The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron
This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mas...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/9/1753 |
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| author | Akanksha Verma Wojciech Sumelka Pramod Kumar Yadav |
| author_facet | Akanksha Verma Wojciech Sumelka Pramod Kumar Yadav |
| author_sort | Akanksha Verma |
| collection | DOAJ |
| description | This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. |
| first_indexed | 2024-03-10T21:54:18Z |
| format | Article |
| id | doaj.art-2af5447bd7964f62946fd8e5ef743a2c |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T21:54:18Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2af5447bd7964f62946fd8e5ef743a2c2023-11-19T13:12:11ZengMDPI AGSymmetry2073-89942023-09-01159175310.3390/sym15091753The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal PerceptronAkanksha Verma0Wojciech Sumelka1Pramod Kumar Yadav2Department of Mathematics, Dyal Singh College, University of Delhi, New Delhi 110003, IndiaInstitute of Structural Analysis, Poznan University of Technology, Piotrowo 5 Street, 60-965 Poznan, PolandDepartment of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj 211004, IndiaThis paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods.https://www.mdpi.com/2073-8994/15/9/1753orthogonal neural networksimulated annealing optimization techniquefractional differential equationsCaputo derivative |
| spellingShingle | Akanksha Verma Wojciech Sumelka Pramod Kumar Yadav The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron Symmetry orthogonal neural network simulated annealing optimization technique fractional differential equations Caputo derivative |
| title | The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron |
| title_full | The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron |
| title_fullStr | The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron |
| title_full_unstemmed | The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron |
| title_short | The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron |
| title_sort | numerical solution of nonlinear fractional lienard and duffing equations using orthogonal perceptron |
| topic | orthogonal neural network simulated annealing optimization technique fractional differential equations Caputo derivative |
| url | https://www.mdpi.com/2073-8994/15/9/1753 |
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