scholarly journals An efficient 4-step block method for solution of first order initial value problems via shifted chebyshev polynomial

2021 ◽  
Vol 1 (2) ◽  
pp. 25-36
Author(s):  
Isah O. ◽  
Salawu S. ◽  
Olayemi S. ◽  
Enesi O.
Keyword(s):  
Exact Solution ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Test Problems ◽  
Block Method ◽  
Initial Value ◽  
Order Zero ◽  
First Order ◽  
Continuous Scheme ◽  
Shifted Chebyshev Polynomial

In this paper, we develop a four-step block method for solution of first order initial value problems of ordinary differential equations. The collocation and interpolation approach is adopted to obtain a continuous scheme for the derived method via Shifted Chebyshev Polynomials, truncated after sufficient terms. The properties of the proposed scheme such as order, zero-stability, consistency and convergence are also investigated. The derived scheme is implemented to obtain numerical solutions of some test problems, the result shows that the new scheme competes favorably with exact solution and some existing methods.

scholarly journals Seventh order hybrid block method for solution of first order stiff systems of initial value problems

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
O. A. Akinfenwa ◽  
R. I. Abdulganiy ◽  
B. I. Akinnukawe ◽  
S. A. Okunuga
Keyword(s):  
Initial Value Problems ◽  
Stiff Systems ◽  
Block Method ◽  
Initial Value ◽  
First Order ◽  
Seventh Order

Continuous linear multistep method for the general solution of first order initial value problems for Volterra integro-differential equations

2018 ◽  
Vol 14 (5) ◽  
pp. 960-969
Author(s):  
Nathaniel Mahwash Kamoh ◽  
Terhemen Aboiyar
Keyword(s):  
General Solution ◽  
Approximation Method ◽  
Legendre Polynomial ◽  
Basis Function ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
Content Type ◽  
First Order ◽  
Predictor Corrector

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

scholarly journals A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Nazreen Waeleh ◽  
Zanariah Abdul Majid
Keyword(s):  
Initial Value Problems ◽  
Adaptive Strategy ◽  
Variable Step Size ◽  
Block Method ◽  
Initial Value ◽  
Step Size ◽  
First Order ◽  
Variable Step ◽  
Improved Performance ◽  
Fifth Order

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs.

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.

scholarly journals New Algorithm for First order Stiff Initial Value Problems

2017 ◽  
Vol 58 (1) ◽  
pp. 19-28 ◽  
Author(s):  
A. O. Adesanya ◽  
R. O. Onsachi ◽  
M. R. Odekunle
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
System Of Nonlinear Equations ◽  
Initial Value ◽  
Unknown Parameters ◽  
First Order ◽  
Continuous Scheme ◽  
Discrete Methods ◽  
Grid Points ◽  
The Stability

AbstractIn this paper, we consider the development and implementation of algorithms for the solution of stiff first order initial value problems. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme that is evaluated at selected grid points to give discrete methods. The stability properties of the method is verified and numerical experiments show that the new method is efficient in handling stiff problems.

scholarly journals Modified Legendre Collocation Block Method for Solving Initial Value Problems of First Order Ordinary Differential Equations

OALib ◽  
2018 ◽  
Vol 05 (07) ◽  
pp. 1-15
Author(s):  
Toyin Gideon Okedayo ◽  
Ayodele Olakiitan Owolanke ◽  
Olaseni Taiwo Amumeji ◽  
Muyiwa Philip Adesuyi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
First Order
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