scholarly journals L-Stable Block Hybrid Numerical Algorithm for First-Order Ordinary Differential Equations

2020 ◽  
pp. 160-165
Author(s):  
B. I. Akinnukawe ◽  
K. O. Muka
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Methods ◽  
Initial Conditions ◽  
Multistep Method ◽  
Stability And Convergence ◽  
First Order ◽  
One Step ◽  
Grid Points ◽  
The Stability

In this work, a one-step L-stable Block Hybrid Multistep Method (BHMM) of order five was developed. The method is constructed for solving first order Ordinary Differential Equations with given initial conditions. Interpolation and collocation techniques, with power series as a basis function, are employed for the derivation of the continuous form of the hybrid methods. The discrete scheme and its second derivative are derived by evaluating at the specific grid and off-grid points to form the main and additional methods respectively. Both hybrid methods generated are composed in matrix form and implemented as a block method. The stability and convergence properties of BHMM are discussed and presented. The numerical results of BHMM have proven its efficiency when compared to some existing methods.

scholarly journals A New L-Stable Third Derivative Hybrid Method for Solving First Order Ordinary Differential Equations

2021 ◽  
pp. 58-69
Author(s):  
Lawrence Osa Adoghe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Stiff Systems ◽  
Convergence Properties ◽  
Main Method ◽  
The Stability ◽  
Third Derivative

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

STABILITY ANALYSIS OF A PROPOSED SCHEME OF ORDER FIVE FOR FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 898
Author(s):  
Sunday Emmanuel Fadugba ◽  
Roseline Bosede Ogunrinde ◽  
Rowland Rotimi Ogunrinde
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Step Length ◽  
Step Method ◽  
First Order ◽  
Absolute Relative Error ◽  
One Step ◽  
Test Equation ◽  
The Stability

This paper presents the stability analysis of a proposed scheme of order five (FCM) for first order Ordinary Differential Equations (ODEs). The proposed FCM is derived by means of an interpolating function of polynomial and exponential forms. The properties of FCM were discussed extensively. The linear stability of FCM in the context of the Third Order One-Step Method (TCM) and Second Order One-Step Method (SCM) for the solution of initial value problems of first order differential equations is presented. The stability region of FCM, TCM and SCM is investigated using the Dahlquist’s test equation. The numerical results obtained via FCM are compared with TCM and SCM. Moreover, by varying the step length, the accuracy and convergence of the methods in terms of the final absolute relative error are measured. The results show that FCM converges faster and more stable than its counterparts.

DERIVATION AND IMPLEMENTATION OF A THREE-STEP HYBRID BLOCK ALGORITHM (TSHBA) FOR DIRECT SOLUTION OF LINEAR AND NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 4 (4) ◽  
pp. 477-483
Author(s):  
O. E. Abolarin ◽  
B. G. Ogunware ◽  
A. F. Adebisi ◽  
S. O. Ayinde
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Multistep Method ◽  
Second Order ◽  
Direct Solution ◽  
Block Method ◽  
Order Error ◽  
Block Algorithm ◽  
Grid Points

The development and application of an implicit hybrid block method for the direct solution of second order ordinary differential equations with given initial conditions is shown in this research. The derivation of the three-step scheme was done through collocation and interpolation of power series approximation to give a continuous linear multistep method. The evaluation of the continuous method at the grid and off grid points formed the discrete block method. The basic properties of the method such as order, error constant, zero stability, consistency and convergence were properly examined. The new block method produced more accurate results when compared with similar works carried out by existing authors on the solution of linear and non-linear second order ordinary differential equations

scholarly journals Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Nur Zahidah Mokhtar ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Method ◽  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Step Method ◽  
One Step ◽  
The Stability ◽  
The One

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in thePE(CE)mmode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

Exact Equations for the Inextensional Deformation of Cantilevered Plates

1979 ◽  
Vol 46 (3) ◽  
pp. 631-636 ◽  
Author(s):  
J. G. Simmonds ◽  
A. Libai
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Independent Set ◽  
Straight Edge ◽  
The Other ◽  
Subsequent Paper ◽  
First Order ◽  
Alternate Forms ◽  
Approximate Equations

A set of first-order ordinary differential equations with initial conditions is derived for the exact, nonlinear, inextensional deformation of a loaded plate bounded by two straight edges and two curved ones. The analysis extends earlier approximate work of Mansfield and Kleeman, Ashwell, and Lin, Lin, and Mazelsky. For a plate clamped along one straight edge and subject to a force and couple along the other, there are 13 differential equations, but an independent set of 9 may be split off. In a subsequent paper, we consider alternate forms of these 9 equations for plates that twist as they deform. Their structure and solutions are compared to Mansfield’s approximate equations and particular attention is given to tip-loaded triangular plates.

A NEW CLASS OF L-STABLE HYBRID ONE-STEP METHODS FOR THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 1 (2) ◽  
pp. 39-44
Author(s):  
Mohammad Mehdizadeh ◽  
Maryam Molayi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Hybrid Methods ◽  
Akzo Nobel ◽  
New Class ◽  
Numerical Studies ◽  
One Step ◽  
Judicious Choice ◽  
Stable Hybrid

In this paper, a class of one-step hybrid methods for the numerical solution of ordinary differential equations (ODEs) are considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a class of method is derived which is shown to be L-stable and so is appropriate for the solution of certain ordinary differential and stiff differential equations. We apply the new method for numerical integration of some famous stiff chemical problems such chemical Akzo-Nobel problem, ROBER problem (suggested by Robertson) and some others which are very popular in numerical studies.

Dynamics of a Basketball Rolling Around the Rim

2005 ◽  
Vol 128 (2) ◽  
pp. 359-364
Author(s):  
C. Q. Liu ◽  
Fang Li ◽  
R. L. Huston
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Coaching Strategies ◽  
Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

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