scholarly journals NEW APPROACH ON THE GENERAL SHAPE EQUATION OF AXISYMMETRIC VESICLES

1999 ◽  
Vol 13 (01) ◽  
pp. 13-18 ◽  
Author(s):  
ZHAN-NING HU
Keyword(s):  
Differential Equation ◽  
Equation System ◽  
Variable Coefficients ◽  
General Shape ◽  
General Constraint ◽  
New Approach ◽  
Shape Equation ◽  
Equilibrium Shapes ◽  
Linear Differential ◽  
Lipid Bilayer Vesicles

The general Helfrich shape equation determined by minimizing the curvature free energy describes the equilibrium shapes of the axisymmetric lipid bilayer vesicles in different conditions. It is a nonlinear differential equation with variable coefficients. In this letter, by analyzing the unique property of the solution, we change this shape equation into a system of the two differential equations. One of them is a linear differential equation. This equation system contains all of the known rigorous solutions of the general shape equation. And the more general constraint conditions are found for the solution of the general shape equation.

EQUIVALENCE OF THE TWO METHODS IN CALCULATING THE EQUILIBRIUM SHAPES OF CYLINDRICAL VESICLES

1999 ◽  
Vol 13 (16) ◽  
pp. 547-553
Author(s):  
SHAOGUANG ZHANG ◽  
ZHONGCAN OUYANG ◽  
JIXING LIU
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
The Other ◽  
General Shape ◽  
Calculus Of Variation ◽  
And Topology ◽  
Shape Equation ◽  
Equilibrium Shapes ◽  
Comparison Of The Results ◽  
The Relationship

So far, two methods are often used in solving the equilibrium shapes of vesicles. One method is by starting with the general shape equation and restricting it to the shapes with particular symmetry. The other method is by assuming the symmetry and topology of the vesicle first and treating it with the calculus of variation to get a set of ordinary differential equations. The relationship between these two methods in the case of cylindrical vesicles, and a comparison of the results are given.

scholarly journals Theoretical Vibration Analysis Regarding Excitation due to Elliptical Shaft Journals in Sleeve Bearings of Electrical Motors

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Ulrich Werner
Keyword(s):  
Differential Equation ◽  
Mathematical Description ◽  
Vibration Analysis ◽  
Equation System ◽  
Electrical Machines ◽  
Differential Equation System ◽  
Kinematic Constraints ◽  
Linear Differential ◽  
Electrical Motors ◽  
Rotor Mass

This paper shows a theoretical vibration analysis regarding excitation due to elliptical shaft journals in sleeve bearings of electrical motors, based on a simplified rotordynamic model. It is shown that elliptical shaft journals lead to kinematic constraints regarding the movement of the shaft journals on the oil film of the sleeve bearings and therefore to an excitation of the rotordynamic system. The solution of the linear differential equation system leads to the mathematical description of the movement of the rotor mass, the shaft journals, and the sleeve bearing housings. Additionally the relative movements between the shaft journals and the bearing housings are deduced, as well as the bearing housing vibration velocities. The presented simplified rotordynamic model can also be applied to rotating machines, other than electrical machines. In this case, only the electromagnetic spring valuecmhas to be put to zero.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

New formula for solution of one class of linear differential equations of the second order with the variable coefficients

2020 ◽  
Vol 8 (3) ◽  
pp. 61-68
Author(s):  
Avyt Asanov ◽  
Kanykei Asanova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Linear Differential Equations ◽  
Common Solution ◽  
The Common ◽  
Linear Differential ◽  
New Formula

Exact solutions for linear and nonlinear differential equations play an important rolein theoretical and practical research. In particular many works have been devoted tofinding a formula for solving second order linear differential equations with variablecoefficients. In this paper we obtained the formula for the common solution of thelinear differential equation of the second order with the variable coefficients in themore common case. We also obtained the new formula for the solution of the Cauchyproblem for the linear differential equation of the second order with the variablecoefficients.Examples illustrating the application of the obtained formula for solvingsecond-order linear differential equations are given.Key words: The linear differential equation, the second order, the variablecoefficients,the new formula for the common solution, Cauchy problem, examples.

scholarly journals Einsteinian Gravitational Field of a Heterogeneous Fluid Sphere

1925 ◽  
Vol 44 ◽  
pp. 72-78
Author(s):  
Jyotirmaya Ghosh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Empty Space ◽  
Variable Coefficients ◽  
Fluid Sphere ◽  
Uniform Density ◽  
Large Radius ◽  
Field Equations ◽  
Linear Differential ◽  
Gravitational Equations

The field-equations of gravitation in Einstein's theory have been solved in the case of an empty space, giving rise to de Sitter's spherical world. In the case of homogeneous matter filling all space, the solution gives Einstein's cylindrical world. The field corresponding to an isolated particle has been obtained by Schwarzchild. He has also obtained a solution for a fluid sphere with uniform density, a problem treated also by Nordström and de Donder. A new solution of the gravitational equations has been obtained in this paper, which corresponds to the field of a heterogeneous fluid sphere, the density at any point being a certain function of the distance of the point from the centre. The law of density is quite simple and such as to give finite density at the centre and gradually diminishing values as the distance from the centre increases, as might be expected of a natural sphere of fluid of large radius. The general problem of the fluid sphere with any arbitrary law of density cannot be solved in exact terms. It will be seen, however, from a theorem obtained in this paper, that the solution depends on a linear differential equation of the second order with variable coefficients involving the density, and thus the laws of density for which the problem admits of exact solution are those for which the above coefficients satisfy the conditions of integrability of the differential equation. An approximate solution for any law of density may be obtained by the method of series.

