Authors:
Amandeep Singh,Sarita Pippal,Akash Kumar,DOI NO:
https://doi.org/10.26782/jmcms.spl.12/2025.08.00006Keywords:
Laplace Transform,Sumudu Transform,Fractional Differential Equations,Riemann-Liouville's Fractional Differential Equations,Abstract
To differentiate itself from other integral transformations, this article presents a novel integral transformation known as the A (or Aman) transform. This transformation was inspired by a thorough study of the effectiveness of the Laplace and Sumudu transforms, specifically with regard to fractional differential equations. Applying these transformations might occasionally make processing their inverse transform challenging. This concept encourages us to reconsider and put in more effort to develop fresh, essential transformations that will make difficult problems easier to tackle. The proposed transformation has been successfully used to solve the Riemann-Liouville and Caputo FDE analytically. The outcomes of using this new approach are in perfect harmony with those of using contemporary methods. This demonstrates the A transform's dependability and efficiency in the analytical resolution of complex mathematical situationsRefference:
I. Abdel Rahim Mahgoub, Mohand M. “The New Integral Transform Mahgoub Transform.” Advances in Theoretical and Applied Mathematics, vol. 11, no. 4, 2016, pp. 391–398. https://www.ripublication.com/atam16/atamv11n4_07.pdf
II. Abdel Rahim Mahgoub, Mohand M. “The New Integral Transform Mohand Transform.” Advances in Theoretical and Applied Mathematics, vol. 12, no. 2, 2017, pp. 113–120. https://www.ripublication.com/atam17/atamv12n2_07.pdf
III. Abdel Rahim Mahgoub, Mohand M. “The New Integral Transform Sawi Transform.” Advances in Theoretical and Applied Mathematics, vol. 14, no. 1, 2019, pp. 81–87. https://www.ripublication.com/atam19/atamv14n1_05.pdf
IV. Aboodh, Khalid Suliman. “The New Integral Transform Aboodh Transform.” Global Journal of Pure and Applied Mathematics, vol. 9, no. 1, 2013, pp. 35–43. https://www.ripublication.com/gjpamv7/gjpamv9n1_04.pdf
V. Belgacem, Rachid, Dumitru Baleanu, and Ahmed Bokhari. “Shehu Transform and Applications to Caputo-Fractional Differential Equations.” International Journal of Analysis and Applications, vol. 17, 2019, no. 6, pp. 917–927. 10.28924/2291-8639-17-2019-917VI.
VI. Bracewell, Ron, and Peter B. Kahn. “The Fourier Transform and Its Applications.” American Journal of Physics, vol. 34, no. 8, 1966, pp. 712–712. 10.1119/1.1973431.
VII. Bulut, H., H. M. Baskonus, and F. B. M. Belgacem. “The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method.” Abstract and Applied Analysis, vol. 4, 2013, pp. 1–6. 10.1155/2013/203875
VIII. Debnath, Lokenath, and Dambaru Bhatta. “Integral Transforms and Their Applications. Chapman and Hall/CRC.” Taylor and Francis Group, 2007. 10.1201/b17670
IX. Elzaki, Tarig M. “The New Integral Transform Elzaki Transform.” Global Journal of Pure and Applied Mathematics, vol. 7, no. 1, 2011, pp. 57–64.
X. Elzaki, Tarig M., and Mourad Chamekh. “Solving Nonlinear Fractional Differential Equations Using a New Decomposition Method.” Universal Journal of Applied Mathematics and Computation, vol. 6, 2018, pp. 27–35.
XI. Erdélyi, A., et al. Tables of Integral Transforms. Vol. 1, McGraw-Hill, 1954.
XII. Goswami, Pranay, and Rubayyi T. Alqahtani. “Solutions of Fractional Differential Equations by Sumudu Transform and Variational Iteration Method.” Journal of Nonlinear Sciences and Applications, 2016. 10.22436/jnsa.009.04.48
XIII. Hailat, Ibrahim, Zarita Zainuddin, and Amirah Azmi. “New Approach of Modifying Laplace Transform Variational Iteration Method to Solve Fourth-Order Fractional Integro-Differential Equations.” Authorea, 30 Jan. 2024. 10.22541/au.170665886.63054022/v1
XIV. Liang, Song, Ranchao Wu, and Liping Chen. “Laplace Transform of Fractional Order Differential Equations.” Electronic Journal of Differential Equations, no. 139, 2015, pp. 1–15. http://ejde.math.txstate.edu/Volumes/2015/139/liang.pdf
XV. Miller, Kenneth S., and Bertram Ross. “An Introduction to the Fractional Calculus and Fractional Differential Equations.” John Wiley and Sons, 1993.
XVI. Moazzam, A., and Z. I. Muhammad. “A New Integral Transform ‘Ali and Zafar’ Transformation and Its Application in Nuclear Physics.” Proceedings of the 19th International Conference on Statistical Sciences, vol. 36, 2022, pp. 177–182. https://www.researchgate.net/publication/361409149.
XVII. Podlubny, Igor. Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198, Academic Press, January 15, 1999. URL: https://archive.org/details/fractionaldiffer00podl_878.
XVIII. Schi, J. L. “The Laplace Transform: Theory and Applications.” Springer, 1999.
XIX. Singh, Amandeep, et al. “Solving Nonlinear Coupled Fractional Partial Differential Equations by ZZ Transform and Adomian Polynomials.” Journal of Mechanics of Continua and Mathematical Sciences, Special Issue no. 11, May 2024, pp. 1–17. 10.26782/jmcms.spl.11/2024.05.00001
XX. Singh, Jagdev, Devendra Kumar, and A. Klman. “Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform.” Abstract and Applied Analysis, 2013. 10.1155/2013/934060
XXI. Sontakke, Bhausaheb R., and Rajashri Pandit. “Convergence Analysis and Approximate Solution of Fractional Differential Equations.” Malaya Journal of Matematik, vol. 7, no. 2, 2019, pp. 338–344. 10.26637/MJM0702/0029
XXII. Tuluce Demiray, Seyma, Hasan Bulut, and Fethi Bin Muhammad Belgacem. “Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations.” Mathematical Problems in Engineering, 2015, Article ID 131690. 10.1155/2015/131690
XXIII. Watugala, G. K. “Sumudu Transform—An Integral Transform to Solve Differential Equations and Control Engineering Problems.” International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, 1993, pp. 35–43. 10.1080/0020739930240105
XXIV. Yang, Xiao-Jun. Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher, 2011.
XXV. Zhao, Weidong, and Shehu Maitama. “New Integral Transform: Shehu Transform—A Generalization of Sumudu and Laplace Transform for Solving Differential Equations.” International Journal of Analysis and Applications, vol. 17, no. 2, 2019, pp. 167–190.
XXVI. Zhao, Weidong, and Shehu M. “Homotopy Perturbation Shehu Transform Method for Solving Fractional Models Arising in Applied Science.” Journal of Applied Mathematics and Computational Mechanics, vol. 20, no. 1, 2021, pp. 71–82.
XXVII. Yang, Yi, and Haiyan Henry Zhang. Fractional Calculus with its Applications in Engineering and Technology. Morgan & Claypool (Springer Nature Imprint), 2019. https://link.springer.com/book/10.1007/978-3-031-79625-8