This study investigates the dynamics of pneumococcal pneumonia using a novel fractal-fractional Susceptible-Carrier-Infected-Recovered model formulated with the Atangana-Baleanu in Caputo(ABC)sense.Unlike traditional ...This study investigates the dynamics of pneumococcal pneumonia using a novel fractal-fractional Susceptible-Carrier-Infected-Recovered model formulated with the Atangana-Baleanu in Caputo(ABC)sense.Unlike traditional epidemiological models that rely on classical or Caputo fractional derivatives,the proposed model incorporates nonlocal memory effects,hereditary properties,and complex transmission dynamics through fractalfractional calculus.The Atangana-Baleanu operator,with its non-singular Mittag-Leffler kernel,ensures a more realistic representation of disease progression compared to classical integer-order models and singular kernel-based fractional models.The study establishes the existence and uniqueness of the proposed system and conducts a comprehensive stability analysis,including local and global stability.Furthermore,numerical simulations illustrate the effectiveness of the ABC operator in capturing long-memory effects and nonlocal interactions in disease transmission.The results provide valuable insights into public health interventions,particularly in optimizing vaccination strategies,treatment approaches,and mitigation measures.By extending epidemiological modeling through fractal-fractional derivatives,this study offers an advanced framework for analyzing infectious disease dynamics with enhanced accuracy and predictive capabilities.展开更多
Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, et...Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.展开更多
In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a h...In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law.The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization.The fully discrete scheme is proved to be stable and convergent with an accuracy of O(τ^(2)+N-r),whereτis the time step-size and N is the polynomial degree.To reduce the memory requirements and computational cost,a fast method is developed,which is based on a globally uniform approximation of the trapezoidal rule for integrals on the real line.The strict convergence of the numerical scheme with this fast method is proved.We present the results of several numerical experiments to verify the effectiveness of the proposed method.Finally,we simulate the unsteady fractional MHD flow and heat transfer of the generalized second-grade fluid through a porous medium.The effects of the relevant parameters on the velocity and temperature are presented and analyzed in detail.展开更多
For the three-dimensional convection-dominated problem of dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Fract...For the three-dimensional convection-dominated problem of dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Fractional steps techniques are needed to convert a multi-dimensional problem into a series of successive one-dimensional problems. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates are adopted. Optimal order estimates are derived to determine the error in the second order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources and predicting the consequences of seawater intrusion and protection projects.展开更多
In this paper,a two dimensional(2D)fractional Black-Scholes(FBS)model on two assets following independent geometric Lévy processes is solved numerically.A high order convergent implicit difference scheme is const...In this paper,a two dimensional(2D)fractional Black-Scholes(FBS)model on two assets following independent geometric Lévy processes is solved numerically.A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established.The fractional derivative is a quasidifferential operator,whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix.In order to speed up calculation and save storage space,a fast bi-conjugate gradient stabilized(FBi-CGSTAB)method is proposed to solve the resultant linear system.Finally,one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique.The pricing of a European Call-on-Min option is showed in the other example,in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.展开更多
For combinatorial system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward and two-d...For combinatorial system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as implicit-explicit difference scheme, calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L 2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources. Keywords: combinatorial system, multilayer dynamics of fluids in porous media, two-class upwind finite difference fractional steps method, convergence, numerical simulation of energy sources.展开更多
In this paper,a high-accuracy numerical scheme is developed for long-time dynamic simulations of 2D and 3D wave propagation phenomena.In the derivation of the present approach,the second order time derivative of the p...In this paper,a high-accuracy numerical scheme is developed for long-time dynamic simulations of 2D and 3D wave propagation phenomena.In the derivation of the present approach,the second order time derivative of the physical quantity in the wave equation is treated as a substitution variable.Based on the temporal discretization with the Krylov deferred correction(KDC)technique,the original wave problem is then converted into the modified Helmholtz equation.The transformed boundary value problem(BVP)in space is efficiently simulated by using the meshless generalized finite difference method(GFDM)with Taylor series after truncating the second and fourth order approximations.The developed scheme is finally verified by four numerical experiments including cases with complicated domains or the temporally piecewise defined source function.Numerical results match with the analytical solutions and results by the COMSOL software,which demonstrates that the developed KDC-GFDM can allow large time-step sizes for wave propagation problems in longtime intervals.展开更多
基金funded by the Research,Development,and Innovation Authority(RDIA)-Kingdom of Saudi Arabia-with grant number 12803-baha-2023-BU-R-3-1-EI.
