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Uniform Order Continuous Block Hybrid Method for the Solution of First Order Ordinary Differential Equations
Abstract
We know that for any numerical method to be efficient and computational reliable, it must be convergent, consistent, and stable. This paper adopted the method of interpolation of the approximate solution and collocation ofits differential system at grid and off grid points to yield a continuous linear multistep method with a constant step size. The continuous linear multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected grid and off grid points to yield a discrete block method. The basic property of this method is verified to be convergent consistent and satisfies the conditions for stability. The method was tested on numerical examples and found to compete favorably withthe existing methods in term of accuracy and error variation.
... The above Problem has been considered in [32] with a uniform block order of 13. Their method was directly employed without starting values. ...
... The results of the derived formulae are presented in Table 4 with the efficiency curves shown in Figure 8. A clear comparison of our derived methods indicates that 7D2PIB2 outperformed 7D2PIB1 of the same order 10, though with minimal comparable performance in accuracy while outperformed such a method of order 13 in [32]. Figure 7 shows the competitive performance of 7D2PIB1, 7D2PIB2 and with such existing methods in [32]. ...
... A clear comparison of our derived methods indicates that 7D2PIB2 outperformed 7D2PIB1 of the same order 10, though with minimal comparable performance in accuracy while outperformed such a method of order 13 in [32]. Figure 7 shows the competitive performance of 7D2PIB1, 7D2PIB2 and with such existing methods in [32]. It is clear that 7D2PIB1 ...
In this research, a class of implicit block methods of a seventh derivative type are examined through interpolation and collocation techniques using finite power series as the basis function. The discrete schemes, which are implicit two-point block methods, are obtained by carefully and unevenly choose collocation points that ensure better methods’ stability via test. However, these schemes require seventh derivative functions unlike other existing numerical formulae. The new methods are found, investigated and proven to be convergent and A-stable. The implementation of methods is achieved by using Newton Raphson’s method. Experiments show the efficiency and accuracy of the developed formulae on different class of first-order initial value problems, including SIR, growth models and Prothero-Robinson oscillatory problem and with comparison to such existing methods. In addition, it is observed that uneven and positioning of collocation points greatly influence the efficiency and accuracy of numerical methods.
... is examined in this paper. The implementation of numerical method in predictor-corrector approach has some weaknesses which include evaluation of more functions per step which 1 leads to computational burden and lots of human effort (James et. al, 2012). In order to overcome the above disadvantages stated, Sagir (2014) established three-step block method without predictor-corrector mode where three-step with a single off-step point were employed as interpolation points for solving first order ordinary differential equations. The numerical results produced when the method was applied to ...
... By comparing the coefficient of h,C 0 =C 1 =C 2 =C 3 =C 4 andC 4+1 = 0 are obtained. Hence, the order of the new method is [4,4,4,4] T with error vector constants ...
... By comparing the coefficient of h,C 0 =C 1 =C 2 =C 3 =C 4 andC 4+1 = 0 are obtained. Hence, the order of the new method is [4,4,4,4] T with error vector constants ...
This paper presents a single-step block method with three generalized off-step points for solving first order ordinary differential equations. The approach employed in developing this new method is interpolating the approximated power series of order four at xn as well as at all off-step points and collocating the derivative of the power series at xn+1. The developed method is zero-stable, consistent, convergent and of order four. The new method has better accuracy than the existing methods when solving first order ordinary differential equations.
... In this article, the numerical solution to the general second order initial value problem of ordinary differential equation The method of reducing (1) to its equivalent system of first order has been found having some setback which includes: wastage of computer time, a lot of human effort and computational burden (see [4], [8] and [2]). Therefore, scholars have paid more attention on the establishment of direct methods for solving higher order ODEs whereby the numerical results generated are better than the method of reduction to system of first order ODEs ( see [15], [11] and [3]). Some of the methods developed include the self -starting Runge -Kutta type which contains many functions to be evaluated per step ( [5] and [13] ) and linear multistep methods which are not self-starting but require little function to evaluate per step [12]. ...
... Multiplying Equation (6) by gives the hybrid block method 11 2 (8) can also written as ...
