Machine Learning for Modeling the Singular Multi-Pantograph Equations

Amirhosein Mosavi; Manouchehr Shokri; Zulkefli Mansor; Sultan Noman Qasem; Shahab S Band; Ardashir Mohammadzadeh

doi:10.3390/e22091041

Machine Learning for Modeling the Singular Multi-Pantograph Equations

Entropy (Basel). 2020 Sep 18;22(9):1041. doi: 10.3390/e22091041.

Authors

Amirhosein Mosavi^{1

2}, Manouchehr Shokri³, Zulkefli Mansor⁴, Sultan Noman Qasem^{5

6}, Shahab S Band^{7

8}, Ardashir Mohammadzadeh⁹

Affiliations

¹ Environmental Quality, Atmospheric Science and Climate Change Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
² Faculty of Environment and Labour Safety, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
³ Faculty of Civil Engineering, Institute of Structural Mechanics (ISM), Bauhaus-Universität Weimar, 99423 Weimar, Germany.
⁴ Fakulti Teknologi dan Sains Maklumat, Universiti Kebangsan Malaysia, Bangi 43600, Selangor, Malaysia.
⁵ Computer Science Department, College of Computer and Information Sciences, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia.
⁶ Computer Science Department, Faculty of Applied Science, Taiz University, Taiz 6803, Yemen.
⁷ Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan.
⁸ Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam.
⁹ Electrical Engineering Department, University of Bonab, Bonab 5551785176, Iran.

Abstract

In this study, a new approach to basis of intelligent systems and machine learning algorithms is introduced for solving singular multi-pantograph differential equations (SMDEs). For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution. The rules of the suggested type-2 fuzzy logic system (T2-FLS) are optimized by the square root cubature Kalman filter (SCKF) such that the proposed fineness function to be minimized. Furthermore, the stability and boundedness of the estimation error is proved by novel approach on basis of Lyapunov theorem. The accuracy and robustness of the suggested algorithm is verified by several statistical examinations. It is shown that the suggested method results in an accurate solution with rapid convergence and a lower computational cost.

Keywords: Lyapunov function; fuzzy systems; singular multi-pantograph differential equations; square root cubature kalman filter; statistical analysis.