Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow
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 OD...
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| Format: | Article |
| Language: | English |
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IEEE
2021-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/9402728/ |
| _version_ | 1818642687574671360 |
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| author | Mohammad T. Haweel O. Zahran Fathi E. Abd El-Samie |
| author_facet | Mohammad T. Haweel O. Zahran Fathi E. Abd El-Samie |
| author_sort | Mohammad T. Haweel |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-16T23:47:01Z |
| format | Article |
| id | doaj.art-80c69cc2b55c47c89959780c4ce256d0 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-12-16T23:47:01Z |
| publishDate | 2021-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-80c69cc2b55c47c89959780c4ce256d02022-12-21T22:11:28ZengIEEEIEEE Access2169-35362021-01-019678746788910.1109/ACCESS.2021.30729449402728Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film FlowMohammad T. Haweel0https://orcid.org/0000-0002-6739-8741O. Zahran1Fathi E. Abd El-Samie2https://orcid.org/0000-0001-8749-9518Department of Electrical Engineering, Shaqra University, Riyadh, Saudi ArabiaDepartment of Electronics and Electrical Communications Engineering, Faculty of Electronic Engineering, Menoufia University, Menouf, EgyptDepartment of Electronics and Electrical Communications Engineering, Faculty of Electronic Engineering, Menoufia University, Menouf, EgyptDifferential 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.https://ieeexplore.ieee.org/document/9402728/Adaptive algorithmsanalytical solutionnumerical solutionordinary differential equationspolynomials |
| spellingShingle | Mohammad T. Haweel O. Zahran Fathi E. Abd El-Samie Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow IEEE Access Adaptive algorithms analytical solution numerical solution ordinary differential equations polynomials |
| title | Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow |
| title_full | Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow |
| title_fullStr | Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow |
| title_full_unstemmed | Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow |
| title_short | Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow |
| title_sort | adaptive polynomial method for solving third order ode with application in thin film flow |
| topic | Adaptive algorithms analytical solution numerical solution ordinary differential equations polynomials |
| url | https://ieeexplore.ieee.org/document/9402728/ |
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