scholarly journals PUISEUX POWER SERIES SOLUTIONS FOR SYSTEMS OF EQUATIONS

2010 ◽  
Vol 21 (11) ◽  
pp. 1439-1459 ◽  
Author(s):  
FUENSANTA AROCA ◽  
GIOVANNA ILARDI ◽  
LUCÍA LÓPEZ DE MEDRANO
Keyword(s):  
Algebraic Curves ◽  
Newton Polygon ◽  
Series Solutions ◽  
Systems Of Equations ◽  
Algebraic Equations ◽  
Puiseux Series ◽  
Tropical Variety ◽  
Power Series Solutions ◽  
Tropical Varieties ◽  
The Ideal

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

The semi-analytic theory of standing waves

1987 ◽  
Vol 411 (1840) ◽  
pp. 123-137 ◽  
Keyword(s):  
Numerical Calculation ◽  
Power Series ◽  
Deep Water ◽  
Standing Waves ◽  
Affirmative Answer ◽  
Rigorous Proof ◽  
Series Solutions ◽  
Wave Problem ◽  
Power Series Solutions ◽  
Formal Power

The algorithm proposed by Schwartz & Whitney ( J. Fluid Mech . 107, 147–171 (1981)) for the numerical calculation of formal power series solutions of the classical standing-wave problem is vindicated by a rigorous proof that resonances do not occur in the calculations. A detailed account of a successful algorithm is given. The analytical question of the convergence of the power series whose coefficients have been calculated remains open. An affirmative answer would be a first demonstration of the existence of standing waves on deep water.

Bright soliton solutions, power series solutions and travelling wave solutions of a (3+1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (06) ◽  
pp. 1850082
Author(s):  
Ding Guo ◽  
Shou-Fu Tian ◽  
Li Zou ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Bright Soliton ◽  
Travelling Wave Solutions ◽  
Series Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Power Series Solutions ◽  
De Vries

In this paper, we consider the (3[Formula: see text]+[Formula: see text]1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili (mKdV-KP) equation, which can be used to describe the nonlinear waves in plasma physics and fluid dynamics. By using solitary wave ansatz in the form of sech[Formula: see text] function and a direct integrating way, we construct the exact bright soliton solutions and the travelling wave solutions of the equation, respectively. Moreover, we obtain its power series solutions with the convergence analysis. It is hoped that our results can provide the richer dynamical behavior of the KdV-type and KP-type equations.

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