We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the k...We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force.In this paper,we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.展开更多
Machine learning(ML)provides a new surrogate method for investigating groundwater flow dynamics in unsaturated soils.Traditional pure data-driven methods(e.g.deep neural network,DNN)can provide rapid predictions,but t...Machine learning(ML)provides a new surrogate method for investigating groundwater flow dynamics in unsaturated soils.Traditional pure data-driven methods(e.g.deep neural network,DNN)can provide rapid predictions,but they do require sufficient on-site data for accurate training,and lack interpretability to the physical processes within the data.In this paper,we provide a physics and equalityconstrained artificial neural network(PECANN),to derive unsaturated infiltration solutions with a small amount of initial and boundary data.PECANN takes the physics-informed neural network(PINN)as a foundation,encodes the unsaturated infiltration physical laws(i.e.Richards equation,RE)into the loss function,and uses the augmented Lagrangian method to constrain the learning process of the solutions of RE by adding stronger penalty for the initial and boundary conditions.Four unsaturated infiltration cases are designed to test the training performance of PECANN,i.e.one-dimensional(1D)steady-state unsaturated infiltration,1D transient-state infiltration,two-dimensional(2D)transient-state infiltration,and 1D coupled unsaturated infiltration and deformation.The predicted results of PECANN are compared with the finite difference solutions or analytical solutions.The results indicate that PECANN can accurately capture the variations of pressure head during the unsaturated infiltration,and present higher precision and robustness than DNN and PINN.It is also revealed that PECANN can achieve the same accuracy as the finite difference method with fewer initial and boundary training data.Additionally,we investigate the effect of the hyperparameters of PECANN on solving RE problem.PECANN provides an effective tool for simulating unsaturated infiltration.展开更多
In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatmen...In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.展开更多
Oil spills continue to generate various issues and concerns regarding their effect and behavior in the marine environment,owing to the related potential for detrimental environmental,economic and social implications.I...Oil spills continue to generate various issues and concerns regarding their effect and behavior in the marine environment,owing to the related potential for detrimental environmental,economic and social implications.It is essential to have a solid understanding of the ways in which oil interacts with the water and the coastal ecosystems that are located nearby.This study proposes a simplified model for predicting the plume-like transport behavior of heavy Bunker C fuel oil discharging downward from an acutely-angled broken pipeline located on the water surface.The results show that the spill overall profile is articulated in three major flow areas.The first,is the source field,i.e.,a region near the origin of the initial jet,followed by the intermediate or transport field,namely,the region where the jet oil flow transitions into an underwater oil plume flow and starts to move horizontally,and finally,the far-field,where the oil re-surface and spreads onto the shore at a significant distance from the spill site.The behavior of the oil in the intermediate field is investigated using a simplified injection-type oil spill model capable of mimicking the undersea trapping and lateral migration of an oil plume originating from a negatively buoyant jet spill.A rectangular domain with proper boundary conditions is used to implement the model.The Projection approach is used to discretize a modified version of the Navier-Stokes equations in two dimensions.A benchmark fluid flow issue is used to verify the model and the results indicate a reasonable relationship between specific gravity and depth as well as agreement with the aerial data and a vertical temperature profile plot.展开更多
2,4-dinitroanisole(DNAN)is a good replacement for 2,4,6-trinitrotoluene(TNT)in melt-cast explosives due to its superior insensitivity.With the increasing use of DNAN-based melt-cast explosives,the prediction of reacti...2,4-dinitroanisole(DNAN)is a good replacement for 2,4,6-trinitrotoluene(TNT)in melt-cast explosives due to its superior insensitivity.With the increasing use of DNAN-based melt-cast explosives,the prediction of reaction violence and hazard assessment of the explosives subjected to shock is of great significance.This study investigated the shock initiation characteristics for a DNAN-based melt-cast explosive,DHFA,using the one-dimensional Lagrangian apparatus.The embedded manganin gauges in the apparatus record the pressure histories at four Lagrangian positions and show that shock-todetonation transition in DHFA needs a high input shock pressure.The experimental data are analyzed to calibrate the Ignition and Growth model.The calibration is performed using an objective function based on both pressure history and the arrival time of shock.Good agreement between experimental and calculated pressure histories indicates the high accuracy of the calibrated parameters with the optimization method.展开更多
