Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

Adebola (2016), Adoghe and Omole (2019), Kuboye and Omar (2015), Olabode and Momoh (2016), Kayode and Adeyeye (2011) and Mohammed and Adeniyi (2014). We adopted the method of collocation and interpolation of the power series approximation as the basis function to generate continuous linear multistep method as other scholars in solving (1). The derivation of continuous linear multistep methods for direct solution of ordinary differential equations have been discussed over the years in literature and these include, among others collocation, interpolation, integration, interpolating polynomials and basis functions such as, Chebyshev polynomials, trigonometric functions, exponential functions. In this paper, we developed half-step third derivative hybrid block method of order four with two cases for direct solution of third order ordinary differential equation which is implemented in block. The method developed evaluates less function per step and circumventing the Dahlquist barrier ’ s by introducing a hybrid points. The paper is organized as follows: First is a discussion of the new half-step third derivative hybrid block method and the materials for the development of the method. This is followed by a consideration of analysis of the basic properties of the new half- step third derivative hybrid block method, which include convergence, stability region, numerical experiments where the efficiency of the derived method is tested on some stiff numerical examples and discussion of results. Lastly, the study concludes by comparing the results obtained with an existing work of Adeyeye and Omar (2018), Adebayo and Adebola (2016), Adoghe and Omole (2019) and Mohammed and Adeniyi (2014). II. D erivation of the N ew H alf -S tep H ybrid B lock M ethod In this section we intend to develop a family of half-step third derivative hybrid method with three hybrid points w and vu , , which are all rational numbers           ∈ 2 1,0 , , wvu of the form                           + ++ ++ + + = + = + =+ ∑ ∑ wnftw vnft v unftu j nft j k j h vu i inyt i tny )( )( )( )( 0 3 , ,0 )( β β β β α (2) )( ,)( ,)( ,)( ,)( ,)( ), (0 tw tv tu t j tv tu t β β β β α α α are polynomials,           + =+ j nxy j ny ,           + + =+ j ny j nxf j nf , h nxx t − = Consider the power series approximate solution of the form j h nxx rs j ia xy                 − ∑ −+ = = 1 0 (3) where 3 = r and 5 = s are the numbers of interpolation and collocation points respectively, is considered to be a solution to (1). The third derivative of (3) gives 3 1 3 3 3 ! ''' − − ∑ −+ = − =                         j h nxx rs j j h jja x y (4) Substituting (4) into (1) gives Half-Step Implicit Linear Multistep Hybrid Block Third Derivative Methods of order Four for the Solution of Third order Ordinary Differential Equations © 2021 Global Journals 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 52 ( F ) Version I N otes