... 1 , 10 5 s r = = into Equation (11) gives the order of the method A to be 4,4,4,4,4 ...
This paper proposes a single-step hybrid block method
with generalized two off-step point for the direct solution of initial
value problem of second order ordinary differential equations. The
uses of power series approximate solution as an interpolation
polynomial at the off points is employed in developing this method,
while its second derivative is collocated at all points in the interval.
Furthermore, some basic properties of the generalized method such
as order, zero stability, consistency and convergence are also
established. In addition, two examples of specific points of the
developed method are considered to solve some initial value
problems of second order ordinary differential equations. The
numerical results confirm that the proposed method produces better
accuracy if compared with the existing methods.
... Diverse method(s) of obtaining approximate solution to general or special equations in the form of ) ,..., ' , , ( (1) have been derived for various classes of linear multistep methods. Amongst these are the methods of Fatunla (1988), Ayinde and Ibijola (2015), Onumanyi et al (1994), Butcher (1993), James et al (2012), , Mohammed and Yahaya (2010), Skwame et al (2012), Ajileye et al (2018). Ogunride et al (2020), Isah et al (2012) and Salawu et al (2020), where (1) is of first order. ...
... Ogunride et al (2020), Isah et al (2012) and Salawu et al (2020), where (1) is of first order. Further development led to approximate solution to (1) where = 2, see ( Adesanya et al (2009), Awoyemi et al (2012), Awoyemi and Kayide (2005), D'Ambrosio et al ( ), Fatunla (1991, Fudziah et al (2009), Jator and Li (2009), Yahaya and Badmus (2009) and Adoghe and Omole (2018) Adesanya et al (2012) and ) and , Adeyeye and Omar (2019) and Luke et al (2020)) respectively. To achieve higher efficiency of derived methods of solution, it is expected that the approximating polynomial agrees with the solution of a given differential equation at good number of points within a finite interval within which the solution to such differential equation is being sort. ...
In this paper, we adopt the general method of interpolation and collocation in Linear Multistep Methods in deriving some numerical schemes for solving second order ordinary differential equations. Different choices of the interpolating function in the form of shifted Legendre, shifted Chebyshev and Lucas polynomials with the same interpolation and collocation points are considered in order to establish uniformity or otherwise of the derived schemes for the various polynomials. Furthermore, probable disparities in the derived schemes for varied choices of interpolation and collocation points are also investigated. Results indicate that all the polynomials yield exactly the same schemes for the same choice of interpolation and collocation points but different schemes for different choices of interpolation and collocation points. However, numerical examples considered showed that all the derived schemes performed exactly in the same manner in terms of accuracy, regardless of the choices of interpolation or collocation points. Nevertheless, the derived schemes perform admirably better when compared with existing methods in literature
... The block methods are self-starting and can be applied to both stiff and non-stiff initial value problem in differential equations. More recently, authors like [6][7][8][9][10] and to mention few, these authors proposed methods ranging from predictor-corrector to hybrid block method for initial value problem in ordinary differential equation. ...
In this paper, we consider the derivation of hybrid block method for the solution of general first order Initial Value Problem (IVP) in Ordinary Differential Equation. We adopted the method of Collocation and Interpolation of power series approximation to generate the continuous formula. The properties and feature of the method are analyzed and some numerical examples are also presented to illustrate the accuracy and effectiveness of the method.
... The block methods are self-starting and can be applied to both stiff and non-stiff initial value problem in differential equations. More recently, authors like [6,7,8,9] and [10] to mention few proposed methods ranging from predictor-corrector to hybrid block method for initial value problem in ordinary differential equation. ...
... Several scholars such as Awoyemi et al. (2007), Badmus and Mishelia (2011), Sunday et al. (2013) developed numerical methods which were implemented in predictor-corrector mode for solving (1). The implementation of numerical method in predictor-corrector approach, however, has some setbacks which include lengthy computational time due to more function evaluations needed per step and computational burden which may affect the accuracy of the method in terms of error (James et al., 2012). In overcoming the setbacks mentioned above, Sagir (2014) developed a three-step block method without predictor where three points with a single off-step point were considered as interpolation points for solving first order ordinary differential equations. ...
