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 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.

The Formation of Implicit Second Order Backward Difference Adam’s Formulae for Solving Stiff Systems of First Order Initial Value Problems of Ordinary Differential Equations

2020 ◽  
pp. 21-29
Author(s):  
Sabo J. ◽  
Kyagya T. Y. ◽  
Ayinde A. M.
Keyword(s):  
Initial Value Problems ◽  
Absolute Stability ◽  
Multistep Method ◽  
Second Order ◽  
Stiff Systems ◽  
Initial Value ◽  
First Order ◽  
Systems Of Odes ◽  
Linear Odes ◽  
Region Of Absolute Stability

The formation of implicit second order backward difference Adam’s formulae for solving stiff systems of ODEs was study in this paper. We used interpolation and collocation in deriving backward differentiae Adam’s formulae. The basic properties of our method was analyzed, and it was found to be consistent, zero-stability and convergent, we further plotted the region of absolute stability and it was shown to be A-stable. Numerical evidences shows that the multistep method develop is very effective method for in handling linear ODEs either initial value problems or boundary value 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

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.

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