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

Mohammad T Haweel,Fathi E Abd El-Samie,O Zahran

doi:10.1109/access.2021.3072944

Mohammad T Haweel, Fathi E Abd El-Samie + Show 1 more

Open Access

https://doi.org/10.1109/access.2021.3072944

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Journal: IEEE Access	Publication Date: Jan 1, 2021
Citations: 23	License type: CC BY 4.0

Affiliation: Menoufia University, Shaqra University

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 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 it achieves 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.

Highlights

Several engineering, chemical, science, and biological applications can be modeled, analyzed and/or solved using ordinary differential equations (ODEs)
Fractionalorder differential equations have been proposed for modeling of modern and classical dynamical systems [6]
Fourth-order composition methods for the numerical integration of Initial Value Problems (IVPs) defined by ODEs for dynamical systems were proposed [14]

Summary

INTRODUCTION

Chemical, science, and biological applications can be modeled, analyzed and/or solved using ordinary differential equations (ODEs). Semi-explicit and semi-implicit multi-step integration methods have been addressed These methods provided good convergence properties with less computational cost, they suffer from a decrease of the numerical stability with the increase of the accuracy order of the applied scheme. Fourth-order composition methods for the numerical integration of IVPs defined by ODEs for dynamical systems were proposed [14]. An adaptive analytical solution for solving IVPs of the third-order ODEs is proposed. Such solution has the advantages of minimizing the drawbacks that occur with the majority of both analytical and numerical solutions, and providing closed-form solutions that are closer to the exact solution in a shorter time.

PROPOSED ADAPTIVE POLYNOMIAL METHOD

APPLICATION OF APM TO THE THIN FILM FLOW PROBLEM

CONCLUSION