Stiffly stable second derivative multistep methods with higher order and improved stability regions

1983 ◽  
Vol 23 (1) ◽  
pp. 75-83 ◽  
Author(s):  
P. C. Chakravarti ◽  
M. S. Kamel
Keyword(s):  
Multistep Methods ◽  
Higher Order ◽  
Second Derivative ◽  
Stability Regions ◽  
Improved Stability

scholarly journals A family of higher order iterations free from second derivative for nonlinear equations inR

2018 ◽  
Vol 330 ◽  
pp. 676-694 ◽  
Author(s):  
Abhimanyu Kumar ◽  
P. Maroju ◽  
R. Behl ◽  
D.K. Gupta ◽  
S.S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Higher Order ◽  
Second Derivative

scholarly journals Determination of Coefficients of High-Order Schemes for Riemann-Liouville Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-21 ◽  
Author(s):  
Rifang Wu ◽  
Hengfei Ding ◽  
Changpin Li
Keyword(s):  
Multistep Methods ◽  
Numerical Algorithms ◽  
Higher Order ◽  
High Order ◽  
Linear Multistep Methods ◽  
High Order Schemes ◽  
Liouville Derivative ◽  
Recursion Formulas ◽  
Fractional Differential

Although there have existed some numerical algorithms for the fractional differential equations, developing high-order methods (i.e., with convergence order greater than or equal to 2) is just the beginning. Lubich has ever proposed the high-order schemes when he studied the fractional linear multistep methods, where he constructed thepth order schemes(p=2,3,4,5,6)for theαth order Riemann-Liouville integral andαth order Riemann-Liouville derivative. In this paper, we study such a problem and develop recursion formulas to compute these coefficients in the higher-order schemes. The coefficients of higher-order schemes(p=7,8,9,10)are also obtained. We first find that these coefficients are oscillatory, which is similar to Runge’s phenomenon. So, they are not suitable for numerical calculations. Finally, several numerical examples are implemented to testify the efficiency of the numerical schemes forp=3,…,6.

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