A new single-step hybrid block method with three off-step points for the solution of first order ordinary differential equations is proposed. The strategy employed to develop this method is interpolating the power series approximate solution at xn and off-step points and collocating the derivative of the power at xn+1. The class of linear multistep method derived is then simultaneously applied to first order ordinary differential equations together with the associated initial conditions. The numerical results generated are found to be better when compared with the existing methods in terms of error. Besides its excellent performance in term of accuracy, this method also possesses good properties of numerical method such as zerostable, consistent and convergent.
... Many researchers have worked on the development of continuous linear multi-step method in finding solution to (1). These scholars proposed methods with different basis functions, among them are, [3,4,5], to mention few. These block methods are self-starting and can directly be applied to stiff problems. ...
The performance of the Block Generalised Multistep Adams and Backward Differentiation Formulae (BGMBDF) as compared with Generalised Multistep Adams and Backward Differentiation Formulae (GMBDF) is presented. These methods are used to solve initial value problems (IVPs) for stiff and nonstiff systems of ordinary differential equations (ODEs). The results obtained show that the BGMBDF reduces the total number of steps in computation.
We construct a self-starting Simpson’s type block hybrid method (BHM) consisting of very closely
accurate members each of order p=q+2 as a block. The higher order members of each were
obtained by increasing the number k in the multi-step collocation (MC) used to derive the k-step
continuous formula (k≥2) through the aid of MAPLE soft wire program. The stability analysis is
poorer as the step size k increases, as expect ed with the linear multi-step methods (lmms),
discrete or continuous. In this paper we identify a continuous hybrid block schemes (CHBS)
through the addition of one off-mesh collocation points in the MC. The (CHBS) is evaluated along
with it’s first derivative where necessary to give continuous hybrid block schemes for a
simultaneous application to the stiff ordinary differ ential equations (SODEs} with initial, boundary
or mixed conditions.
In this note, we describe the construction of two-step Block Simpson type method with non-uniform order, in continuous approximation form, with large regions of absolute stability. All the discrete schemes used in the block method come from a single continuous formulation. When used in block form, the suggested approach is self- starting, accurate and efficient. Numerical result is included to further justify our present method. JONAMP Vol. 11 2007: pp. 261-268
A multistep collocation technique is used for the direct solution of second-order ordinary differential equations. The approach is used to obtain multiple finite differential methods which are combined as simultaneous numerical integrators to form some block methods which are self-starting. The stability and convergence of the block methods are investigated, and the methods are found to be zero-stable, consistent and hence convergent. The block methods derived are tested on standard electric circuit and mechanical problems to illustrate the accuracy and desirability of our new method.
New block methods of order two and three for the numerical solution of initial value problems are derived. The matrix coefficients of these methods are chosen such that low powers of the blocksize appear in the principal local truncation errors.
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.
Some uniform order block methods for the solution of first ordinary differential equation
- A Badmus
- D Mishehia
Badmus, A.M and Mishehia, D.W (2011), "Some uniform order block methods for the solution of first ordinary differential
equation", J. N.A.M. P, 19, 149-154
Solution of first order system of ordering differential equation by finite difference methods with arbitrary
- J Fatokun
- Onumanyi
- U Serisena
Fatokun, J, Onumanyi, P and Serisena, U.V (2005), "Solution of first order system of ordering differential equation by finite
difference methods with arbitrary". J.N.A.M.P, 30-40.
They proposed a hybrid method of order seven and adopted classical RungeKutta method to provide the starting values. The new method gave better approximation because the proposed method is self-starting and does not require starting values. Problem 2 was solved by Badmus and Mishelia
- V Discussion
- The Result We
V. DISCUSSION OF THE RESULT
We have considered two numerical examples to test the efficiency of our method. Problem 1 was
solved by Areoet al. (2012). They proposed a hybrid method of order seven and adopted classical RungeKutta
method to provide the starting values. The new method gave better approximation because the proposed method
is self-starting and does not require starting values. Problem 2 was solved by Badmus and Mishelia (2012). They
adopted self-starting block methods of order six. Our method gave better approximation because the iteration
per step in the new method was lower than the method proposed by Badmus and Mishelia (2012)