Pub Date : 2023-11-03DOI: 10.1186/s13662-023-03790-4
Tao Guo, Aiguo Xiao, Junjie Wang, Xueyang Li
Abstract In this paper, we consider the Fourier spectral method and numerical investigation for a class of modified Zakharov system with high-order space fractional quantum correction. First, the numerical scheme of the system is developed with periodic boundary condition based on the Crank–Nicolson/leap-frog methods in time and the Fourier spectral method in space. Moreover, it is shown that the scheme preserves simultaneously mass and energy conservation laws. Second, we analyze stability and convergence of the numerical scheme. Last, the numerical experiments are given, and the results show the correctness of theoretical results and the efficiency of the conservative scheme.
{"title":"Conservative Fourier spectral method for a class of modified Zakharov system with high-order space fractional quantum correction","authors":"Tao Guo, Aiguo Xiao, Junjie Wang, Xueyang Li","doi":"10.1186/s13662-023-03790-4","DOIUrl":"https://doi.org/10.1186/s13662-023-03790-4","url":null,"abstract":"Abstract In this paper, we consider the Fourier spectral method and numerical investigation for a class of modified Zakharov system with high-order space fractional quantum correction. First, the numerical scheme of the system is developed with periodic boundary condition based on the Crank–Nicolson/leap-frog methods in time and the Fourier spectral method in space. Moreover, it is shown that the scheme preserves simultaneously mass and energy conservation laws. Second, we analyze stability and convergence of the numerical scheme. Last, the numerical experiments are given, and the results show the correctness of theoretical results and the efficiency of the conservative scheme.","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":"44 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135820332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1186/s13662-023-03789-x
Toni Schneidereit, Michael Breuß
Abstract A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent differential equations, one can apply the recently developed method of domain segmentation. That is, the domain may be split into several subdomains, on which the optimisation problem is solved. In classic adaptive numerical methods, the mesh as well as the domain may be refined or decomposed, in order to improve the accuracy. Also, the degree of approximation accuracy may be adapted. Therefore, it is desirable to transfer such important and successful strategies to the field of neural-network-based solutions. In the presented work, we propose a novel adaptive neural approach to meet this aim for solving time-dependent problems. To this end, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks employed are by default small, we propose a means to adjust also the number of neurons in an adaptive way. We introduce conditions to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary. Results are provided for several initial-value problems that illustrate important computational properties of the method.
{"title":"Adaptive neural-domain refinement for solving time-dependent differential equations","authors":"Toni Schneidereit, Michael Breuß","doi":"10.1186/s13662-023-03789-x","DOIUrl":"https://doi.org/10.1186/s13662-023-03789-x","url":null,"abstract":"Abstract A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent differential equations, one can apply the recently developed method of domain segmentation. That is, the domain may be split into several subdomains, on which the optimisation problem is solved. In classic adaptive numerical methods, the mesh as well as the domain may be refined or decomposed, in order to improve the accuracy. Also, the degree of approximation accuracy may be adapted. Therefore, it is desirable to transfer such important and successful strategies to the field of neural-network-based solutions. In the presented work, we propose a novel adaptive neural approach to meet this aim for solving time-dependent problems. To this end, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks employed are by default small, we propose a means to adjust also the number of neurons in an adaptive way. We introduce conditions to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary. Results are provided for several initial-value problems that illustrate important computational properties of the method.","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":"40 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135218956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1186/s13662-023-03787-z
Hirotada Honda
Abstract Recently, differential equation-based neural networks have been actively studied. This paper discusses the universal approximation property of a neural network that is based on a nonlinear partial differential equation (PDE) of the parabolic type. Based on the assumption that the activation function is non-polynomial and Lipschitz continuous, and applying the theory of the difference method, we show that an arbitrary continuous function on any compact set can be approximated using the output of the network with arbitrary precision. Additionally, we present an estimate of the order of accuracy with respect to △ t and △ x .