文摘This study investigates the dynamics of pneumococcal pneumonia using a novel fractal-fractional Susceptible-Carrier-Infected-Recovered model formulated with the Atangana-Baleanu in Caputo(ABC)sense.Unlike traditional epidemiological models that rely on classical or Caputo fractional derivatives,the proposed model incorporates nonlocal memory effects,hereditary properties,and complex transmission dynamics through fractalfractional calculus.The Atangana-Baleanu operator,with its non-singular Mittag-Leffler kernel,ensures a more realistic representation of disease progression compared to classical integer-order models and singular kernel-based fractional models.The study establishes the existence and uniqueness of the proposed system and conducts a comprehensive stability analysis,including local and global stability.Furthermore,numerical simulations illustrate the effectiveness of the ABC operator in capturing long-memory effects and nonlocal interactions in disease transmission.The results provide valuable insights into public health interventions,particularly in optimizing vaccination strategies,treatment approaches,and mitigation measures.By extending epidemiological modeling through fractal-fractional derivatives,this study offers an advanced framework for analyzing infectious disease dynamics with enhanced accuracy and predictive capabilities.
文摘Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.
基金supported by the Project of the National Key R&D Program(Grant No.2021YFA1000202)National Natural Science Foundation of China(Grant Nos.12120101001,12001326 and 12171283)+2 种基金Natural Science Foundation of Shandong Province(Grant Nos.ZR2021ZD03,ZR2020QA032 and ZR2019ZD42)China Postdoctoral Science Foundation(Grant Nos.BX20190191 and 2020M672038)the Startup Fund from Shandong University(Grant No.11140082063130)。
文摘In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law.The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization.The fully discrete scheme is proved to be stable and convergent with an accuracy of O(τ^(2)+N-r),whereτis the time step-size and N is the polynomial degree.To reduce the memory requirements and computational cost,a fast method is developed,which is based on a globally uniform approximation of the trapezoidal rule for integrals on the real line.The strict convergence of the numerical scheme with this fast method is proved.We present the results of several numerical experiments to verify the effectiveness of the proposed method.Finally,we simulate the unsteady fractional MHD flow and heat transfer of the generalized second-grade fluid through a porous medium.The effects of the relevant parameters on the velocity and temperature are presented and analyzed in detail.
基金Project supported by the Major State Basic Research Program of China (No.G1999032803)the National Tackling Key Problems Program (No.20050200069)the National Natural Science Foundation of China (Nos.10372052, 10271066)the Doctoral Foundation of Ministry of Education of China (No.20030422047).
文摘For the three-dimensional convection-dominated problem of dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Fractional steps techniques are needed to convert a multi-dimensional problem into a series of successive one-dimensional problems. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates are adopted. Optimal order estimates are derived to determine the error in the second order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources and predicting the consequences of seawater intrusion and protection projects.
基金supported by the Natural Science Foundation of Fujian Province2017J01555,2017J01502,2017J01557 and 2019J01646the National NSF of China 11201077+1 种基金China Scholarship Fundthe Natural Science Foundation of Fujian Provincial Department of Education JAT160274
文摘In this paper,a two dimensional(2D)fractional Black-Scholes(FBS)model on two assets following independent geometric Lévy processes is solved numerically.A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established.The fractional derivative is a quasidifferential operator,whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix.In order to speed up calculation and save storage space,a fast bi-conjugate gradient stabilized(FBi-CGSTAB)method is proposed to solve the resultant linear system.Finally,one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique.The pricing of a European Call-on-Min option is showed in the other example,in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.
基金This work was supported by the Major State Basic Research Program of China(Grant No. 1990328) the National Tackling Key Problem Program, the National Natural Science Foundation of China (Grant Nos. 19871051 and 19972039) the Doctorate Foundation
文摘For combinatorial system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as implicit-explicit difference scheme, calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L 2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources. Keywords: combinatorial system, multilayer dynamics of fluids in porous media, two-class upwind finite difference fractional steps method, convergence, numerical simulation of energy sources.
基金supported by the National Natural Science Foundation of China(Grant No.11802165)the Natural Science Foundation of Shandong Province of China(Grant No.ZR2017BA003)the China Postdoctoral Science Foundation(Grant No.2019M650158).
文摘In this paper,a high-accuracy numerical scheme is developed for long-time dynamic simulations of 2D and 3D wave propagation phenomena.In the derivation of the present approach,the second order time derivative of the physical quantity in the wave equation is treated as a substitution variable.Based on the temporal discretization with the Krylov deferred correction(KDC)technique,the original wave problem is then converted into the modified Helmholtz equation.The transformed boundary value problem(BVP)in space is efficiently simulated by using the meshless generalized finite difference method(GFDM)with Taylor series after truncating the second and fourth order approximations.The developed scheme is finally verified by four numerical experiments including cases with complicated domains or the temporally piecewise defined source function.Numerical results match with the analytical solutions and results by the COMSOL software,which demonstrates that the developed KDC-GFDM can allow large time-step sizes for wave propagation problems in longtime intervals.
