ONE STEP BLOCK METHOD FOR THE DIRECT SOLUTION OF THIRD ORDER INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

2016 ◽  
Vol 99 (6) ◽  
pp. 945-958
Author(s):  
Ra’ft Abdelrahim ◽  
Z. Omar
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Direct Solution ◽  
Block Method ◽  
Initial Value ◽  
Third Order ◽  
One Step

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 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 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.

scholarly journals Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Mohammad Alkasassbeh ◽  
Zurni Omar
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Approximate Function ◽  
The Given ◽  
Third Derivative

A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at {xn,xn+r} while its second and third derivatives are collocated at all points {xn,xn+r,xn+s,xn+t,xn+1} in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy.

scholarly journals An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lee Ken Yap ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Film Flow ◽  
Direct Solution ◽  
Thin Film Flow ◽  
Block Method ◽  
Third Order ◽  
Block Form ◽  
The Stability

The block hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in block form to provide the approximation at five points concurrently. The stability properties of the block method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow.

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