A New Block-Predictor Corrector Algorithm for the Solution of y’’’=f(x, y, y’, y’’)

Abstract

We consider direct solution to third order ordinary differential equations in this paper. Method of collection and interpolation of the power series approximant of single variable is considered to derive a linear multistep method (LMM) with continuous coefficient. Block method was later adopted to generate the independent solution at selected grid points. The properties of the block viz: order, zero stability and stability region are investigated. Our method was tested on third order ordinary differential equation and found to give better result when compared with existing methods.

ONE-TWELVETH STEP CONTINUOUS BLOCK METHOD FOR THE SOLUTION OF $ y'''=f(x,y,y',y'') $

“…Absolute error of the approximate solution are then compared with the existing methods. In particular, the comparison are made with those proposed by Olabode [16] and Adesanya et al [2] Discussion of the results of the methods are also done here.…”

“…Conventionally, higher order ordinary differential equations are solved directly by the predictorcorrector method where separate predictors are developed to implement the corrector and Taylor series expansion adopted to provide the starting values. Predictor-corrector methods are extensively studied by [1][2][3][4][5][6][7][8][9][10][11][12]. These authors proposed linear multistep methods with continuous coefficient, which have advantage of evaluation at all points within the grid over the proposed method in [4] The major setbacks of predictor-corrector method are extensively discussed by [6].…”

4 -step 5-point hybrid block method for the direct solution of second order initial value problems

“…The application of hybrid methods for the numerical approximation of (1.1) has been applied in predictor-corrector modes [8], block-predictor corrector [2] and direct block methods [3,4]. Although, this approach involves more function steps computation, the results obtained when solving numerical problems shows great advantage, especially when in comparison to previous methods of the same order or higher order as displayed in this article.…”

Hybrid Block Method for Direct Numerical Approximation of Second Order Initial Value Problems Using Taylor Series Expansions

“…Block multistep methods have evolved with time for the direct solution of second order initial value problems (Abdelrahim and Omar, 2016;Jator, 2007;Adesanya et al, 2012) and the concept of introducing evaluation at offgrid points birthing the introduction of hybrid block methods for numerical approximation of (1) have also been investigated into.…”

“…Hence, the main objective of this article which is the introduction of a new hybrid block method for numerically approximating (1) with better accuracy when compared to previously existing methods in terms of error. The methodology for developing the hybrid method is a new and easy to understand apptroach which differs from the commonly used interpolation (Jator, 2007;Adesanya et al, 2012;Kuboye and Omar, 2015) and numerical integration approach (See et al, 2016;Abdul Majid et al, 2012;Omar and Suleiman, 2016) for developing multistep methods.…”