scholarly journals Direct Method of Stability of Null Solution for Higher Order Ordinary Differential Equations

2016 ◽  
Vol 05 (03) ◽  
pp. 412-415
Author(s):  
芝祥 王
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Direct Method ◽  
Higher Order ◽  
Null Solution

PAINLEVÉ ANALYSIS AND NEW ANALYTIC SOLUTIONS FOR (1+1)-DIMENSIONAL HIGHER-ORDER BROER–KAUP SYSTEM WITH SYMBOLIC COMPUTATION

2014 ◽  
Vol 28 (14) ◽  
pp. 1450067 ◽  
Author(s):  
XIAO-NAN LI ◽  
GUANG-MEI WEI ◽  
YU-PING LIU ◽  
YUE-QIAN LIANG ◽  
XIANG-HUA MENG
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformation ◽  
Direct Method ◽  
Analytic Solutions ◽  
Higher Order ◽  
Painlevé Test ◽  
Backlund Transformation ◽  
Truncated Painlevé Expansion ◽  
Similarity Reductions

The (1+1)-dimensional higher-order Broer–Kaup (HBK) system is investigated in this paper. Painlevé test shows that there are two solution branches, one of which has the resonance -2. And an auto-Bäcklund transformation is obtained by the truncated Painlevé expansion. The new analytic solutions are presented by means of the auto-Bäcklund transformation, including the periodic and soliton-like solutions. Similarity reductions for the HBK system are given out to two ordinary differential equations (ODEs) through CK direct method.

Pathwise convergent higher order numerical schemes for random ordinary differential equations

2007 ◽  
Vol 463 (2087) ◽  
pp. 2929-2944 ◽  
Author(s):  
Peter E Kloeden ◽  
Arnulf Jentzen
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Time Variable ◽  
Numerical Schemes ◽  
Hölder Continuous ◽  
Usual Order

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

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