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.

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.

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

A New Method for Finding Solutions of Linear Partial Differential Equation

2011 ◽  
Vol 219-220 ◽  
pp. 675-679
Author(s):  
Yan Tang ◽  
Mao Chang Qin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Differential Equation ◽  
Variable Coefficient ◽  
Linear Partial Differential Equation ◽  
New Method ◽  
Partial Differential ◽  
Special Solutions ◽  
Linear Differential

A useful technique is adopted to study special solutions of a general Black and Scholes equation in this letter. Several kinds of new special solutions are obtained. This method is effective for finding special solutions of linear differential equation with variable coefficient.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

Sign in / Sign up

Export Citation Format

Share Document