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.
Highlights
A special third-order differential equation (ODE) of the form y (x) = f (x, y (x)), y (x0) = α, (1)y (x0) = β, y (x0) = γ, x ≥ x0 which is not explicitly dependent on the first derivative y(x) and the second derivative y(x) of the solution is frequently found in many physical problems such as electromagnetic waves, thin film flow, and gravity driven flows
Y (x0) = β, y (x0) = γ, x ≥ x0 which is not explicitly dependent on the first derivative y(x) and the second derivative y(x) of the solution is frequently found in many physical problems such as electromagnetic waves, thin film flow, and gravity driven flows
Awoyemi and Idowu [5] and Jator [6] proposed a class of hybrid collocation methods for the direct solution of higher-order ordinary differential equations (ODEs)
Summary
The solution to (1) can be obtained by reducing it to an equivalent first-order system which is three times the dimension and can be solved using a standard Runge-Kutta method or a multistep method. Awoyemi and Idowu [5] and Jator [6] proposed a class of hybrid collocation methods for the direct solution of higher-order ordinary differential equations (ODEs). Samat and Ismail [7] developed an embedded hybrid method for solving special second-order ODEs. Waeleh et al [2] developed a block multistep method which can directly solve general thirdorder equations; on the other hand, Ibrahim et al [8] developed a multistep method that can directly solve stiff third-order differential equations. Senu et al [9] derived the Runge-Kutta-Nystrom method for solving second-order ODEs directly. The method produces yn+1, yn+1, and yn+1 to approximate y(xn+1), y(xn+1), and y(xn+1), where yn+1 is the computed solution and y(xn+1) is the exact solution
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