A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem

Table 4

Table comparing values of the numerical solution using a fifth-order Runge-Kutta method (DOPRI), fourth-order Runge-Kutta method (RK4), and our new RKD5 method at 𝑥∈[0,0.2,0.4,0.6,0.8,1.0] taking ℎ=0.1 and 𝑘=3 with the initial conditions 𝑦(0)=𝑦^′(0)=𝑦^′′(0)=1.

𝑥	RK4	DOPRI	RKD5
0.0	1.0000000000	1.0000000000	1.0000000000
0.2	1.2211559590	1.2211551421	1.2211550887
0.4	1.4881067401	1.4881052848	1.4881049238
0.6	1.8042645823	1.8042625503	1.8042615558
0.8	2.1715254210	2.1715228023	2.1715208324
1.0	2.5909615178	2.5909582657	2.5909549758