Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

Abstract

Differential equations are commonly used to model several engineering, science, and biological applications. Unfortunately, finding analytical solutions for solving higher-order Ordinary Differential Equations (ODEs) is a challenge. Numerical methods represent a leading candidate for solving such ODEs. This work presents an innovated adaptive technique that uses polynomials to solve linear or nonlinear third-order ODEs. The proposed technique adapts the coefficients of the polynomial to obtain an explicit analytical solution. A signed least mean square algorithm is exploited to enhance the adaptation process and to decrease both computational requirements and time. The efficiency of the proposed Adaptive Polynomial Method (APM) is illustrated through six well-known examples. The proposed technique is compared with recent analytical and numerical methods to validate its effectiveness in terms of Mean Square Error (MSE) and computation time. An application in a thin film flow system is modeled to a third-order ODE. The proposed technique is compared with recent numerical and analytical methods in solving the thin film flow equation, and the proposed technique has achieved better results. Furthermore, the proposed technique provides an analytical solution with an increased dynamic range and much lower computational time than those of the conventional numerical methods.

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

“…According to Duffy and Wilson (1997), thin film flow can describe the dynamic balance between surface tension and viscous force in the thin film layer without gravity. Recently, various direct methods have been developed to solve particular problems (Ghawadri et al, 2018;Lee et al, 2020;Jikantoro et al, 2018;Haweel et al, 2018). The thin film flow problem can be represented by Equation 46: (46) where u(t) implies the cartesian coordinate system in flowing fluid, and we express f(u(t)) in various terms:…”