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Two-step Hybrid Block Method for the Numerical Solution of Third Order Ordinary Differential Equations
Abstract
A new zero-stable two-step hybrid block method for solving third order initial value problems of ordinary differential equations directly is derived and proposed. In the derivation of the method, the assumed power series solution is interpolated at the initial and the hybrid points while its third ordered derivative is collocated at all the nodal and off-step points in the interval of consideration. The relevant properties of the method were examined and the method was found to be zero-stable, consistent and convergent. A comparison of the results by the method with the exact solutions and other results in literature shows that the method is accurate, simple and effective in solving the class of problems considered.
Development of 3-Point Block Method with one off-step point using Backward
Differentiation Formula (BDF) is presented in this paper to find out the solution for stiff Ordinary
Differential Equation (ODEs). By considering the Backward Differentiation Formulas (BBDF),
the block method has been derived. It is well known that BBDF is used for solving stiff ODEs.
The strategy for the development of this process is to compute three solution values with one off-step point concurrently to each iteration. One off-step point is added in the implicit BBDF method
for better accuracy. Derivation of the formulae and consistency properties are generated in this
paper. Numerically the proposed method with order five is achieved as a result. Mathematica
software has been used for the derivation and consistency of the method.
In this paper, we propose a new One Step Hybrid method
for numerical integration of third order ordinary differential equation. The
method was developed via collocation and interpolation techniques of the
power series approximation. The accuracy and stability of the proposed
method were well discussed and their applicability to some initial value problems
is also considered. The new hybrid method was found to give better
approximation than the existing method
n this paper, fourth-order Improved Runge-Kutta method (IRKD) for directly solving a special third-order ordinary differential equation is constructed. The fourth order IRKD method has a lower number of function evaluations compared with the fourth order Runge-Kutta method. The stability polynomial of the method is given. Numerical comparisons are also performed using the existing Runge-Kutta method after reducing the problems into a system of first-order equations and solving them, and direct RKD method for solving special third order ordinary differential equations. Numerical examples are presented to illustrate the efficiency and the accuracy of the new method in terms of number of function evaluations as well as max absolute error.
In this study, a simple and Taylor series-based method known as differential transformation method (DTM) is used to solve initial-value problems involving third-order ordinary differential equations. We introduced briefly the concept of DTM and applied it to obtain the solution of three numerical examples for demonstration. The results are compared with the existing ones in literature and it is concluded that results yielded by DTM converge to the analytical solution more rapidly with few terms.
In this paper, we developed a new continuous block method using the approach of collocation of the differential system and interpolation of the power series approximate solution. A constant step length within a half step interval of integration was adopted. We evaluated at grid and off grid points to get a continuous linear multistep method. The continuous linear multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent and zero stable hence convergent. The new method was tested on real life problems namely: SIR model, Growth model and Mixture Model. The results were found to compete favorably with the existing methods in terms of accuracy and error bound.
A block linear multistep method for solving special third order initial value problems of ordinary differential equations is presented in this paper. The approach of collocation approximation is adopted in the derivation of the scheme and then the scheme is applied as simultaneous integrator to special third order initial value problem of ordinary differential equations. This implementation strategy is more accurate and efficient than those given when the same scheme is applied over overlapping intervals in predictor- corrector mode. Furthermore, the new block method possesses the desirable feature of Runge-Kutta method of being self-starting and eliminates the use predictor-corrector method. Experimental results confirm the superiority of the new scheme over the existing methods.
In this article we use Adomian decomposition method, which is a well-known method for solving functional equations now-a-days, to solve systems of differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one. Theoretical considerations are being discussed, and convergence of the method for theses systems is addressed. Some examples are presented to show the ability of the method for linear and non-linear systems of differential equations.
In this paper, we develop an order six block method using method of collocation and interpolation of power series approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continous hybrid linear multistep method is solved for the independent solutions to give a continous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method was investigated and found to be zero stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing method.
A family of hybrid linear multistep methods with two nonstep points is proposed for the direct solution of second order initial value problems. The methods are applied in block form as simultaneous numerical integrators over non-overlapping intervals. We give specific cases for step number k=2,3,4 to illustrate the process. Numerical results obtained using the proposed methods reveal that they are highly competitive with existing methods in the literature.
We consider the third-order nonlinear differential equation u''' (t) = f (t, u (t), u'(t), u" (t)), a.e. t is an element of (0, 1), satisfying u(0) = u'(0) = u"(1) = 0 and u(0) = u'(1) = u"(1) = 0, where f : [0, 1] x R-3 -> R is L-p-Caratheodory, 1 <= p < infinity. We obtain the existence of at least one positive solution using the Leray-Schauder Continuation Principle for each set of boundary conditions by separately considering the cases p > 1 and p = 1. (c) 2006 Elsevier Ltd. All rights reserved.
In this paper, a new block method of the second order is presented to solve initial value problems numerically. This method is similar to the block trapezoidal rule [S. Abbas and L.M. Delves, Parallel solution of ODE's by one step block methods, Report CSMR, University of Liverpool, 1989.], where the low power of the block size appears in the principal local truncation error. Direct comparison with the related results of the block trapezoidal rule has been outlined.
Discrete variable methods in ODE
- P Henrici
P. Henrici, Discrete variable methods in ODE. New York, USA: John Wiley & Sons, 1962.
A six-step continuous multistep method for the solution of general fourth order initial value problems of ordinary differential equations
- D O Awoyemi
- S J Kayode
- L O Adoghe
D. O. Awoyemi, S. J. Kayode and L. O. Adoghe, A six-step continuous multistep method for the solution of general fourth order
initial value problems of ordinary differential equations, Journal of Natural Sciences Research, 5(5), 2015, 131 -139.
Numerical solution of third order ordinary differential equations using a seven-step block method
- J O Okuboye
- Z Omar
J. O. Okuboye and Z. Omar, Numerical solution of third order ordinary differential equations using a seven-step block method,
International Journal of Mathematical Analysis, 9(15), 2015, 743 -745, doi.org/10.12988/ijma.2015.5125.