On the Application of A-Transform in the Fractional Calculus
Abstract
This paper investigates the application of the A-Transform within the framework of fractional calculus. The main objective is to establish analytical formulas for the A-Transform of key fractional operators and to demonstrate its capability in solving fractional differential and integrodifferential equations. The study derives simplified expressions for the Mittag–Leffler function, the Caputo derivative, and the mixed Caputo–Volterra equation using the A-Transform. In addition, a numerical scheme based on integral reformulation and Gauss–Jacobi quadrature is developed to handle non-homogeneous fractional models involving multiple Caputo derivatives and non-polynomial forcing terms. The results show that the proposed analytical and numerical approaches provide accurate, stable, and efficient solutions, confirming the A-Transform as a useful tool for the analysis of fractional-order systems.
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References
R. M. Pandey, A. Chandola, L. Rathour, and V. N. Mishra, “Fractional calculus operators applied on m-parameter mittag-leffler function and bessel-maitland function with applications,” in AIP Conference Proceedings, vol. 3005, p. 020020, AIP Publishing LLC, 2024. https://doi.org/10.1063/5.0210377.
K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity,” in Scientific computing in chemical engineering II: computational fluid dynamics, reaction engineering, and molecular properties, pp. 217– 224, Springer, 1999. https://link.springer.com/chapter/10.1007/978-3-642-60185-9_24.
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, vol. 204. elsevier, 2006. https://sutlib2.sut.ac.th/sut_contents/H103746.pdf.
W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007. https://doi.org/10.1016/j.jmaa.2006.10.040.
D. Das, B. C. Ghosh, L. Rathour, S. K. Sahani, L. N. Mishra, and V. N. Mishra, “Dislocated quasi modular metric spaces and its applications in fixed point theory via a-class and c-class function,” in AIP Conference Proceedings, vol. 3005, p. 020033, AIP Publishing LLC, 2024. https://doi.org/10.1063/5.0210617.
M. Sharma, S. Chaudhary, L. Rathour, and V. N. Mishra, “A green fractional transportation system under dual hesitant fermatean fuzzy configuration with safety factor,” in Optimization and Computing using Intelligent Data-Driven Approaches for Decision-Making, pp. 270–282, CRC Press. https://www.taylorfrancis.com/chapters/edit/10.1201/9781003536796-12.
L. Rathour, D. Obradovic, L. N. Mishra, K. Khatri, and V. N. Mishra, “Application of the function in basic mathematics,” International Journal of Advanced Science and Engineering, vol. 9, no. 4, pp. 3144–3150, 2023. https://doi.org/10.29294/IJASE.9.4.2023.3144-3150.
G. Farid, S. Mehmood, L. Rathour, M. Elamin, H. U. M. Ahamd, and N. Yahia, “Gr¨uss type k-fractional integral operator inequalities and allied results,” International Journal of Analysis and Applications, vol. 21, pp. 133–133, 2023. https://doi.org/10.28924/2291-8639-21-2023-133.
G. Farid, S. B. Akbar, L. Rathour, and L. N. Mishra, “Riemann-liouville fractional versions of hadamard inequality for strongly (α, m)-convex functions,” Korean Journal of Mathematics, vol. 29, no. 4, pp. 687–704, 2021. https://doi.org/10.11568/kjm.2021.29.4.687.
S. A. Murad and S. M. Rasheed, “Application of a domian decomposition method for solving fractional differential equation,” Journal of Education and Science, vol. 22, no. 1, pp. 93–0, 2009. https://ojs.uomosul.edu.iq/index.php/edusj/article/view/42082.
B. Alqahtani, H. Aydi, E. Karapınar, and V. Rakoˇcevi´c, “A solution for volterra fractional integral equations by hybrid contractions,” Mathematics, vol. 7, no. 8, p. 694, 2019. https://doi.org/10.3390/math7080694.
U. Lepik, “Solving fractional integral equations by the haar wavelet method,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 468–478, 2009. https://doi.org/10.1016/j.amc.2009.04.015.
R. Garrappa, “Numerical solution of fractional differential equations: A survey and a software tutorial,” Mathematics, vol. 6, no. 2, p. 16, 2018. https://doi.org/10.3390/math6020016.
A. Jannelli, “Numerical solutions of fractional differential equations arising in engineering sciences,” Mathematics, vol. 8, no. 2, p. 215, 2020. https://doi.org/10.3390/math8020215.
