On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

Abstract

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.

Direct integrator of block type methods with additional derivative for general third order initial value problems

“…The set of formulae in equations 14, (15), (16), (23), (24) and (25) is denoted as 2PSDIBM(3) method.…”

“…The proposed method has order p if C 0 = C 1 = :::C p = C p + 1 = C p + 2 = 0, C p + 3 6 ¼ 0: Therefore, C p + 3 is the error constant and C p + 3 h p + 3 Z p + 3 ð Þ t n ð Þ is the principal local truncation error at the point t n . The two-point block method given by (14), (15), (16), (23), (24) and (25) can be written in a matrix as follows: , , By substituting these matrices into equation 37we have…”

“…Also by substituting equations (14), (15) and (16) into (25), we have y n + 2 = y n + 2hy 0 n + 2h 2 y 00 n + h 3 Write the formulae in (14), (15), (16), (39), (40), and (41) matrix form and the zero-stability polynomial of the method can be obtained from The zero-stability polynomial of the two-point block method is, :…”

“…Noted too that the methods can be used to solve physical problems such as the well-known thin film flow problem, 15 Blasius equation 16 and the nonlinear Genesio equation, 17 and we can treat it as further research.…”

A sixth‐order improved Runge–Kutta direct method for special third‐order ordinary differential equations

“…The problem is integrated in the interval [0,5] with step sizes h = 1/2 i , i = 2, …, 5. The results are shown in Figure 2.Problem We consider the homogeneous linear IVP Its exact solution is y ( x ) = e − x [19]. The problem is integrated in the interval [0,5] with step sizes h = 1/2 i , i = 2, …, 5.…”