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

Abstract

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

Approximate Analytical Solution of a Third-Order IVP arising in Thin Film Flows driven by Surface Tension

“…RK4: the four-stage fourth order Runge-Kutta method as in [4], 3. RKD5: the three-stage fifth order direct Runge-Kutta (RKD) method derived in [18].…”

“…In [18], the authors introduce a new method called RKD5 that requires less function evaluations than the RK4 and DOPRI methods, essentially because the reduced system 5.1 is three times the dimension. From [7,8,20] the exact solution of (1.1) is given in parametric form as…”

“…Approximate analytical solution is derived and compared to the results obtained from the Adomian decomposition method (ADM) proposed in [20], to the exact analytical solution [7,8], to a fifth order Runge-Kutta method (DOPRI), a fourth order Runge-Kutta method (RK4), a three-stage fifth order Runge-Kutta method (RKD5) developed in [18]. A very good agreement and accuracy is observed.…”

Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving y′′′x=fGhawadri1, Senu2, Ismail3 et al. 2018Journal of Applied MathematicsSelf Cite

“…However, we can use these reductions to determine an efficient way to solve (1) numerically. Here, we are focusing on the cases = 2 and = 3 (see Mechee et al [22]). …”

Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4) = f(x, u, u′, u″)

“…Jain et al [21] developed finite difference approach to solve ODEs of order four, all the methods discussed above are multi-step in nature. On the other hand, Mechee et al [22,23], constructed a RK-based method for solving special third-order ODEs directly. Senu et al [24] developed embedded explicit RKT method to directly solve special ODEs of order three.…”