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
An unsteady viscous fluid flow with Dufour and Soret effect,which results in heat and mass transfer due to upward and downward motion of flexible rotating disk,has been studied.The upward or downward motion of non rot...An unsteady viscous fluid flow with Dufour and Soret effect,which results in heat and mass transfer due to upward and downward motion of flexible rotating disk,has been studied.The upward or downward motion of non rotating disk results in two dimensional flow,while the vertical action and rotation of the disk results in three dimensional flow.By using an appropriate transformation the governing equations are transformed into the system of ordinary differential equations.The system of ordinary differential equations is further converted into first order differential equation by selecting suitable variables.Then,we generalize the model by using the Caputo derivative.The numerical result for the fractional model is presented and validated with Runge Kutta order 4 method for classical case.The compared results are presented in Table and Figures.It is concluded that the fractional model is more realistic than that of classical one,because it simulates the fluid behavior at each fractional value rather than the integral values.展开更多
In this paper, we introduce a numerical treatment using generalized Euler method (GEM) for the non-linear programming problem which is governed by a system of fractional differential equations (FDEs). The appeared fra...In this paper, we introduce a numerical treatment using generalized Euler method (GEM) for the non-linear programming problem which is governed by a system of fractional differential equations (FDEs). The appeared fractional derivatives in these equations are in the Caputo sense. We compare our numerical solutions with those numerical solutions using RK4 method. The obtained numerical results of the optimization problem model show the simplicity and the efficiency of the proposed scheme.展开更多
In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ 3 is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturb...In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ 3 is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The current results of this paper are in excellent agreement with those provided by Chandrasekar et al. [1] and thereby illustrate the reliability and the performance of the differential transform method. We have also compared the results with the classical Runge-Kutta 4 (RK4) Method.展开更多
This study employs the Buongiorno model to explore nanoparticle migration in a mixed convection second-grade fluid over a slendering(variable thickness)stretching sheet.The convective boundary conditions are applied t...This study employs the Buongiorno model to explore nanoparticle migration in a mixed convection second-grade fluid over a slendering(variable thickness)stretching sheet.The convective boundary conditions are applied to the surface.In addition,the analysis has been carried out in the presence of Joule heating,slips effects,thermal radiation,heat generation and magnetohydrodynamic.This study aimed to understand the complex dynamics of these nanofluids under various external influences.The governing model has been developed using the flow assumptions such as boundary layer approximations in terms of partial differential equations.Governing partial differential equations are first reduced into ordinary differential equations and then numerically solved using the Runge-Kutta-Fehlberg method(RK4)in conjunction with a shooting scheme.Our results indicate significant increases in Nusselt and Sherwood numbers by up to 14.6%and 23.2%,respectively,primarily due to increases in the Brownian motion parameter and thermophoresis parameter.Additionally,increases in the magnetic field parameter led to a decrease in skin friction coefficients by 37.5%.These results provide critical insights into optimizing industrial processes such as chemical production,automotive cooling systems,and energy generation,where efficient heat andmass transfer are crucial.Buongiornomodel;velocity-slip effects;Joule heating;convective boundary conditions;Runge-Kutta-Fehlberg method(RK4).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An unsteady viscous fluid flow with Dufour and Soret effect,which results in heat and mass transfer due to upward and downward motion of flexible rotating disk,has been studied.The upward or downward motion of non rotating disk results in two dimensional flow,while the vertical action and rotation of the disk results in three dimensional flow.By using an appropriate transformation the governing equations are transformed into the system of ordinary differential equations.The system of ordinary differential equations is further converted into first order differential equation by selecting suitable variables.Then,we generalize the model by using the Caputo derivative.The numerical result for the fractional model is presented and validated with Runge Kutta order 4 method for classical case.The compared results are presented in Table and Figures.It is concluded that the fractional model is more realistic than that of classical one,because it simulates the fluid behavior at each fractional value rather than the integral values.
文摘In this paper, we introduce a numerical treatment using generalized Euler method (GEM) for the non-linear programming problem which is governed by a system of fractional differential equations (FDEs). The appeared fractional derivatives in these equations are in the Caputo sense. We compare our numerical solutions with those numerical solutions using RK4 method. The obtained numerical results of the optimization problem model show the simplicity and the efficiency of the proposed scheme.
文摘In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ 3 is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The current results of this paper are in excellent agreement with those provided by Chandrasekar et al. [1] and thereby illustrate the reliability and the performance of the differential transform method. We have also compared the results with the classical Runge-Kutta 4 (RK4) Method.
文摘This study employs the Buongiorno model to explore nanoparticle migration in a mixed convection second-grade fluid over a slendering(variable thickness)stretching sheet.The convective boundary conditions are applied to the surface.In addition,the analysis has been carried out in the presence of Joule heating,slips effects,thermal radiation,heat generation and magnetohydrodynamic.This study aimed to understand the complex dynamics of these nanofluids under various external influences.The governing model has been developed using the flow assumptions such as boundary layer approximations in terms of partial differential equations.Governing partial differential equations are first reduced into ordinary differential equations and then numerically solved using the Runge-Kutta-Fehlberg method(RK4)in conjunction with a shooting scheme.Our results indicate significant increases in Nusselt and Sherwood numbers by up to 14.6%and 23.2%,respectively,primarily due to increases in the Brownian motion parameter and thermophoresis parameter.Additionally,increases in the magnetic field parameter led to a decrease in skin friction coefficients by 37.5%.These results provide critical insights into optimizing industrial processes such as chemical production,automotive cooling systems,and energy generation,where efficient heat andmass transfer are crucial.Buongiornomodel;velocity-slip effects;Joule heating;convective boundary conditions;Runge-Kutta-Fehlberg method(RK4).
