For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods...For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .展开更多
The energy of a spring with a well-distributed mass ms is theoretically studied in this paper.The solution of the wave equation is derived in detail,and then the kinetic energy and potential energy of the spring are s...The energy of a spring with a well-distributed mass ms is theoretically studied in this paper.The solution of the wave equation is derived in detail,and then the kinetic energy and potential energy of the spring are studied with the wave equation,as well as the kinetic energy of the oscillating mass M.The kinetic energy and potential energy of the spring,and total energy are numerically simulated for different ratios ms/M with considering the spring’s mass,which makes the property of energy of the oscillating system understood easily.展开更多
Planck’s radiation law provides an equation for the intensity of the electromagnetic radiation from a physical body as a function of frequency and temperature. The frequency that corresponds to the maximum intensity ...Planck’s radiation law provides an equation for the intensity of the electromagnetic radiation from a physical body as a function of frequency and temperature. The frequency that corresponds to the maximum intensity is a function of temperature. At a specific temperature, for the frequencies correspond to much less than the maximum intensity, an equation was derived in the form of the Lambert <em>W</em> function. Numerical calculations validate the equation. A new form of solution for the Euler’s transcendental equation was derived in the form of the Lambert <em>W</em> function with logarithmic argument. Numerical solutions to the Euler’s equation were determined iteratively and iterative convergences were investigated. Numerical coincidences with physical constants were explored.展开更多
The model of laminated wave turbulence puts forth a novel computational problem–construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order 10^(12) and more.The equations...The model of laminated wave turbulence puts forth a novel computational problem–construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order 10^(12) and more.The equations to be solved in integers are resonant conditions for nonlinearly interacting waves and their form is defined by the wave dispersion.It is established that for the most common dispersion as an arbitrary function of a wave-vector length two different generic algorithms are necessary:(1)one-class-case algorithm for waves interacting through scales,and(2)two-class-case algorithm for waves interacting through phases.In our previous paper we described the one-class-case generic algorithm and in our present paper we present the two-classcase generic algorithm.展开更多
文摘For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .
基金supported by the program for Higher educational Quality engineering projects of Anhui Province(2018zygc062)Anhui Provincial Natural Science Foundation(1808085MA20 and 1708085MA10)+3 种基金Excellent Young Talents in University of Anhui Province(gxyq2017027)the key Scientific Research Foundation of Anhui Provincial Education Department(KJ2019A0564,KJ2019A0580,KJ2018A0366,Wdxy2018jyxm008 and Wdxy2018jyxm009)Excellent course of Anhui Provincial Education Department(2017kfk061)Wisdom classroom of Anqing Normal University(2018aqnuzhkt008).
文摘The energy of a spring with a well-distributed mass ms is theoretically studied in this paper.The solution of the wave equation is derived in detail,and then the kinetic energy and potential energy of the spring are studied with the wave equation,as well as the kinetic energy of the oscillating mass M.The kinetic energy and potential energy of the spring,and total energy are numerically simulated for different ratios ms/M with considering the spring’s mass,which makes the property of energy of the oscillating system understood easily.
文摘Planck’s radiation law provides an equation for the intensity of the electromagnetic radiation from a physical body as a function of frequency and temperature. The frequency that corresponds to the maximum intensity is a function of temperature. At a specific temperature, for the frequencies correspond to much less than the maximum intensity, an equation was derived in the form of the Lambert <em>W</em> function. Numerical calculations validate the equation. A new form of solution for the Euler’s transcendental equation was derived in the form of the Lambert <em>W</em> function with logarithmic argument. Numerical solutions to the Euler’s equation were determined iteratively and iterative convergences were investigated. Numerical coincidences with physical constants were explored.
文摘The model of laminated wave turbulence puts forth a novel computational problem–construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order 10^(12) and more.The equations to be solved in integers are resonant conditions for nonlinearly interacting waves and their form is defined by the wave dispersion.It is established that for the most common dispersion as an arbitrary function of a wave-vector length two different generic algorithms are necessary:(1)one-class-case algorithm for waves interacting through scales,and(2)two-class-case algorithm for waves interacting through phases.In our previous paper we described the one-class-case generic algorithm and in our present paper we present the two-classcase generic algorithm.
