An Efficient Zero-Stable Numerical Method for Fourth-Order Differential Equations

Table 2

Comparison of errors in Kayode [3] and new method (2.6) for Problem 2.

𝑌	𝑎^𝑗𝑠-exact	𝑎𝑎01=𝑦𝑛−𝑎1𝑥𝑛−𝑎2𝑥^2𝑛−𝑎3𝑥^3𝑛−𝑎4𝑥^4𝑛−𝑎5𝑥^5𝑛−𝑎6𝑥^6𝑛−𝑎7𝑥^7𝑛,=1ℎ𝑦^𝑛+1−𝑎4−𝑦𝑛−𝑎^22𝑥^𝑛+ℎ−𝑎33𝑥^2𝑛+3ℎ𝑥𝑛+ℎ^2−𝑎64𝑥^3𝑛+6ℎ𝑥^2𝑛+4ℎ^2𝑥𝑛+ℎ^3−𝑎^55𝑥^4𝑛+10ℎ𝑥^3𝑛+10ℎ^2𝑥𝑛+5ℎ^3𝑥𝑛+ℎ^4−𝑎76𝑥^5𝑛+15ℎ𝑥^4𝑛+20ℎ^2𝑥^3𝑛+15ℎ^3𝑥^2𝑛+6ℎ^4𝑥𝑛+ℎ^5𝑎27𝑥^6𝑛+21ℎ𝑥^5𝑛+35ℎ^2𝑥^4𝑛+35ℎ^3𝑥^3𝑛+21ℎ^4𝑥^2𝑛7ℎ^5𝑥𝑛+ℎ^6,=12ℎ^2𝑦^𝑛+2−𝑎4−2𝑦𝑛+1+𝑦𝑛−𝑎^33𝑥^𝑛+3ℎ−𝑎66𝑥^2𝑛+12ℎ𝑥𝑛+7ℎ^2−𝑎^510𝑥^3𝑛+30ℎ𝑥^2𝑛+35ℎ^2𝑥𝑛+15ℎ^3−𝑎715𝑥^4𝑛+60ℎ𝑥^3𝑛+105ℎ^2𝑥^2𝑛+90ℎ^3𝑥𝑛+31ℎ^4𝑎321𝑥^5𝑛+105ℎ𝑥^4𝑛+245ℎ^2𝑥^3𝑛+315ℎ^3𝑥^2𝑛+217ℎ^4𝑥𝑛+63ℎ^5,=16ℎ^33𝑦^𝑛+3−𝑎5−3𝑦𝑛+23𝑦𝑛+1−𝑦𝑛−𝑎^44𝑥^𝑛+6ℎ−𝑎710𝑥^2𝑛+30ℎ𝑥𝑛+25ℎ^2−𝑎^620𝑥^3𝑛+90ℎ𝑥^2𝑛+150ℎ^2𝑥𝑛+90ℎ^3𝑎435𝑥^4𝑛+210ℎ𝑥^3𝑛+525ℎ^2𝑥^2𝑛+630ℎ^3𝑥𝑛+301ℎ^4,=𝑓^𝑛+124−5𝑎^5−35𝑎7(𝑥𝑛+ℎ)−15𝑎6𝑥^2𝑛+2ℎ𝑥𝑛+ℎ^2𝑎5𝑥^3𝑛+3ℎ𝑥^2𝑛+3ℎ^2𝑥𝑛+ℎ^3,=1240ℎ𝑓^𝑛+3−𝑓𝑛+1−6𝑎^6𝑥^𝑛+2ℎ−21𝑎7𝑥^2𝑛+4ℎ𝑥𝑛+91𝑎621ℎ^2,=12880ℎ𝑓^𝑛+5𝑎−2𝑓𝑛+3+𝑓𝑛+1−7𝑎^6𝑥^𝑛+3ℎ7,=120160ℎ12𝑦^𝑛+4−4𝑦𝑛+3+6𝑦𝑛+2−4𝑦𝑛+1+𝑦𝑛+ℎ^4𝑓^𝑛+5−8𝑓𝑛+3−5𝑓𝑛+1.(A.1)-computed with new method (2.6)	Errors in Kayode [3]	Errors in new method (2.6)
0.103125	0.11192647D+01	0.11192647D+01	4.6842974D-11	8.1714635D-12
0.206250	0.12715995D+01	0.12715995D+01	2.0687163D-10	1.4680923D-10
0.306250	0.1452111D+01	0.14521109D+01	9.0421981D-10	7.8813045D-10
0.406250	0.16662167D+01	0.16662169D+01	2.9137912D-09	2.6795135D-09
0.506250	0.19153471D+01	0.19153471D+01	7.5114031D-09	7.0615016D-09
0.603125	0.21915816D+01	0.21915816D+01	1.6231339D-08	1.5438411D-08
0.703125	0.25144403D+01	0.25144403D+01	3.2237145D-08	3.0900764D-08
0.803125	0.28775164D+01	0.28775164D+01	5.8891835D-08	5.6759556D-08
0.903125	0.32829362D+01	0.32829362D+01	1.0079927D-07	9.7549608D-08
1.003125	0.37330495D+01	0.37330495D+01	1.6373663D-07	1.5897083D-07