Black-Scholes Model (B-SM) simulates the dynamics of financial market and contains instruments such as options and puts which are major indices requiring solution. B-SM is known to estimate the correct prices of Europ...Black-Scholes Model (B-SM) simulates the dynamics of financial market and contains instruments such as options and puts which are major indices requiring solution. B-SM is known to estimate the correct prices of European Stock options and establish the theoretical foundation for Option pricing. Therefore, this paper evaluates the Black-Schole model in simulating the European call in a cash flow in the dependent drift and focuses on obtaining analytic and then approximate solution for the model. The work also examines Fokker Planck Equation (FPE) and extracts the link between FPE and B-SM for non equilibrium systems. The B-SM is then solved via the Elzaki transform method (ETM). The computational procedures were obtained using MAPLE 18 with the solution provided in the form of convergent series.展开更多
The Bayesian structural equation model integrates the principles of Bayesian statistics, providing a more flexible and comprehensive modeling framework. In exploring complex relationships between variables, handling u...The Bayesian structural equation model integrates the principles of Bayesian statistics, providing a more flexible and comprehensive modeling framework. In exploring complex relationships between variables, handling uncertainty, and dealing with missing data, the Bayesian structural equation model demonstrates unique advantages. Therefore, Bayesian methods are used in this paper to establish a structural equation model of innovative talent cognition, with the measurement of college students’ cognition of innovative talent being studied. An in-depth analysis is conducted on the effects of innovative self-efficacy, social resources, innovative personality traits, and school education, aiming to explore the factors influencing college students’ innovative talent. The results indicate that innovative self-efficacy plays a key role in perception, social resources are significantly positively correlated with the perception of innovative talents, innovative personality tendencies and school education are positively correlated with the perception of innovative talents, but the impact is not significant.展开更多
The manuscript introduces an “ab initio” quantum model to deduce the Maxwell equations. After general considerations and laying out the model’s theoretical framework, these equations can be derived alongside a broa...The manuscript introduces an “ab initio” quantum model to deduce the Maxwell equations. After general considerations and laying out the model’s theoretical framework, these equations can be derived alongside a broad variety of other results. Specifically, a corollary of the present model proposes a possible mechanism underlying the formation of magnetic monopoles and allows estimating their formation energy in order of magnitude.展开更多
A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearize...A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearized under the nonlinear transformation. Various exact solutions of the WBK model equations are obtained via the nonlinear transformation with the aid of solutions for the linear equation.展开更多
A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency ...A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.展开更多
A new lattice Bhatnagar-Gross-Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional spa...A new lattice Bhatnagar-Gross-Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods. Satisfactory results are obtained by the numerical simulations.展开更多
A stable high-order Runge-Kutta discontinuous Galerkin(RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is k...A stable high-order Runge-Kutta discontinuous Galerkin(RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional(2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.展开更多
In the present paper, the longitudinal dynamic flight stability properties of two model insects are predicted by an approximate theory and computed by numerical sim- ulation. The theory is based on the averaged model ...In the present paper, the longitudinal dynamic flight stability properties of two model insects are predicted by an approximate theory and computed by numerical sim- ulation. The theory is based on the averaged model (which assumes that the frequency of wingbeat is sufficiently higher than that of the body motion, so that the flapping wings' degrees of freedom relative to the body can be dropped and the wings can be replaced by wingbeat-cycle-average forces and moments); the simulation solves the complete equations of motion coupled with the Navier-Stokes equations. Comparison between the theory and the simulation provides a test to the validity of the assumptions in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency (164 Hz) and the other is a model hawkmoth which has relatively low wingbeat frequency (26 Hz). The results show that the averaged model is valid for the hawkmoth as well as for the dronefly. Since the wingbeat frequency of the hawkmoth is relatively low (the characteristic times of the natural modes of motion of the body divided by wingbeat period are relatively large) compared with many other insects, that the theory based on the averaged model is valid for the hawkmoth means that it could be valid for many insects.展开更多
Purpose: To review some of the basic models, differential equations and solutions, both analytic and numerical, which produce time courses for the fractions of Susceptible (S), Infectious (I) and Recovered (R) fractio...Purpose: To review some of the basic models, differential equations and solutions, both analytic and numerical, which produce time courses for the fractions of Susceptible (S), Infectious (I) and Recovered (R) fractions of the population during the epidemic and/or endemic conditions. Methods: Two and three-compartment models with analytic solutions to the proposed linear differential equations as well as models based on the non-linear differential equations first proposed by Kermack and McKendrick (KM) [1] a century ago are considered. The equations reviewed include the ability to slide between so-called Susceptible-Infected-Recovered (SIR), Susceptible-Infectious-Susceptible (SIS), Susceptible-Infectious (SI) and Susceptible-Infectious-Recovered-Susceptible (SIRS) models, effectively moving from epidemic to endemic characterizations of infectious disease. Results: Both the linear and KM model yield typical “curves” of the infected fraction being sought “to flatten” with the effects of social distancing/masking efforts and/or pharmaceutical interventions. Demonstrative applications of the solutions to fit real COVID-19 data, including linear and KM SIR fit data from the first 100 days following “lockdown” in the authors’ locale and to the total number of cases in the USA over the course of 1 year with SI and SIS models are provided. Conclusions: COVID-19 took us all by surprise, all wondering how to help. Spreading a basic understanding of some of the mathematics used by epidemiologists to model infectious diseases seemed like a good place to start and served as the primary purpose for this tutorial.展开更多
Due to high cost of fixing failures, safety concerns, and legal liabilities, organizations need to produce software that is highly reliable. Software reliability growth models have been developed by software developer...Due to high cost of fixing failures, safety concerns, and legal liabilities, organizations need to produce software that is highly reliable. Software reliability growth models have been developed by software developers in tracking and measuring the growth of reliability. Most of the Software Reliability Growth Models, which have been proposed, treat the event of software fault detection in the testing and operational phase as a counting process. Moreover, if the size of software system is large, the number of software faults detected during the testing phase becomes large, and the change of the number of faults which are detected and removed through debugging activities becomes sufficiently small compared with the initial fault content at the beginning of the testing phase. Therefore in such a situation, we can model the software fault detection process as a stochastic process with a continuous state space. Recently, Artificial Neural Networks (ANN) have been applied in software reliability growth prediction. In this paper, we propose an ANN based software reliability growth model based on Ito type of stochastic differential equation. The model has been validated, evaluated and compared with other existing NHPP model by applying it on actual failure/fault removal data sets cited from real software development projects. The proposed model integrated with the concept of stochastic differential equation performs comparatively better than the existing NHPP based model.展开更多
基金supported by the National Natural Science Foundation of China (12001033)。
文摘We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force.In this paper,we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.
基金funding support from the science and technology innovation Program of Hunan Province(Grant No.2023RC1017)Hunan Provincial Postgraduate Research and Innovation Project(Grant No.CX20220109)National Natural Science Foundation of China Youth Fund(Grant No.52208378).
文摘Machine learning(ML)provides a new surrogate method for investigating groundwater flow dynamics in unsaturated soils.Traditional pure data-driven methods(e.g.deep neural network,DNN)can provide rapid predictions,but they do require sufficient on-site data for accurate training,and lack interpretability to the physical processes within the data.In this paper,we provide a physics and equalityconstrained artificial neural network(PECANN),to derive unsaturated infiltration solutions with a small amount of initial and boundary data.PECANN takes the physics-informed neural network(PINN)as a foundation,encodes the unsaturated infiltration physical laws(i.e.Richards equation,RE)into the loss function,and uses the augmented Lagrangian method to constrain the learning process of the solutions of RE by adding stronger penalty for the initial and boundary conditions.Four unsaturated infiltration cases are designed to test the training performance of PECANN,i.e.one-dimensional(1D)steady-state unsaturated infiltration,1D transient-state infiltration,two-dimensional(2D)transient-state infiltration,and 1D coupled unsaturated infiltration and deformation.The predicted results of PECANN are compared with the finite difference solutions or analytical solutions.The results indicate that PECANN can accurately capture the variations of pressure head during the unsaturated infiltration,and present higher precision and robustness than DNN and PINN.It is also revealed that PECANN can achieve the same accuracy as the finite difference method with fewer initial and boundary training data.Additionally,we investigate the effect of the hyperparameters of PECANN on solving RE problem.PECANN provides an effective tool for simulating unsaturated infiltration.
文摘In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.
文摘Oil spills continue to generate various issues and concerns regarding their effect and behavior in the marine environment,owing to the related potential for detrimental environmental,economic and social implications.It is essential to have a solid understanding of the ways in which oil interacts with the water and the coastal ecosystems that are located nearby.This study proposes a simplified model for predicting the plume-like transport behavior of heavy Bunker C fuel oil discharging downward from an acutely-angled broken pipeline located on the water surface.The results show that the spill overall profile is articulated in three major flow areas.The first,is the source field,i.e.,a region near the origin of the initial jet,followed by the intermediate or transport field,namely,the region where the jet oil flow transitions into an underwater oil plume flow and starts to move horizontally,and finally,the far-field,where the oil re-surface and spreads onto the shore at a significant distance from the spill site.The behavior of the oil in the intermediate field is investigated using a simplified injection-type oil spill model capable of mimicking the undersea trapping and lateral migration of an oil plume originating from a negatively buoyant jet spill.A rectangular domain with proper boundary conditions is used to implement the model.The Projection approach is used to discretize a modified version of the Navier-Stokes equations in two dimensions.A benchmark fluid flow issue is used to verify the model and the results indicate a reasonable relationship between specific gravity and depth as well as agreement with the aerial data and a vertical temperature profile plot.
基金Scientific Research Foundation for High-level Talents of Anhui University of Science and Technology(Grant No.2021yjrc38)Anhui Provincial Natural Science Foundation(Grant No.2208085QA27)+1 种基金National Natural Science Foundation of China(Grant Nos.11972046,12002266)the authors would like to thank these foundations for financial support.
文摘2,4-dinitroanisole(DNAN)is a good replacement for 2,4,6-trinitrotoluene(TNT)in melt-cast explosives due to its superior insensitivity.With the increasing use of DNAN-based melt-cast explosives,the prediction of reaction violence and hazard assessment of the explosives subjected to shock is of great significance.This study investigated the shock initiation characteristics for a DNAN-based melt-cast explosive,DHFA,using the one-dimensional Lagrangian apparatus.The embedded manganin gauges in the apparatus record the pressure histories at four Lagrangian positions and show that shock-todetonation transition in DHFA needs a high input shock pressure.The experimental data are analyzed to calibrate the Ignition and Growth model.The calibration is performed using an objective function based on both pressure history and the arrival time of shock.Good agreement between experimental and calculated pressure histories indicates the high accuracy of the calibrated parameters with the optimization method.
文摘Black-Scholes Model (B-SM) simulates the dynamics of financial market and contains instruments such as options and puts which are major indices requiring solution. B-SM is known to estimate the correct prices of European Stock options and establish the theoretical foundation for Option pricing. Therefore, this paper evaluates the Black-Schole model in simulating the European call in a cash flow in the dependent drift and focuses on obtaining analytic and then approximate solution for the model. The work also examines Fokker Planck Equation (FPE) and extracts the link between FPE and B-SM for non equilibrium systems. The B-SM is then solved via the Elzaki transform method (ETM). The computational procedures were obtained using MAPLE 18 with the solution provided in the form of convergent series.
文摘The Bayesian structural equation model integrates the principles of Bayesian statistics, providing a more flexible and comprehensive modeling framework. In exploring complex relationships between variables, handling uncertainty, and dealing with missing data, the Bayesian structural equation model demonstrates unique advantages. Therefore, Bayesian methods are used in this paper to establish a structural equation model of innovative talent cognition, with the measurement of college students’ cognition of innovative talent being studied. An in-depth analysis is conducted on the effects of innovative self-efficacy, social resources, innovative personality traits, and school education, aiming to explore the factors influencing college students’ innovative talent. The results indicate that innovative self-efficacy plays a key role in perception, social resources are significantly positively correlated with the perception of innovative talents, innovative personality tendencies and school education are positively correlated with the perception of innovative talents, but the impact is not significant.
文摘The manuscript introduces an “ab initio” quantum model to deduce the Maxwell equations. After general considerations and laying out the model’s theoretical framework, these equations can be derived alongside a broad variety of other results. Specifically, a corollary of the present model proposes a possible mechanism underlying the formation of magnetic monopoles and allows estimating their formation energy in order of magnitude.
文摘A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearized under the nonlinear transformation. Various exact solutions of the WBK model equations are obtained via the nonlinear transformation with the aid of solutions for the linear equation.
文摘A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 70271069 and 60073044).
文摘A new lattice Bhatnagar-Gross-Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods. Satisfactory results are obtained by the numerical simulations.
基金Project supported by the National Natural Science Foundation of China(No.11302017)
文摘A stable high-order Runge-Kutta discontinuous Galerkin(RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional(2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.
基金supported by the National Natural Science Foundation of China (10732030) and the 111 Project (B07009)
文摘In the present paper, the longitudinal dynamic flight stability properties of two model insects are predicted by an approximate theory and computed by numerical sim- ulation. The theory is based on the averaged model (which assumes that the frequency of wingbeat is sufficiently higher than that of the body motion, so that the flapping wings' degrees of freedom relative to the body can be dropped and the wings can be replaced by wingbeat-cycle-average forces and moments); the simulation solves the complete equations of motion coupled with the Navier-Stokes equations. Comparison between the theory and the simulation provides a test to the validity of the assumptions in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency (164 Hz) and the other is a model hawkmoth which has relatively low wingbeat frequency (26 Hz). The results show that the averaged model is valid for the hawkmoth as well as for the dronefly. Since the wingbeat frequency of the hawkmoth is relatively low (the characteristic times of the natural modes of motion of the body divided by wingbeat period are relatively large) compared with many other insects, that the theory based on the averaged model is valid for the hawkmoth means that it could be valid for many insects.
文摘Purpose: To review some of the basic models, differential equations and solutions, both analytic and numerical, which produce time courses for the fractions of Susceptible (S), Infectious (I) and Recovered (R) fractions of the population during the epidemic and/or endemic conditions. Methods: Two and three-compartment models with analytic solutions to the proposed linear differential equations as well as models based on the non-linear differential equations first proposed by Kermack and McKendrick (KM) [1] a century ago are considered. The equations reviewed include the ability to slide between so-called Susceptible-Infected-Recovered (SIR), Susceptible-Infectious-Susceptible (SIS), Susceptible-Infectious (SI) and Susceptible-Infectious-Recovered-Susceptible (SIRS) models, effectively moving from epidemic to endemic characterizations of infectious disease. Results: Both the linear and KM model yield typical “curves” of the infected fraction being sought “to flatten” with the effects of social distancing/masking efforts and/or pharmaceutical interventions. Demonstrative applications of the solutions to fit real COVID-19 data, including linear and KM SIR fit data from the first 100 days following “lockdown” in the authors’ locale and to the total number of cases in the USA over the course of 1 year with SI and SIS models are provided. Conclusions: COVID-19 took us all by surprise, all wondering how to help. Spreading a basic understanding of some of the mathematics used by epidemiologists to model infectious diseases seemed like a good place to start and served as the primary purpose for this tutorial.
文摘Due to high cost of fixing failures, safety concerns, and legal liabilities, organizations need to produce software that is highly reliable. Software reliability growth models have been developed by software developers in tracking and measuring the growth of reliability. Most of the Software Reliability Growth Models, which have been proposed, treat the event of software fault detection in the testing and operational phase as a counting process. Moreover, if the size of software system is large, the number of software faults detected during the testing phase becomes large, and the change of the number of faults which are detected and removed through debugging activities becomes sufficiently small compared with the initial fault content at the beginning of the testing phase. Therefore in such a situation, we can model the software fault detection process as a stochastic process with a continuous state space. Recently, Artificial Neural Networks (ANN) have been applied in software reliability growth prediction. In this paper, we propose an ANN based software reliability growth model based on Ito type of stochastic differential equation. The model has been validated, evaluated and compared with other existing NHPP model by applying it on actual failure/fault removal data sets cited from real software development projects. The proposed model integrated with the concept of stochastic differential equation performs comparatively better than the existing NHPP based model.