scholarly journals A hybrid two-step method for direct solutions of general second order initial value problem

2021 ◽  
Keyword(s):  
Initial Value Problem ◽  
Second Order ◽  
Initial Value ◽  
Step Method

EXPLICIT EIGHTH ORDER TWO-STEP METHODS WITH NINE STAGES FOR INTEGRATING OSCILLATORY PROBLEMS

2006 ◽  
Vol 17 (06) ◽  
pp. 861-876 ◽  
Author(s):  
Ch. TSITOURAS
Keyword(s):  
Initial Value Problem ◽  
Second Order ◽  
Initial Value ◽  
Step Method ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
Phase Errors ◽  
Matlab Program ◽  
Algebraic Order

We present a new explicit hybrid two step method for the solution of second order initial value problem. It costs only nine function evaluations per step and attains eighth algebraic order so it is the cheapest in the literature. Its coefficients are chosen to reduce amplification and phase errors. Thus the method is well suited for facing problems with oscillatory solutions. After implementing a MATLAB program, we proceed with numerical tests that justify our effort.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

AN IMPLEMENTATION OF THE COLLOCATION METHOD FOR INITIAL VALUE PROBLEMS

2013 ◽  
Vol 04 (02) ◽  
pp. 1350006 ◽  
Author(s):  
RAFAEL G. CAMPOS ◽  
FRANCISCO DOMÍNGUEZ MOTA
Keyword(s):  
Collocation Method ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Polynomial Interpolation ◽  
Second Order ◽  
Initial Value ◽  
Differentiation Matrix

An implementation of the standard collocation method based on polynomial interpolation is presented in a matrix framework in this paper. The underlying differentiation matrix can be partitioned to yield a superconvergent implicit multistep-like method to solve the initial value problem numerically. The first- and second-order versions of this method are L-stable.

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