scholarly journals Implicit five-step block method with generalised equidistant points for solving fourth order linear and non-linear initial value problems

2019 ◽  
Vol 10 (4) ◽  
pp. 881-889
Author(s):  
Oluwaseun Adeyeye ◽  
Zurni Omar
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
Order Linear ◽  
Non Linear ◽  
Equidistant Points

scholarly journals Implicit Two-Point Block Method for Solving Fourth-Order Initial Value Problem Directly with Application

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Reem Allogmany ◽  
Fudziah Ismail ◽  
Zanariah Abdul Majid ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Approximation ◽  
Stability And Convergence ◽  
Block Method ◽  
Initial Value ◽  
Order Zero ◽  
First Derivative ◽  
Hermite Interpolating Polynomial

This paper proposes an implicit block method with two-point to directly solve the fourth-order Initial Value Problems (IVPs). The implicit block method is derived by adopting Hermite interpolating polynomial as the basis function, incorporating the first derivative of ft,y,y′,y′′,y′′′ to enhance the solution’s accuracy. A block formulation is presented to acquire the numerical approximation at two points simultaneously. The introduced method’s basic properties, including order, zero stability, and convergence, are presented. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed method compared with those of the several existing methods. Application in ship dynamics is also presented which yield impressive results for the proposed two-point block method.

scholarly journals Generalized Hybrid One-Step Block Method Involving Fifth Derivative for Solving Fourth-Order Ordinary Differential Equation Directly

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Mohammad Alkasassbeh ◽  
Zurni Omar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Initial Value Problems ◽  
Block Method ◽  
Initial Value ◽  
Order Ordinary Differential Equation ◽  
One Step ◽  
Approximate Function

A general one-step three-hybrid (off-step) points block method is proposed for solving fourth-order initial value problems of ordinary differential equations directly. A power series approximate function is employed for deriving this method. The approximate function is interpolated at xn,xn+r,xn+s,xn+t while its fourth and fifth derivatives are collocated at all points xi, i=0,r,s,t,1, in the interval of approximation. Several fourth-order initial value problems of ordinary differential equations are then solved to compare the performance of the proposed method with the derived methods. The analysis of the method reveals that the method is consistent and zero stable concluding that the method is also convergent. The numerical results demonstrate the superiority of the new method over the existing ones in terms of error.

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 Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
Emmanuel A Areo ◽  
Nosimot O Adeyanju ◽  
Sunday J Kayode
Keyword(s):  
Initial Value Problems ◽  
Hybrid Methods ◽  
Multistep Method ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Order Error ◽  
Block Methods ◽  
Grid Points ◽  
Step Number

This research proposes the derivation of a class of hybrid methods for solution of second order initial value problems (IVPs) in block mode. Continuous linear multistep method of two cases with step number k = 4 is developed by interpolating the basis function at certain grid points and collocating the differential system at both grid and off-grid points. The basic properties of the method including order, error constant, zero stability, consistency and convergence were investigated. In order to examine the accuracy of the methods, some differential problems of order two were solved and results generated show a better performance when comparison is made with some current methods.Keywords- Block Method, Hybrid Points, Initial Value Problems, Power Series, Second Order 

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