Analytic solutions of an iterative differential equation under Brjuno condition

2009 ◽  
Vol 25 (9) ◽  
pp. 1469-1482 ◽  
Author(s):  
Jian Liu ◽  
Jian Guo Si
Keyword(s):  
Differential Equation ◽  
Analytic Solutions ◽  
Brjuno Condition

scholarly journals Analytic Solutions for a Functional Differential Equation Related to a Traffic Flow Model

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Houyu Zhao
Keyword(s):  
Differential Equation ◽  
Traffic Flow ◽  
Functional Differential Equation ◽  
Flow Model ◽  
Analytic Solutions ◽  
Auxiliary Equation ◽  
Traffic Flow Model ◽  
Near Resonance ◽  
Brjuno Condition ◽  
Functional Differential

We study the existence of analytic solutions of a functional differential equation(z(s)+α)2z'(s)=β(z(s+z(s))-z(s))which comes from traffic flow model. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constantλat resonance, that is, at a root of the unity, but also thoseλnear resonance under the Brjuno condition.

Analytic Solutions of a Nonhomogeneous Iterative Functional Differential Equation

2011 ◽  
Vol 474-476 ◽  
pp. 2155-2160
Author(s):  
Ling Xia Liu
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Analytic Solutions ◽  
Diophantine Condition ◽  
Auxiliary Equation ◽  
State Dependent ◽  
Near Resonance ◽  
Brjuno Condition ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

This paper is concerned with an iterative functional differential equation with state-dependent delay As well as in previous works, we reduce this problem with the Schroeder transformation to obtain auxiliary equation. For technical reasons, in previous work the constantgiven in the Schroeder transformation, is required to fulfill that is off the unit circle or lies on the circle with the Diophantine condition. In this paper, we discuss not only thoseat a root of the unity, but also those near resonance under the Brjuno condition.

scholarly journals Analytic Solutions of an Iterative Functional Differential Equation near Resonance

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Tongbo Liu ◽  
Hong Li
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Unknown Function ◽  
Analytic Solutions ◽  
Complex Field ◽  
Second Order Differential Equations ◽  
Near Resonance ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function in the complex field . By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function + = , we get its local invertible analytic solutions.

scholarly journals Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

2019 ◽  
Vol 27 (2) ◽  
pp. 171-185 ◽  
Author(s):  
Astha Chauhan ◽  
Rajan Arora
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Analytic Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Liouville Fractional Derivative ◽  
Symmetry Generators ◽  
Symmetry Method

AbstractIn this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

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