one order higher convergence. However, these methods have a higher computational complexity, which we aim to reduce by introducing economic extended processed methods. In this case we show the lower computational complexity and prove the second-order convergence. In the end, we test the analyzed methods in three models: a large-scale linear model, a piecewise-linear model of flutter and the heat conduction equation. Runtimes and errors are also compared.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"230 ","pages":"Pages 438-455"},"PeriodicalIF":4.4,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129479","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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