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

Table 1

Table comparing values of the numerical solution, 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=0.01 and 𝑘=2 with the initial conditions 𝑦(0)=𝑦^′(0)=𝑦^′′(0)=1.

𝑥	Exact solution	DOPRI	RK4	RKD5
0.0	1.000000000	1.0000000000	1.0000000000	1.0000000000
0.2	1.221211030	1.2212100045	1.2212100046	1.2212100045
0.4	1.488834893	1.4888347799	1.4888347800	1.4888347799
0.6	1.807361404	1.8073613977	1.8073613978	1.8073613977
0.8	2.179819234	2.1798192339	2.1798192341	2.1798192339
1.0	2.608275822	2.6082748676	2.6082748678	2.6082748676