A class of linear multi-step method adapted to general oscillatory second-order initial value problems

2017 ◽  
Vol 56 (1-2) ◽  
pp. 561-591 ◽  
Author(s):  
Jiyong Li ◽  
Xianfen Wang ◽  
Ming Lu
Keyword(s):  
Initial Value Problems ◽  
Second Order ◽  
Initial Value ◽  
Step Method

DISSIPATIVE TRIGONOMETRICALLY-FITTED METHODS FOR SECOND ORDER IVPs WITH OSCILLATING SOLUTION

2002 ◽  
Vol 13 (10) ◽  
pp. 1333-1345 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Oscillating Solution ◽  
Second Order ◽  
Initial Value ◽  
Step Method ◽  
Numerical Examples ◽  
Oscillating Solutions ◽  
Trigonometrical Fitting ◽  
Trigonometrically Fitted Methods

In this paper a dissipative trigonometrically-fitted two-step explicit hybrid method is developed. This method is based on a dissipative explicit two-step method developed recently by Papageorgiou, Tsitouras and Famelis.6 Numerical examples show that the procedure of trigonometrical fitting is the only way in one to produce efficient dissipative methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions.

scholarly journals Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

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