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

a) Specification of the New Half-step Third Derivative Method The family of hybrid half-step method with three off-grid hybrid points u,v and w which are rational numbers. Equation (7) is evaluated at the non-interpolating points               + + 2 1 , n xwnx and (8) at all points               + + + + 2 1 , , , , n xwnxvnxunxnx , which produces the following general equations in block form 3 2 1 RDRCRBLYA + + = (9) Case One: We consider the case where           = = = 6 1 ,7 1 ,8 1 w v u equation (9) becomes                                                                                                                                                                                                                                                                                                                                                                                                                                           = − = = − − = + + + + + + + + + + + + = − − − − − − − − − − − = nf R C R h h h h h h h h h h B n y n y n y n y n y n y n y n y n y n y n y n y Ly h h h h h h h h h h h h h h h h h h h h A 2 , 206524416 3 38277037h - 0 2788079616 3 64551877h 68841472 3 158321h 516311040 3 1191359h 1032622080 3 32189281h - 147517440 3 3132947h - 20 1254635827 3 9470009h 0 1445670912 3 297119h 1032622080 3 21251h 688414720 3 411933h 550731776 3 833821h - 40 6691391078 3 764401h , n'y' ny' ny 1 , 00 2 112 00 2 112 0 0 2 112 00 2 112 10 2 112 00 41 00 3 11 00 1 00 1 01 15 00 2 15 00 18 1 , 2 1 '' 6 1 '' 7 1 '' 8 1 '' 2 1 ' 6 1 ' 7 1 ' 8 1 ' 2 1 6 1 7 1 8 1 , 1000000000 2 784 2 896 0100000000 2 784 2 896 0010000000 2 784 2 896 0001000000 2 784 2 896 0000000000 2 784 2 896 0000100000 343 384 0000010000 3 245 3 256 0000001000 63 64 0000000100 49 48 0000000000 49 64 0000000010 2 147 80 0000000001 18 49 9 16 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 56 ( F ) Version I N otes