V. Mechanical integration of the linear differential equations of the second order with variable coefficients

1876 ◽  
Vol 24 (164-170) ◽  
pp. 269-271 ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Mathematical Physics ◽  
Variable Coefficients ◽  
Second Order ◽  
Sectional Area ◽  
Practical Solution ◽  
Hanging Chain ◽  
Linear Differential

Every linear differential equation of the second order may, as is known, be reduced to the form d / dx (1/P du / dx ) = u , . . . . . . (1) where P is any given function of x . On account of the great importance of this equation in mathematical physics (vibrations of a non-uniform stretched cord, of a hanging chain, water in a canal of non-uniform breadth and depth, of air in a pipe of non-uniform sectional area, conduction of heat along a bar of non-uniform fiction or non-uniform conductivity, Laplace’s differential equation of the tides, &c. &c.), I have long endeavoured to obtain a means of faciliiting its practical solution.

scholarly journals VI. Mechanical integration of the general linear differential equation of any order with variable coefficients

1876 ◽  
Vol 24 (164-170) ◽  
pp. 271-275 ◽  
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Variable Coefficients ◽  
General Linear ◽  
The Third ◽  
Independent Variable ◽  
Linear Differential

Take any number i of my brother’s disk-, globe-, and cylinder-integrators, and make an integrating chain of them thus:—Connect the cylinder of the first so as to give a motion equal to its own to the fork of the second. Similarly connect the cylinder of the second with the fork of the third, and so on. Let g 1 , g 2 , g 3 , up to g i be the positions of the globes at any time. Let infinitesimal motions P 1 dx P 2 dx P 3 dx .... be given simultaneously to all the disks ( dx denoting an infinitesimal motion of some part of the mechanism whose displacement it is convenient to take as independent variable). The motions ( d k 1 , d k 2 , . . . d ki ) of the cylinders thus produced are d k 1 = g 1 P 1 dx , d k 2 = g 2 P 2 dx ,... d k i = g i P i dx ... (1) But, by the connexions between the cylinders and forks which move the globes, d K 1 = dg 2 , d K 2 = dg 3 , . . . d K i-1 = dg i , and therefore

scholarly journals New Approach for Solving Partial Differential Equations Based on Collocation Method

2020 ◽  
Vol 18 ◽  
pp. 118-128
Author(s):  
Alaa Almosawi ◽  
Luma N. M. Tawfiq
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Collocation Method ◽  
Linear Differential Operator ◽  
Partial Differential ◽  
Restrictive Assumption ◽  
New Approach ◽  
Linear Differential

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion. Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.

scholarly journals Parallel Solving Method for the Variable Coefficient Nonlinear Equation

2022 ◽  
Vol 16 ◽  
pp. 264-271
Author(s):  
Liling Shen
Keyword(s):  
Differential Equation ◽  
Nonlinear Equation ◽  
Linear Differential Equation ◽  
Nonlinear Differential Equation ◽  
Nonlinear Equations ◽  
Variable Coefficient ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Traveling Wave Solution ◽  
Linear Differential

In view of the inaccuracy of traditional methods for solving nonlinear equations with variable coefficients in parallel, a new method for solving nonlinear equations with variable coefficients is proposed. Using the generalized symmetry group, the variable coefficient of the equation is taken as a new variable which is the same as the state of the original actual physical field. Some relations between variable coefficient equations and their solutions are found. This paper analyzes the meaning of linear differential equation and nonlinear differential equation, the difference between linear differential equation and nonlinear differential equation and their role in physics, and the necessity of solving nonlinear differential equation. By solving the nonlinear equation with variable coefficients, it can be seen that the general methods to solve the nonlinear equation include scattering inversion, Backlund transform and traveling wave solution. Based on the existing methods for solving nonlinear equations with variable coefficients, the homogeneous balance method is combined with the improved truncated expansion method, truncated expansion method and function reduction method, and the Hopf Cole transform and trial function are combined respectively. The above three methods are used to solve nonlinear equations with variable coefficients. Based on KdV Painleve principle, a parallel method for solving nonlinear equations with variable coefficients is proposed. Finally, it is proved that the method is accurate and effective for the parallel solution of nonlinear equations with variable coefficients.

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