A New Numerical Integrator for the Solution of General Second Order Ordinary Differential Equations

Five Steps Block Predictor-Block Corrector Method for the Solution of <i>y''</i> = <i>f</i> (<i>x</i>,<i>y</i>,<i>y'</i>)

“…In [1] we compared our method with the existing methods like the block and block predictor-corrector and the results re-affirms the claim of [10] that though block predictor-corrector method takes longer time to implement, it gives better approximation than the block method. In this paper we extended the step length considered in [1] and considered varying the number of interpolation points to observe the effect on the performance of the method.…”

“…Also vital to this paper is the concept of block predictor-corrector method (Milne approach). This method formed a bridge between the predictor-corrector method and block method [4] [10] [13]. In [1] we stated that results generated at an overlapping interval affect the accuracy of the method and the nature of the model cannot be determined at the selected grid points.…”

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

“…Some of these models do not always have theoretical solutions, thus numerical methods are often employed to solve them. Researchers in most cases always use method of reduction of higher order ODEs into system of first order ODEs to solve (1). This technique though quite good, is bedeviled with many problems such tediousness, complexity of the method, waste of time, and the need for large computer storage memories because of too many auxiliary functions, etc.…”

“…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].…”

An Eight Order Two-Step Taylor Series Algorithm for the Numerical Solutions of Initial Value Problems of Second Order Ordinary Differential Equations

“…Actually, considerable attention has been devoted to solving ordinary differential equation of higher order directly without reduction for instance: methods of linear multistep method (LMM) were considered by Lambert and Watson [8], Dormand and El-Mikkawy [9], El-Mikkawy and ElDesouky [10] and Awoyemi [11] [12] [13] [14]. Subsequently, LMM was independently proposed by Kayode [15], Onumanyi et al [16] and Adesanya et al [17] in the predictor-corrector mode, based on collocation method. These authors proposed LMM with continuous coefficients where they adopted Taylor series algorithm to supply the starting values.…”