Two one-step methods to solve the initial value problem

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.

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

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.10.11.6 ◽

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.

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

International Journal of Natural and Applied Sciences ◽

10.4314/ijonas.v3i4.36221 ◽

2009 ◽

Vol 3 (4) ◽

Author(s):

VU Aihie

Keyword(s):

Initial Value Problem ◽

Arbitrary Order ◽

Initial Value ◽

Step Method ◽

One Step

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Development of a New One-Step Scheme for the Solution of Initial Value Problem (IVP) in Ordinary Differential Equations

International Journal of Theoretical and Applied Mathematics ◽

10.11648/j.ijtam.20170302.12 ◽

2017 ◽

Vol 3 (2) ◽

pp. 58 ◽

Cited By ~ 1

Author(s):

Fadugba Sunday Emmanuel

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Initial Value Problem ◽

Initial Value ◽

One Step

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Continuous One Step Linear Multi-Step Hybrid Block Method for the Solution of First Order Linear and Nonlinear Initial Value Problem of Ordinary Differential Equations

10.5772/intechopen.95619 ◽

2021 ◽

Author(s):

Kamoh Nathaniel ◽

Kumleng Geoffrey ◽

Sunday Joshua

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Initial Value Problem ◽

Initial Value Problems ◽

Algebraic Equations ◽

Initial Value ◽

Closed Form Solutions ◽

First Order ◽

One Step ◽

Collocation Approach

In this paper, a collocation approach for solving initial value problem of ordinary differential equations (ODEs) of the first order is presented. This approach consists of reducing the problem to a set of linear multi-step algebraic equations by approximating the ODE with a shifted Legendre polynomial basis function to determine the unknown constants. The proposed method is simple and efficient; it approximates the solutions very closely to the closed form solutions. Some problems were considered using Maple Software to illustrate the simplicity, efficiency and accuracy of the method. The results obtained revealed that the hybrid method can be suitable candidate for all forms of first order initial value problems of ordinary differential equations.

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