R. T. Matoog, A. M. Mahdy, M. A. Abdou, and D. S. Mohamed, “A computational method for solving nonlinear fractional integral equations,” Fractal and Fractional, vol. 8, no. 11, p. 663, 2024. https://doi.org/10.3390/fractalfract8110663.
H. Jafari, “A new general integral transform for solving integral equations,” Journal of Advanced Research, vol. 32, pp. 133–138, 2021. https://doi.org/10.1016/j.jare.2020.08.016.
Y. Luchko and V. Kiryakova, “The mellin integral transform in fractional calculus,” Fractional Calculus and Applied Analysis, vol. 16, no. 2, pp. 405–430, 2013. https://doi.org/10.2478/s13540-013-0025-8.
S. Sattaso, K. Nonlaopon, H. Kim, and S. Al-Omari, “Certain solutions of abel’s integral equations on distribution spaces via distributional g α-transform,” Symmetry, vol. 15, no. 1, p. 53, 2022. https://doi.org/10.3390/sym15010053.
H. Liu and G. Cao, “Effectiveness of the young-laplace equation at nanoscale,” Scientific reports, vol. 6, no. 1, p. 23936, 2016. https://www.nature.com/articles/srep23936.
A. S. Hasan, “The a-variational iteration transform,” Journal of Applied Computer Science and Mathematics, vol. 19, no. 38, pp. 22–27, 2025. https://doi.org/10.4316/JACSM.202501004.
A. Munir, A. Qayyum, L. Rathour, G. Atta, S. S. Supadi, and U. Ali, “A study on milne-type inequalities for a specific fractional integral operator with applications,” Korean Journal of Mathematics, vol. 32, no. 2, pp. 297–314, 2024. https://doi.org/10.11568/kjm.2024.32.2.297.
F. Mainardi, “Why the mittag-leffler function can be considered the queen function of the fractional calculus?,” Entropy, vol. 22, no. 12, 2020. https://www.mdpi.com/1099-4300/22/12/1359.
G. Farid, S. B. Akbar, L. Rathour, L. N. Mishra, and V. N. Mishra, “Riemann-liouville fractional versions of hadamard inequality for strongly m-convex functions,” International Journal of Analysis and Applications, vol. 20, pp. 5–5, 2022. https://doi.org/10.28924/2291-8639-20-2022-5.
S. S. Redhwan, S. L. Shaikh, and M. S. Abdo, “Some properties of sadik transform and its applications of fractional-order dynamical systems in control theory,” Advances in the Theory of Nonlinear Analysis and its Application, vol. 4, p. 51–66, 2020. https://doi.org/10.31197/atnaa.647503.
J. Sousa, T. Abdeljawad, and D. Oliveira, “Mild and classical solutions for fractional evolution differential equation,” arXiv preprint arXiv:1908.04948, 2019. https://doi.org/10.48550/arXiv.1908.04948.
S. Jahanshahi, E. Babolian, D. F. Torres, and A. Vahidi, “A fractional gauss–jacobi quadrature rule for approximating fractional integrals and derivatives,” Chaos, Solitons & Fractals, vol. 102, pp. 295–304, 2017. https://doi.org/10.1016/j.chaos.2017.04.034.
D. S. Dimitrov, “Gauss–jacobi quadrature method for approximating fractional integrals and derivatives,” Applied Mathematics and Computation, vol. 274, no. 1, p. 395–405, 2016. https://doi.org/10.1016/j.amc.2015.12.034.
A. Turan Dincel, S. N. Tural Polat, and P. Sahin, “Hermite wavelet method for nonlinear fractional differential equations,” Fractal and Fractional, vol. 7, no. 5, p. 346, 2023. https://doi.org/10.3390/fractalfract7050346.
H. M. Arafa, M. A. Ramadan, and N. Althobaiti, “A numerical technique based on bernoulli wavelet operational matrices for solving a class of fractional order differential equations,” Fractal and Fractional, vol. 7, no. 8, p. 604, 2023. https://doi.org/10.3390/fractalfract7080604.
N. A. Khan, M. Ali, A. Ara, I. Alraddadi, and H. Ahmad, “A haar wavelet operational matrix method for fractional derivatives with non-singular kernel,” in Operations Research Forum, vol. 6, p. 118, Springer, 2025. https://link.springer.com/article/10.1007/s43069-025-00502-4.
E. A. Al-Taai, “A new operational matrix approach based on legendre wavelets for fractional differential equations,” International Journal of Applied Mathematics, vol. 38, no. 8s, pp. 1167–1184, 2025. https://doi.org/10.12732/ijam.v38i8s.633.
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