The paper is devoted to the study of the gravitational collapse within the framework of the spherically symmetric problem in the Newton theory and general relativity on the basis of the pressure-free model of the cont...The paper is devoted to the study of the gravitational collapse within the framework of the spherically symmetric problem in the Newton theory and general relativity on the basis of the pressure-free model of the continuum. In application to the Newton gravitation theory, the analysis consists of three stages. First, we assume that the gravitational force is determined by the initial sphere radius and constant density and does not change in the process of the sphere collapse. The obtained analytical solution allows us to find the collapse time in the first approximation. Second, we construct the step-by-step process in which the gravitational force at a given time moment depends on the current sphere radius and density. The obtained numerical solution specifies the collapse time depending on the number of steps. Third, we find the exact value of the collapse time which is the limit of the step-by-step solutions and study the collapse and the expansion processes in the Newton theory. In application to general relativity, we use the space model corresponding to the special four-dimensional space which is Euclidean with respect to space coordinates and Riemannian with respect to the time coordinate only. The obtained solution specifies two possible scenarios. First, sphere contraction results in the infinitely high density with the finite collapse time, which does not coincide with the conventional result corresponding to the Schwarzschild geometry. Second, sphere expansion with the velocity which increases with a distance from the sphere center and decreases with time.展开更多
本文提出了一种新型数值求解方法,该方法将四阶Runge-Kutta法与Newton迭代法相结合,旨在高效求解流体力学中的Blasius方程边值问题。首先,我们将Blasius方程转化为一组一阶微分方程组,并采用四阶Runge-Kutta法(RK4)进行数值求解。随后,...本文提出了一种新型数值求解方法,该方法将四阶Runge-Kutta法与Newton迭代法相结合,旨在高效求解流体力学中的Blasius方程边值问题。首先,我们将Blasius方程转化为一组一阶微分方程组,并采用四阶Runge-Kutta法(RK4)进行数值求解。随后,引入Newton迭代法动态调整初始条件,以确保满足边界条件的要求。实验结果表明,与传统的打靶法和结合打靶法的四阶Runge-Kutta法(SRK)进行对比实验,新方法在迭代次数和计算时间上均展现显著优势,同时求解精度也得到提升。In this paper, we propose a novel numerical solution method that combines the fourth-order Runge-Kutta method with the Newton iterative method, aiming to efficiently solve the margin problem of Blasius equation in fluid mechanics. Firstly, we transform the Blasius equation into a set of first-order differential equations and solve it numerically using the fourth-order Runge-Kutta method (RK4). Subsequently, the Newton iteration method is dynamically introduced to adjust the initial conditions to ensure that the boundary conditions are satisfied. The experimental results show that the new method exhibits significant advantages in terms of the number of iterations and computation time, as well as improved solution accuracy, in comparison experiments with the traditional shooting method and the Runge-Kutta method (SRK) combined with the improved shooting method.展开更多
In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.[1]). We find that the three formulas lead to the ...In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.[1]). We find that the three formulas lead to the extension of Newton-Raphson’s method and Horner’s method at the same time. This shows originality of Japanese native mathematics (Wasan) in the Edo era (1600- 1867). Suzuki (Ref.[2]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 1 introduces Murase’s three solutions of the cubic equation of the hearth. Section 2 explains the Horner’s method. We give the generalization of three formulas and the relation between these formulas and Horner’s method. Section 3 gives definitions of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), general recurrence formula of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), and general recurrence formula of the extension of Murase-Newton’s method (the extension of Tsuchikura-Horiguchi’s method) concerning n-degree polynomial equation. Section 4 is contents of the title of this paper.展开更多
基金Project supported by Key Industrial Projects of Major Science and Technology Projects of Zhejiang(No.2009C11023)Foundation of Zhejiang Educational Committee(No.Y200907886)Major High-Tech Industrialization Project of Jiaxing(No.2009BY10004)
文摘The paper is devoted to the study of the gravitational collapse within the framework of the spherically symmetric problem in the Newton theory and general relativity on the basis of the pressure-free model of the continuum. In application to the Newton gravitation theory, the analysis consists of three stages. First, we assume that the gravitational force is determined by the initial sphere radius and constant density and does not change in the process of the sphere collapse. The obtained analytical solution allows us to find the collapse time in the first approximation. Second, we construct the step-by-step process in which the gravitational force at a given time moment depends on the current sphere radius and density. The obtained numerical solution specifies the collapse time depending on the number of steps. Third, we find the exact value of the collapse time which is the limit of the step-by-step solutions and study the collapse and the expansion processes in the Newton theory. In application to general relativity, we use the space model corresponding to the special four-dimensional space which is Euclidean with respect to space coordinates and Riemannian with respect to the time coordinate only. The obtained solution specifies two possible scenarios. First, sphere contraction results in the infinitely high density with the finite collapse time, which does not coincide with the conventional result corresponding to the Schwarzschild geometry. Second, sphere expansion with the velocity which increases with a distance from the sphere center and decreases with time.
文摘本文提出了一种新型数值求解方法,该方法将四阶Runge-Kutta法与Newton迭代法相结合,旨在高效求解流体力学中的Blasius方程边值问题。首先,我们将Blasius方程转化为一组一阶微分方程组,并采用四阶Runge-Kutta法(RK4)进行数值求解。随后,引入Newton迭代法动态调整初始条件,以确保满足边界条件的要求。实验结果表明,与传统的打靶法和结合打靶法的四阶Runge-Kutta法(SRK)进行对比实验,新方法在迭代次数和计算时间上均展现显著优势,同时求解精度也得到提升。In this paper, we propose a novel numerical solution method that combines the fourth-order Runge-Kutta method with the Newton iterative method, aiming to efficiently solve the margin problem of Blasius equation in fluid mechanics. Firstly, we transform the Blasius equation into a set of first-order differential equations and solve it numerically using the fourth-order Runge-Kutta method (RK4). Subsequently, the Newton iteration method is dynamically introduced to adjust the initial conditions to ensure that the boundary conditions are satisfied. The experimental results show that the new method exhibits significant advantages in terms of the number of iterations and computation time, as well as improved solution accuracy, in comparison experiments with the traditional shooting method and the Runge-Kutta method (SRK) combined with the improved shooting method.
文摘In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.[1]). We find that the three formulas lead to the extension of Newton-Raphson’s method and Horner’s method at the same time. This shows originality of Japanese native mathematics (Wasan) in the Edo era (1600- 1867). Suzuki (Ref.[2]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 1 introduces Murase’s three solutions of the cubic equation of the hearth. Section 2 explains the Horner’s method. We give the generalization of three formulas and the relation between these formulas and Horner’s method. Section 3 gives definitions of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), general recurrence formula of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), and general recurrence formula of the extension of Murase-Newton’s method (the extension of Tsuchikura-Horiguchi’s method) concerning n-degree polynomial equation. Section 4 is contents of the title of this paper.
