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

For case two of the new half-step third derivative method,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               + + + + − − − − − − − − − + + + + + + + + + + + + = 2 1 3 1 4 1 5 1 24 h 64 27h 3 2h - 192 125h 243 2h - 6 h 81 32h - 243 125h 128 h 1024 135h 24 11h 3072 1625h 9375 74h - 6250 837h 9375 4576h - 75 38h 720 2h 320 2 27h 5 2h- 576 2 125h 10935 2 22h - 1080 2 37h 3645 2 416h - 17496 2 2125h 23040 2 31h - 20480 2 459h 240 2 19h - 36864 2 2875h 140625 2 134h - 125000 2 1971h 15625 2 864h - 9000 2 467h 20160 3 13h 17920 3 243h 420 3 17h - 32256 3 1375h 15309 34h 45360 3 197h 8505 3 128h - 244944 3 3625h 1290240 3 157h - 229376 3 459h 26880 3 187h - 2064384 3 13375h 4921875 3 316h - 8750000 3 9153h 1640625 3 5888h 630000 3 2039h 96 5h - 18 h 1536 85h 18750 1039h 40 2h- 14580 2 233h 30720 2 349h 62500 2 537h 5376 3 31h 102060 3 239h 1720320 3 2069h 6562500 3 4619h '' ' 1 0 0 1 0 0 1 0 0 1 0 0 2 1 0 3 1 0 4 1 0 5 1 0 8 2 2 1 18 2 3 1 32 2 4 1 50 2 5 1 2 1 '' 3 1 '' 4 1 '' 5 1 '' 2 1 ' 3 1 ' 4 1 ' 5 1 ' 2 1 3 1 4 1 5 1 100000000000 010000000000 001000000000 000100000000 000010000000 000001000000 000000100000 000000010000 000000001000 000000000100 000000000010 000000000001 : n f n f n f n f nf ny ny ny h h h h h h h h h h h h n y n y n y n y n y n y n y n y n y n y n y n y hxyL (14) expanding (14) in Taylor series and comparing the coefficient of h gives 07 ... 3 2 1 0 = = = = = = C CCCC and T e e e e e e e e e e e e C               − − − − − − − − − − − − = 07 2338 .7,07 7623 .1,07 718 .1,07 7296 .1,08 7171 .6,08 4681 .4 ,08 0195 .3,08 1577 .2,08 5501 .1,09 897 .5,09 7803 .2,09 4857 .1 8 Hence our method is of order four (4). b) Consistency The hybrid block method (13) and (14) is said to be consistent if it has an order more than or equal to one. Therefore, the new half-step third derivative hybrid block method is consistent. c) Zero Stability of Our Method Definition 2: A block method (10) and (11) is said to be zero-stable if as 0 → h , the root k i z i )1(1 , = of the first characteristic polynomial ( ) 0 = z ρ that is ( ) 0 det 0 )( =      = ∑ = − k j i k i zA z ρ Satisfies 1 ≤ i z and for those roots with i z =1, multiplicity must not exceed two. Hence, the new half-step third derivative hybrid block method is zero-stable. d) Regions of Absolute Stability For case one: The stability polynomial for half step with three off step point gives 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 6 ( F ) © 2021 Global Journals Version I N otes