scholarly journals CONTINUOUS BLOCK METHOD FOR THE SOLUTION OF SECOND ORDER INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATION

2013 ◽  
Vol 83 (3) ◽  
Author(s):  
A.A. James ◽  
A.O. Adesanya ◽  
S. Joshua
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Initial Value Problems ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Continuous Block

scholarly journals A Functionally-Fitted Block Numerov Method for Solving Second-Order Initial-Value Problems with Oscillatory Solutions

2021 ◽  
Vol 18 (6) ◽  
Author(s):  
R. I. Abdulganiy ◽  
Higinio Ramos ◽  
O. A. Akinfenwa ◽  
S. A. Okunuga
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Initial Value Problems ◽  
Second Order ◽  
Initial Value ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Oscillatory Solutions ◽  
Type Method ◽  
Convergent Method

AbstractA functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, $$y''=f \left( x,y,y' \right) $$ y ′ ′ = f x , y , y ′ , it is a fourth order convergent method for the special second-order ordinary differential equation, $$y''=f \left( x,y\right) $$ y ′ ′ = f x , y . Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.

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 

scholarly journals Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

2020 ◽  
Vol 4 (1) ◽  
pp. 448-455
Author(s):  
Mulugeta Andualem ◽  
◽  
Atinafu Asfaw ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Initial Value ◽  
Transform Method ◽  
Adomian Decomposition

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

scholarly journals The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

2019 ◽  
pp. 1-9
Author(s):  
Y. Skwame ◽  
J. Sabo ◽  
M. Mathew
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Direct Solution ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Better Than

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

scholarly journals Numerical Simulation of One Step Block Method for Treatment of Second Order Forced Motions in Mass-Spring Systems

2021 ◽  
pp. 1-11
Author(s):  
J. Sabo ◽  
T. Y. Kyagya ◽  
W. J. Vashawa
Keyword(s):  
Numerical Simulation ◽  
Initial Value Problems ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Mathematical Problems ◽  
One Step ◽  
Grid Points ◽  
The One ◽  
Mass Spring

This paper discuss the numerical simulation of one step block method for treatment of second order forced motions in mass-spring systems of initial value problems. The one step block method has been developed with the introduction of off-mesh point at both grid and off- grid points using interpolation and collocation procedure to increase computational burden which may jeopardize the accuracy of the method in terms of error. The basic properties of the one step block method was established and numerical analysis shown that the one step block method was found to be consistent, convergent and zero-stable. The one step block method was simulated on three highly stiff mathematical problems to validate the accuracy of the block method without reduction, and obviously the results shown are more accurate over the existing method in literature.

scholarly journals Generalized (ψ,α,β)—Weak Contractions for Initial Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 266 ◽  
Author(s):  
Piyachat Borisut ◽  
Poom Kumam ◽  
Vishal Gupta ◽  
Naveen Mani
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Initial Value Problems ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Weak Contraction ◽  
Initial Value ◽  
Order Ordinary Differential Equation ◽  
Ordered Metric Spaces ◽  
First Order

A class of generalized ( ψ , α , β ) —weak contraction is introduced and some fixed-point theorems in a framework of partially ordered metric spaces are proved. The main result of this paper is applied to a first-order ordinary differential equation to find its solution.

scholarly journals Functionally Fitted Block Method for Solving the General Oscillatory Second-Order Initial Value Problems and Hyperbolic Partial Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
S. N. Jator ◽  
F. F. Ngwane ◽  
N. O. Kirby
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Hyperbolic Partial Differential Equations ◽  
Fixed Frequency ◽  
Block Method ◽  
Initial Value ◽  
Partial Differential ◽  
Predictor Corrector

We present a block hybrid functionally fitted Runge–Kutta–Nyström method (BHFNM) which is dependent on the stepsize and a fixed frequency. Since the method is implemented in a block-by-block fashion, the method does not require starting values and predictors inherent to other predictor-corrector methods. Upon deriving our method, stability is illustrated, and it is used to numerically solve the general second-order initial value problems as well as hyperbolic partial differential equations. In doing so, we demonstrate the method’s relative accuracy and efficiency.

An optimized two-step hybrid block method for solving general second order initial-value problems

2015 ◽  
Vol 72 (4) ◽  
pp. 1089-1102 ◽  
Author(s):  
Higinio Ramos ◽  
Z. Kalogiratou ◽  
Th. Monovasilis ◽  
T. E. Simos
Keyword(s):  
Initial Value Problems ◽  
Second Order ◽  
Block Method ◽  
Initial Value

scholarly journals Hybrid Third Derivative Block Method for the Solution of General Second Order Initial Value Problems with Generalized One Step Point

2019 ◽  
Vol 12 (3) ◽  
pp. 1199-1214
Author(s):  
Ra'ft Abdelrahim ◽  
Z. Omar ◽  
O. Ala’yed ◽  
B. Batiha
Keyword(s):  
Power Series ◽  
Convergence Analysis ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Procedure ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Third Derivative

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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