{"title":"Universal approximation property of a continuous neural network based on a nonlinear diffusion equation","authors":"Hirotada Honda","doi":"10.1186/s13662-023-03787-z","DOIUrl":"https://doi.org/10.1186/s13662-023-03787-z","url":null,"abstract":"Abstract Recently, differential equation-based neural networks have been actively studied. This paper discusses the universal approximation property of a neural network that is based on a nonlinear partial differential equation (PDE) of the parabolic type. Based on the assumption that the activation function is non-polynomial and Lipschitz continuous, and applying the theory of the difference method, we show that an arbitrary continuous function on any compact set can be approximated using the output of the network with arbitrary precision. Additionally, we present an estimate of the order of accuracy with respect to △ t and △ x .","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":"22 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135218209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-23DOI: 10.1186/s13662-023-03788-y
Run Yang, Jianglin Zhao
Abstract The dynamics of a discrete Holling–Tanner model with Beddington–DeAngelis functional response is studied. The permanence and local stability of fixed points for the model are derived. The center manifold theorem and bifurcation theory are used to show that the model can undergo flip and Hopf bifurcations. Codimension-two bifurcation associated with 1:2 resonance is analyzed by applying the bifurcation theory. Numerical simulations are performed not only to verify the correctness of theoretical analysis but to explore complex dynamical behaviors such as period-6, 7, 10, 12 orbits, a cascade of period-doubling, quasi-periodic orbits, and the chaotic sets. The maximum Lyapunov exponents validate the chaotic dynamical behaviors of the system. The feedback control method is considered to stabilize the chaotic orbits. These complex dynamical behaviors imply that the coexistence of predator and prey may produce very complex patterns.
{"title":"Bifurcation and chaos in a discrete Holling–Tanner model with Beddington–DeAngelis functional response","authors":"Run Yang, Jianglin Zhao","doi":"10.1186/s13662-023-03788-y","DOIUrl":"https://doi.org/10.1186/s13662-023-03788-y","url":null,"abstract":"Abstract The dynamics of a discrete Holling–Tanner model with Beddington–DeAngelis functional response is studied. The permanence and local stability of fixed points for the model are derived. The center manifold theorem and bifurcation theory are used to show that the model can undergo flip and Hopf bifurcations. Codimension-two bifurcation associated with 1:2 resonance is analyzed by applying the bifurcation theory. Numerical simulations are performed not only to verify the correctness of theoretical analysis but to explore complex dynamical behaviors such as period-6, 7, 10, 12 orbits, a cascade of period-doubling, quasi-periodic orbits, and the chaotic sets. The maximum Lyapunov exponents validate the chaotic dynamical behaviors of the system. The feedback control method is considered to stabilize the chaotic orbits. These complex dynamical behaviors imply that the coexistence of predator and prey may produce very complex patterns.","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":"41 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135405671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-10DOI: 10.1186/s13662-023-03786-0
Jie Ran, Jixiu Qiu, Yonghui Zhou
Abstract In this paper, a stochastic discrete fractional-order chaotic system with short memory is proposed, which possesses two equilibrium points. With the help of the Lyapunov function theory, some sufficient conditions for the stability in probability of the two equilibrium points are given. Secondly, the effects of fractional order and memory steps on the stability of the system are discussed. Finally, the path dynamical behavior of the system is investigated using numerical methods such as Lyapunov exponents, bifurcation diagram, phase diagram, and 0–1 test. The numerical simulation results validate the findings.
{"title":"Stability and dynamics of a stochastic discrete fractional-order chaotic system with short memory","authors":"Jie Ran, Jixiu Qiu, Yonghui Zhou","doi":"10.1186/s13662-023-03786-0","DOIUrl":"https://doi.org/10.1186/s13662-023-03786-0","url":null,"abstract":"Abstract In this paper, a stochastic discrete fractional-order chaotic system with short memory is proposed, which possesses two equilibrium points. With the help of the Lyapunov function theory, some sufficient conditions for the stability in probability of the two equilibrium points are given. Secondly, the effects of fractional order and memory steps on the stability of the system are discussed. Finally, the path dynamical behavior of the system is investigated using numerical methods such as Lyapunov exponents, bifurcation diagram, phase diagram, and 0–1 test. The numerical simulation results validate the findings.","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":"1 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":"136295583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-09DOI: 10.1186/s13662-023-03785-1
Soyoon Bak, Yonghyeon Jeon
Abstract This paper introduces a new perspective of the traditional view on the velocity of each physical particle in the coupled Burgers’ equation in the backward semi-Lagrangian method (BSLM). The proposed methods reduce the number of Cauchy problems to be solved by observing a single virtual characteristic curve with a velocity. This can drastically reduce the computational cost of determining the departure point. Then, we solve the derived system reflected by the single virtual characteristic curve. Moreover, an efficient strategy for the derived linear system of equations is provided. Four examples are tested to demonstrate the adaptability and efficiency of the proposed method. The test results show that the proposed method has third- and fourth-order accuracy in time and space, respectively. In addition, compared with the existing method of solving the problem along two particles with different velocities, we confirm that the proposed method significantly reduces computational cost while maintaining accuracy well.
