Trigonometrically fitted multi-step RKN methods for second-order oscillatory initial value problems

2018 ◽  
Vol 320 ◽  
pp. 740-753
Author(s):  
Jiyong Li ◽  
Shuo Deng
Keyword(s):  
Initial Value Problems ◽  
Second Order ◽  
Initial Value ◽  
Oscillatory Initial Value Problems

scholarly journals Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
N. Senu ◽  
F. Ismail ◽  
S. Z. Ahmad ◽  
M. Suleiman
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Scientific Literature ◽  
Hybrid Methods ◽  
Second Order ◽  
Numerical Results ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Initial Value Problems

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods.

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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