Comments on a one step method of arbitrary order for initial value problem

2009 ◽  
Vol 3 (4) ◽  
Author(s):  
VU Aihie
Keyword(s):  
Initial Value Problem ◽  
Arbitrary Order ◽  
Initial Value ◽  
Step Method ◽  
One Step

scholarly journals Hybrid Third Derivative Block Method for the Solution of General Second Order Initial Value Problems with Generalized One Step Point

2019 ◽  
Vol 12 (3) ◽  
pp. 1199-1214
Author(s):  
Ra'ft Abdelrahim ◽  
Z. Omar ◽  
O. Ala’yed ◽  
B. Batiha
Keyword(s):  
Power Series ◽  
Convergence Analysis ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Procedure ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Third Derivative

This paper deals with two-step hybrid block method with one generalized off-step points for solving second order initial value problem. In derivation of this method, power series of order nine are interpolated at the first two step points while its second and third derivatives are collocated at all point in the selected interval. The new developed method is employed to solve several problems of second order initial value problems. Convergence analysis of the new method alongside numerical procedure is established. The performance of the proposed method is found to be more accurate than existing method available in the literature when solving the same problems.

THREE POINTS BLOCK EMBEDDED DIAGONALLY IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING ODES

2021 ◽  
Vol 10 (11) ◽  
pp. 3449-3460
Author(s):  
Y.F. Rahim ◽  
M.E.H. Hafidzuddin
Keyword(s):  
Initial Value Problem ◽  
Numerical Results ◽  
Kutta Method ◽  
Initial Value ◽  
Stiff System ◽  
The Third ◽  
Runge Kutta Method ◽  
One Step ◽  
Runge Kutta

Block Embedded Diagonally Implicit Runge-Kutta (BEDIRK4(3)) me- thod derived using Butcher analysis and equi-distribution of error approach is outperformed standard Runge-Kutta (RK) formulae. BEDIRK4(3) method produces approximation to the solution of initial value problem (IVP) at a block of three points simultaneously. The standard one step RK3(2) method is used to approximate the solution at the first point of the block. At the second points the solution is approximated using RK4(2) method which is generated by the previous research. The same approach is used to obtain the solution at the third point. The code for this method was built and the algorithm developed is suitable for solving stiff system. The efficiency of the method is supported by some numerical results.

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.

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