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

3 1 3 3 3 ! '' , , − − ∑ −+ = − =                           j h nxx rs j j h jja yyxf (5) Collocating (5) at all points 2 1, , , ,0 , wvu ssnx = + and Interpolating Equation (3) at vu rrnx , ,0 , = + , gives a system of non linear equation of the form U AX = (6) Where , 7,6,5,4,3,2,1,0 TaaaaaaaaA     = , 2 1 , , , , , , , T n fwnf vnfunfnf vnyunyny U                 + + + + + + =                                                                           = 3!4 5) 2 1(!7 3!3 4) 2 1(!6 3!2 3) 2 1(!5 3!1 2) 2 1(!4 3!0 2 1!3 0 00 3!4 5!7 3!3 4!6 3!2 3!5 3!1 2!4 3!0 !3 0 00 3!4 5!7 3!3 4!6 3!2 3!5 3!1 2!4 3!0 !3 0 00 3!4 5!7 3!3 4!6 3!2 3!5 3!1 2!4 3!0 !3 0 00 0 0 0 0 3!0 !3 0 00 7 6 5 4 3 2 1 7 6 5 4 3 2 1 0 0 0 0 0 0 01 h h h h h h w h w h w h w h w h v h v h v h v h v h u h u h u h u h u h v v v v v v v u u u u u uu X Solving (6) for sa i ' using Gaussian elimination method, gives a continuous hybrid linear multistep method of the form wvukknfk jnf i j h jny vu j j xy ,, , 2/1 0 3 ,,0 = + ++ ∑ = ++ ∑ = =                     β β α (7) Differentiating (7) twice yields                     + ∑ = ++ ∑ = ++ ∑ = = jnf j j jnf wvuj j h j ny vu j j h xp 2/1 0 ,, ,,0 2 1 '' β β α (8) We then impose the following conditions on       xy in (3) for j ny j nxy + = +           and j nf j nxy + = +           ''' The coefficient of vu j jny , ,0 , = + and 2 1, , , ,0 , wvu j jnf = + gives                               + + + ++ ++ + + = + =+ ∑ 2 1 )( 2 1 )( )( )( )(0 3 , ,0 )( n ft wnftw vnftv unftu nft h vu i inyt i tny β β β β β α Half-Step Implicit Linear Multistep Hybrid Block Third Derivative Methods of order Four for the Solution of Third order Ordinary Differential Equations 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 53 ( F ) © 2021 Global Journals Version I N otes