A New Approach for Solving Fractional Differential Equations Incorporating Ramadan Group Transform and Machine Learning

Authors

  • Prabakaran Raghavendran Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology image/svg+xml
  • Tharmalingam Gunasekar Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology image/svg+xml
  • Saikat Gochhait Symbiosis International University image/svg+xml

DOI:

https://doi.org/10.4108/eetiot.7134

Keywords:

fractional-order differential equation, Mittag-Leffler function, gamma function, Ramadan Group transform of the fractional derivative

Abstract

This paper examines various types of fractional differential equations using fractional calculus methods. It extends the classical Frobenius method and introduces key theorems that apply the Ramadan Group transform and other techniques. Additionally, the research incorporates machine learning, specifically neural networks, to solve these equations. The paper demonstrates that machine learning can enhance the solution process through data generation, model design, and optimization. Examples provided illustrate how combining traditional methods with machine learning can effectively solve fractional differential equations.

Downloads

<br data-mce-bogus="1"> <br data-mce-bogus="1">

References

[1] Ahmadi, S. A. P., Hosseinzadeh, H., & Cherati, A. Y. (2019). A New Integral Transform for Solving Higher Order Linear Ordinary Differential Equations. Nonlinear Dynamics and Systems Theory, 19(2), 243-252. DOI: https://doi.org/10.1007/s40819-019-0712-1

[2] Belgacem, F. B. M. (2006). Introducing and analyzing deeper Sumudu properties. Nonlinear Studies, 13(1), 23-41. DOI: https://doi.org/10.1155/JAMSA/2006/91083

[3] Elzaki, T. M. (2011). The New Integral Transform Elzaki Transform. Global Journal of Pure and Applied Mathematics, 7(1), 57-64.

[4] Fadhil, R. A. (2017). Convolution for Kamal and Mahgoub transforms. Bulletin of Mathematics and Statistics Research, 5(4), 11-16.

[5] Gunasekar, Th., Raghavendran, P., Santra, Sh. S., Sajid, M. (2024). Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations. Journal of Mathematics and Computer Science, 34(4), 361-380. DOI: https://doi.org/10.22436/jmcs.034.04.04

[6] Khan, Z. H., & Khan, W. A. (2008). N-Transform Properties and Applications. NUST Journal of Engineering Sciences, 1(1), 127-133.

[7] Kim, H. (2017). On the form and properties of an integral transform with strength in integral transforms. Far East Journal of Mathematical Sciences, 102(11), 2831-2844. DOI: https://doi.org/10.17654/MS102112831

[8] Kim, H. (2017). The intrinsic structure and properties of Laplace-typed integral transforms. Mathematical Problems in Engineering, 2017, Article ID 1762729. DOI: https://doi.org/10.1155/2017/1762729

[9] Medina, G. D., Ojeda, N. R., Pereira, Jh., & Romero, L. G. (2017). Fractional Laplace Transform and Fractional Calculus. International Mathematics, 12(20), 991-1000. DOI: https://doi.org/10.12988/imf.2017.71194

[10] Mahgoub, M. A., & Mohand, M. (2019). The new integral transform "Sawi Transform". Advances in Theoretical and Applied Mathematics, 14(1), 81-87.

[11] Podlubny, I. (1999). Fractional Order Systems and PID Controllers. IEEE Transactions on Automatic Control, 44, 208-214. DOI: https://doi.org/10.1109/9.739144

[12] Raghavendran, P., Gunasekar, T. and Gochhait, S. (2024). A Study on Advanced Techniques for Fractional Differential Equations: Integrating the Frobenius Method, ZZ Transform, and Diverse Machine Learning Approaches. In 2024 5th International Conference on Data Analytics for Business and Industry (ICDABI), 30-34. IEEE. DOI: https://doi.org/10.1109/ICDABI63787.2024.10800264

[13] Ramadan, M. A., Raslan, K. R., El-Danaf, T. S., & Hadhoud, A. R. (2016). On a new general integral transform: some properties and remarks. Journal of Mathematical and Computational Science, 6(1), 103-109.

[14] Schiff, J. L. (2013). The Laplace Transform: Theory and Applications. Springer Science and Business Media.

[15] Spiegel, M. R. (1965). Laplace Transform (C. Cerit & S. Eraslan, Trans.). Bilimsel Kitaplar Yaymnevi.

[16] Watugala, G. K. (1993). Sumudu transform: a new integral transform to solve differential equations and control engineering problems. International Journal of Mathematical Education in Science and Technology, 24(1), 35-43. DOI: https://doi.org/10.1080/0020739930240105

[17] Radhakrishnan, G., Pattamatta, A. and Srinivasan, B. (2024). Distributed Physics-Informed machine learning strategies for two-phase flows. International Journal of Multiphase Flow, 177, 104861. DOI: https://doi.org/10.1016/j.ijmultiphaseflow.2024.104861

[18] Zununjan, Z., Turghan, M.A., Sattar, M., Kasim, N., Emin, B. and Abliz, A. (2024). Combining the fractional order derivative and machine learning for leaf water content estimation of spring wheat using hyper-spectral indices. Plant Methods, 20(1), 1-24. DOI: https://doi.org/10.1186/s13007-024-01224-0

[19] Larijani, A. and Dehghani, F. (2023). An efficient optimization approach for designing machine models based on combined algorithm. FinTech, 3(1), 40-54. DOI: https://doi.org/10.3390/fintech3010003

[20] Abdollahi, Z., Mohseni Moghadam, M., Saeedi, H. and Ebadi, M.J. (2022). A computational approach for solving fractional Volterra integral equations based on two-dimensional Haar wavelet method. International journal of computer mathematics, 99(7), 1488-1504. DOI: https://doi.org/10.1080/00207160.2021.1983549

[21] Alam, M., Haq, S., Ali, I., Ebadi, M.J. and Salahshour, S. (2023). Radial basis functions approximation method for time-fractional fitzhugh–nagumo equation. Fractal and Fractional, 7(12), 882. DOI: https://doi.org/10.3390/fractalfract7120882

[22] Avazzadeh, Z., Hassani, H., Ebadi, M.J. and Eshkaftaki, A.B. (2024). A new approach of generalized shifted Vieta-Fibonacci polynomials to solve nonlinear variable order time fractional Burgers-Huxley equations. Physica Scripta, 99(12), 125258. DOI: https://doi.org/10.1088/1402-4896/ad8fde

Downloads

Published

04-03-2025

How to Cite

[1]
P. Raghavendran, T. Gunasekar, and S. Gochhait, “A New Approach for Solving Fractional Differential Equations Incorporating Ramadan Group Transform and Machine Learning”, EAI Endorsed Trans IoT, vol. 11, Mar. 2025.