scholarly journals An explicit third-order one-step method for autonomous scalar initial value problems of first order based on quadratic Taylor approximation

2020 ◽  
Vol 13 (2) ◽  
pp. 231-255
Author(s):  
Thomas Krainer ◽  
Chenzhang Zhou
Keyword(s):  
Initial Value Problems ◽  
Initial Value ◽  
Step Method ◽  
Third Order ◽  
First Order ◽  
Taylor Approximation ◽  
One Step

scholarly journals One–Step Hybrid Block Scheme for the Numerical Approximation for Solution of Third Order Initial Value Problems

2021 ◽  
pp. 51-61
Author(s):  
J. Sabo ◽  
T. Y. Kyagya ◽  
M. Solomon
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
Third Order ◽  
First Order ◽  
Block Algorithm ◽  
One Step ◽  
Linear Block

In this research, we have proposed the simulation of linear block algorithm for modeling third order highly stiff problem without reduction to a system of first order ordinary differential equation, to address the weaknesses in reduction method. The method is derived using the linear block method through interpolation and collocation. The basic properties of the block method were recovered and was found to be consistent, convergent and zero-stability. The new block method is been applied to model third order initial value problems of ordinary differential equations without reducing the equations to their equivalent systems of first order ordinary differential equations. The result obtained on the process on some sampled modeled third order linear problems give better approximation than the existing methods which we compared our result with.

scholarly journals Direct integrator of block type methods with additional derivative for general third order initial value problems

2020 ◽  
Vol 12 (10) ◽  
pp. 168781402096618
Author(s):  
Mohammed Yousif Turki ◽  
Fudziah Ismail ◽  
Norazak Senu ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Order System ◽  
Block Type ◽  
Initial Value ◽  
Third Order ◽  
First Order ◽  
Block Methods ◽  
Hermite Interpolating Polynomial

This paper presents the construction of the two-point and three-point block methods with additional derivatives for directly solving [Formula: see text]. The proposed block methods are formulated using Hermite Interpolating Polynomial and approximate the solution of the problem at two or three-point concurrently. The block methods obtain the numerical solutions directly without reducing the equation into the first order system of initial value problems (IVPs). The order and zero-stability of the proposed methods are also investigated. Numerical results are presented and comparisons with other existing block methods are made. The performance shows that the proposed methods are very efficient in solving the general third order IVPs.

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

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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