{"title":"Algorithms for coupled Burgers’ equations by sharing characteristic curves within BSLM","authors":"Soyoon Bak, Yonghyeon Jeon","doi":"10.1186/s13662-023-03785-1","DOIUrl":"https://doi.org/10.1186/s13662-023-03785-1","url":null,"abstract":"Abstract This paper introduces a new perspective of the traditional view on the velocity of each physical particle in the coupled Burgers’ equation in the backward semi-Lagrangian method (BSLM). The proposed methods reduce the number of Cauchy problems to be solved by observing a single virtual characteristic curve with a velocity. This can drastically reduce the computational cost of determining the departure point. Then, we solve the derived system reflected by the single virtual characteristic curve. Moreover, an efficient strategy for the derived linear system of equations is provided. Four examples are tested to demonstrate the adaptability and efficiency of the proposed method. The test results show that the proposed method has third- and fourth-order accuracy in time and space, respectively. In addition, compared with the existing method of solving the problem along two particles with different velocities, we confirm that the proposed method significantly reduces computational cost while maintaining accuracy well.","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":"2016 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135095804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-22DOI: 10.1186/s13662-023-03784-2
Roumen Anguelov, Jean M.-S. Lubuma
Abstract We present two results on the analysis of discrete dynamical systems and finite difference discretizations of continuous dynamical systems, which preserve their dynamics and essential properties. The first result provides a sufficient condition for forward invariance of a set under discrete dynamical systems of specific type, namely time-reversible ones. The condition involves only the boundary of the set. It is a discrete analog of the widely used tangent condition for continuous systems ( viz. the vector field points either inwards or is tangent to the boundary of the set). The second result is nonstandard finite difference (NSFD) scheme for dynamical systems defined by systems of ordinary differential equations. The NSFD scheme preserves the hyperbolic equilibria of the continuous system as well as their stability. Further, the scheme is time reversible and, through the first result, inherits from the continuous model the forward invariance of the domain. We show that the scheme is of second order, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes. The application of these results, including some numerical simulations for invariant sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type epidemiological model, which may have arbitrary large number of infective or recovered/removed compartments.
{"title":"Forward invariant set preservation in discrete dynamical systems and numerical schemes for ODEs: application in biosciences","authors":"Roumen Anguelov, Jean M.-S. Lubuma","doi":"10.1186/s13662-023-03784-2","DOIUrl":"https://doi.org/10.1186/s13662-023-03784-2","url":null,"abstract":"Abstract We present two results on the analysis of discrete dynamical systems and finite difference discretizations of continuous dynamical systems, which preserve their dynamics and essential properties. The first result provides a sufficient condition for forward invariance of a set under discrete dynamical systems of specific type, namely time-reversible ones. The condition involves only the boundary of the set. It is a discrete analog of the widely used tangent condition for continuous systems ( viz. the vector field points either inwards or is tangent to the boundary of the set). The second result is nonstandard finite difference (NSFD) scheme for dynamical systems defined by systems of ordinary differential equations. The NSFD scheme preserves the hyperbolic equilibria of the continuous system as well as their stability. Further, the scheme is time reversible and, through the first result, inherits from the continuous model the forward invariance of the domain. We show that the scheme is of second order, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes. The application of these results, including some numerical simulations for invariant sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type epidemiological model, which may have arbitrary large number of infective or recovered/removed compartments.","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-30DOI: 10.1186/s13662-023-03783-3
Jaegi Jeon, Jeonggyu Huh, Geonwoo Kim
{"title":"An analytical approach to the pricing of an exchange option with default risk under a stochastic volatility model","authors":"Jaegi Jeon, Jeonggyu Huh, Geonwoo Kim","doi":"10.1186/s13662-023-03783-3","DOIUrl":"https://doi.org/10.1186/s13662-023-03783-3","url":null,"abstract":"","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44185915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-17DOI: 10.1186/s13662-023-03782-4
M. Alghanmi, Shahad Alqurayqiri
{"title":"Existence results for fractional neutral functional differential equations with infinite delay and nonlocal boundary conditions","authors":"M. Alghanmi, Shahad Alqurayqiri","doi":"10.1186/s13662-023-03782-4","DOIUrl":"https://doi.org/10.1186/s13662-023-03782-4","url":null,"abstract":"","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45521720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-16DOI: 10.1186/s13662-023-03781-5
J. Shin, R. Miyazaki, Dohan Kim
{"title":"Stability regions of discrete linear periodic systems with delayed feedback controls","authors":"J. Shin, R. Miyazaki, Dohan Kim","doi":"10.1186/s13662-023-03781-5","DOIUrl":"https://doi.org/10.1186/s13662-023-03781-5","url":null,"abstract":"","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45362677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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