‎A comprehensive study of the general double Mellin transform: Definitions and applications

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Daykondi University, Nili, Afghanistan

2 Department of Mathematics, Yazd University, Yazd, Iran

Abstract

    This paper introduces a novel double integral transform and highlights its theoretical and practical significance in solving complex mathematical problems. The fundamental definitions, key properties, and its relationships with existing integral transforms are systematically developed. The novelty of the proposed transform lies in its generalized structure, which extends the classical Mellin framework and provides a unified approach for handling a broader class of functions. Several applications are presented, including the derivation of Mellin transforms for elementary functions and the solution of partial and fractional differential equations. In addition, a comparative analysis with existing generalized Mellin transforms is carried out to demonstrate the advantages and efficiency of the proposed method. Finally, a comprehensive table of transforms for functions frequently arising in mathematics, physics, and engineering is provided to facilitate practical implementation.

Keywords

Main Subjects

References

[1] Akrami, M. H., Poya, A., & Zirak, M. A. (2024). New general single, double and triple conformable integral transforms: De nitions, properties and applications. Partial Differential Equations in Applied Mathematics, 12, 100991. https://doi.org/10.1016/j.padiff.2024.100991
[2] Andrews, G. E., Askey, R., Roy, R., Roy, R., & Askey, R. (1999). Special functions (Vol.71, pp. xvi+-664). Cambridge: Cambridge university press. https://doi.org/10.1016/j.padiff.2024.100991
[3] Aziz, T. (2022). Generalized Mellin transform and its applications in fractional calculus. Computational & Applied Mathematics, 41(3), 1. https://doi.org/10.1007/s40314-022-01802-9
[4] Belmor, S., Ravichandran, C., & Jarad, F. (2020). Nonlinear generalized fractional differential equations with generalized fractional integral conditions. Journal of Taibah University for Science, 14(1), 114-123. https://doi.org/10.1080/16583655.2019.1709265
[5] Butzer, P. L., & Jansche, S. (1997). A direct approach to the Mellin transform. Journal of Fourier analysis and applications, 3(4), 325-376. https://doi.org/10.1007/BF02649101
[6] Debnath, L., & Bhatta, D. (2016). Integral transforms and their applications. Chapman and Hall/CRC. https://doi.org/10.1201/9781420010916
[7] Elzaki, T. M. (2011). The new integral transform Elzaki transform. Global Journal of pure and applied mathematics, 7(1), 57-64.
[8] Erden, S., & Uyanik, N. (2025). Fractional inequalities involving double integrals of Riemann-Liouville for higher-order partial di erential functions. Thermal Science, 29(4 Part B), 3013-3022. https://doi.org/10.2298/tsci2504013e
[9] Erdogan, E., Kocabas, S., & Dernek, N. (2019). Some results on the generalized Mellin transforms and applications. Konuralp Journal of Mathematics, 7(1), 175-181. https://izlik.org/JA73AR45LW
[10] Fitouhi, A., Bettaibi, N., & Brahim, K. (2006). The Mellin transform in quantum calculus. Constructive Approximation, 23(3), 305-323. https://doi.org/10.1007/s00365-005-0597-6
[11] Goufo, E. F. D., Ravichandran, C., & Birajdar, G. A. (2021). Self-similarity techniques for chaotic attractors with many scrolls using step series switching. Mathematical Modelling and Analysis, 26(4), 591-611. https://doi.org/10.3846/mma.2021.13126
[12] Jafari, H. (2021). A new general integral transform for solving integral equations. Journal of Advanced Research, 32, 133-138. https://doi.org/10.1016/j.jare.2020.08.016
[13] Jia, H., Nie, Y., & Zhao, Y. (2025). General Conformable Fractional Double Laplace-Sumudu Transform and its Application. J. Appl. Anal. Comput, 15, 9-20. https://doi.org/10.11948/20220344
[14] Jumarie, G. (2008). Fourier's transform of fractional order via Mittag-leer function and modi ed Riemann-Liouville derivative. Journal of applied mathematics and informatics, 26(5-6), 1101-1121. https://doi.org/10.1016/j.aml.2009.05.011
[15] Khan Zafar, H., & Khan Waqar, A. (2008). N-Transform-Properties and Applications, NUST J. Engg. Sci, 1(1), 127-133. https://doi.org/10.24949/njes.v1i1.69
[16] Kilbas, A. A., Luchko, Y. F., Martinez, H., & Trujillo, J. J. (2010). Fractional Fourier transform in the framework of fractional calculus operators. Integral Transforms and Special Functions, 21(10), 779-795. https://doi.org/10.1080/10652461003676099
[17] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional di erential equations (Vol. 204). elsevier. https://doi.org/10.1016/S0304-0208(06)80001-0
[18] Kumar, V. (2020). On a generalized fractional Fourier transform. Palestine Journal of Mathematics, 9(2), 903-907. https://doi.org/10.2478/mjpaa-2021-0030
[19] Levit, B. Y. (2025). Minimax linear estimation on the half-line via Mellin transform. Theory of Probability & Its Applications, 70(2), 185-199. https://doi.org/10.1137/S0040585X97T992331
[20] Mahor, T. C., Mishra, R., & Jain, R. (2020). Fractionalization of Fourier sine and Fourier cosine transforms and their applications. International journal of scienti c and technology research 9, 4.
[21] Romero, L., Cerutti, R., & Luque, L. (2011). A new Fractional Fourier Transform and convolutions products. International Journal of Pure and Applied Mathematics, 66(4), 397-408.
[22] Salehi, Y., & Schiesser, W. E. (2022). Numerical integration of space fractional partial di erential equations: vol 2-applications from classical integer PDEs. Springer Nature. https://doi.org/10.1007/978-3-031-08961-5
[23] Upadhyaya, L. M. (2019). Introducing the Upadhyaya Integral Transform Bulletin of Pure & Applied Sciences-Mathematics, 38E(1) 471{510. https://doi.org/10.5958/2320-3226.2019.00051.1
[24] Vasileva, T., & Vasileva, O. (2009). Application Mellin transforms to the Black-Scholes equations. Mathematical Physics and Computer Modeling, (12), 55-63.
[25] