Several numerical methods of differential equations and their applications in ballistic calculation are discussed for the purpose of simplification of the dynamic differential equations of projectile trajectory.Progra...Several numerical methods of differential equations and their applications in ballistic calculation are discussed for the purpose of simplification of the dynamic differential equations of projectile trajectory.Program simulations of Euler method,Heun method,lassic fourth-order Runge Kutta（RK4）method,ABM method and Hamming method are achieved based on Matlab.In addtion,the approximate solutions,local truncation errors and calculation time of the dynamic differential equations are obtained.By analyzing the simultaion results,the advantages and disadvantages of these methods are compared,which provides a basis for choice of ballistic calculation methods.展开更多
The Runge-Kutta discontinuous Galerkin finite element method （RK-DGFEM） is introduced to solve the classical resonator problem in the time domain. DGFEM uses unstructured grid discretization in the space domain and ...The Runge-Kutta discontinuous Galerkin finite element method （RK-DGFEM） is introduced to solve the classical resonator problem in the time domain. DGFEM uses unstructured grid discretization in the space domain and it is explicit in the time domain. Consequently it is a best mixture of FEM and finite volume method （FVM）. RK-DGFEM can obtain local high-order accuracy by using high-order polynomial basis. Numerical experiments of transverse magnetic （TM） wave propagation in a 2-D resonator are performed. A high-order Lagrange polynomial basis is adopted. Numerical results agree well with analytical solution. And different order Lagrange interpolation polynomial basis impacts on simulation result accuracy are discussed. Computational results indicate that the accuracy is evidently improved when the order of interpolation basis is increased. Finally, L^2 errors of different order polynomial basis in RK-DGFEM are presented. Computational results show that L^2 error declines exponentially as the order of basis increases.展开更多
We report a new method for calculating transmission coefficients across arbitrary potential barriers based on the Runge-Kutta method. A numerical solution of the Schrodinger equation is calculated using the Runge-Kutt...We report a new method for calculating transmission coefficients across arbitrary potential barriers based on the Runge-Kutta method. A numerical solution of the Schrodinger equation is calculated using the Runge-Kutta method,and a new model is established to analyze the numerical results to find the transmission coefficient. This technique is applied to various cases, such as parabolic potential barrier and double-barrier structures. Transmission probability with high precision is obtained and discussed. The tunnelling current density through a MOS structure is also explored and the result coincides with the Fowler-Nordheim model,which indicates the applicability of our method.展开更多
The steady, laminar, incompressible and two dimensional micropolar flow between two porous disks was investigated using optimal homotopy asymptotic method(OHAM) and fourth order Runge–Kutta numerical method. Comparis...The steady, laminar, incompressible and two dimensional micropolar flow between two porous disks was investigated using optimal homotopy asymptotic method(OHAM) and fourth order Runge–Kutta numerical method. Comparison between OHAM and numerical method shows that OHAM is an exact and high efficient method for solving these kinds of problems. The results are presented to study the velocity and rotation profiles for different physical parameters such as Reynolds number, vortex viscosity parameter, spin gradient viscosity and microinertia density parameter. As an important outcome, the magnitude of the microrotation increases with an increase in the values of injection velocity while it decreases by increasing the values of suction velocity.展开更多
A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration o...A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.展开更多
The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method （...The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method （LWPM）. Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge-Kutta method of order four （RK-4）. Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.展开更多
In this paper a definitely new analytical technique, predictor homotopy analysis method （PHAM）, is employed to solve the problem of two-dimensional nanofluid flow through expanding or contracting gaps with permeable...In this paper a definitely new analytical technique, predictor homotopy analysis method （PHAM）, is employed to solve the problem of two-dimensional nanofluid flow through expanding or contracting gaps with permeable walls. Moreover, comparison of the PHAM results with numerical results obtained by the shooting method coupled with a Runge- Kutta integration method as well as previously published study results demonstrates high accuracy for this technique. The fluid in the channel is water containing different nanoparticles： silver, copper, copper oxide, titanium oxide, and aluminum oxide. The effects of the nanoparticle volume fraction, Reynolds number, wall expansion ratio, and different types of nanoparticles on the flow are discussed.展开更多
Optimal control theory is applied to a system of ordinary differential equations modeling banana Xanthomonas wilt within plantations. The objective is to reduce the proportion of infected plants by use of controls rep...Optimal control theory is applied to a system of ordinary differential equations modeling banana Xanthomonas wilt within plantations. The objective is to reduce the proportion of infected plants by use of controls representing two types of preventive methods： vector and contaminated tool prevention. The optimal controls are characterized in terms of the optimality system, which is solved analytically and numerically for several scenarios.展开更多
文摘Several numerical methods of differential equations and their applications in ballistic calculation are discussed for the purpose of simplification of the dynamic differential equations of projectile trajectory.Program simulations of Euler method,Heun method,lassic fourth-order Runge Kutta（RK4）method,ABM method and Hamming method are achieved based on Matlab.In addtion,the approximate solutions,local truncation errors and calculation time of the dynamic differential equations are obtained.By analyzing the simultaion results,the advantages and disadvantages of these methods are compared,which provides a basis for choice of ballistic calculation methods.
文摘The Runge-Kutta discontinuous Galerkin finite element method （RK-DGFEM） is introduced to solve the classical resonator problem in the time domain. DGFEM uses unstructured grid discretization in the space domain and it is explicit in the time domain. Consequently it is a best mixture of FEM and finite volume method （FVM）. RK-DGFEM can obtain local high-order accuracy by using high-order polynomial basis. Numerical experiments of transverse magnetic （TM） wave propagation in a 2-D resonator are performed. A high-order Lagrange polynomial basis is adopted. Numerical results agree well with analytical solution. And different order Lagrange interpolation polynomial basis impacts on simulation result accuracy are discussed. Computational results indicate that the accuracy is evidently improved when the order of interpolation basis is increased. Finally, L^2 errors of different order polynomial basis in RK-DGFEM are presented. Computational results show that L^2 error declines exponentially as the order of basis increases.
文摘We report a new method for calculating transmission coefficients across arbitrary potential barriers based on the Runge-Kutta method. A numerical solution of the Schrodinger equation is calculated using the Runge-Kutta method,and a new model is established to analyze the numerical results to find the transmission coefficient. This technique is applied to various cases, such as parabolic potential barrier and double-barrier structures. Transmission probability with high precision is obtained and discussed. The tunnelling current density through a MOS structure is also explored and the result coincides with the Fowler-Nordheim model,which indicates the applicability of our method.
文摘The steady, laminar, incompressible and two dimensional micropolar flow between two porous disks was investigated using optimal homotopy asymptotic method(OHAM) and fourth order Runge–Kutta numerical method. Comparison between OHAM and numerical method shows that OHAM is an exact and high efficient method for solving these kinds of problems. The results are presented to study the velocity and rotation profiles for different physical parameters such as Reynolds number, vortex viscosity parameter, spin gradient viscosity and microinertia density parameter. As an important outcome, the magnitude of the microrotation increases with an increase in the values of injection velocity while it decreases by increasing the values of suction velocity.
基金This work is supported by the National Natural Science Foundation of China （No.21373064）, the Program for Innovative Research Team of Guizhou Province （No.QKTD[2014]4021）, and the Natural Sci- entific Foundation from Guizhou Provincial Department of Education （No.ZDXK[2014]IS）. All the calculations were performed at Guizhou Provincial High- Performance Computing Center of Condensed Mate- rials and Molecular Simulation in Guizhou Education University.
文摘A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.
文摘The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method （LWPM）. Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge-Kutta method of order four （RK-4）. Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.
文摘In this paper a definitely new analytical technique, predictor homotopy analysis method （PHAM）, is employed to solve the problem of two-dimensional nanofluid flow through expanding or contracting gaps with permeable walls. Moreover, comparison of the PHAM results with numerical results obtained by the shooting method coupled with a Runge- Kutta integration method as well as previously published study results demonstrates high accuracy for this technique. The fluid in the channel is water containing different nanoparticles： silver, copper, copper oxide, titanium oxide, and aluminum oxide. The effects of the nanoparticle volume fraction, Reynolds number, wall expansion ratio, and different types of nanoparticles on the flow are discussed.
文摘Optimal control theory is applied to a system of ordinary differential equations modeling banana Xanthomonas wilt within plantations. The objective is to reduce the proportion of infected plants by use of controls representing two types of preventive methods： vector and contaminated tool prevention. The optimal controls are characterized in terms of the optimality system, which is solved analytically and numerically for several scenarios.
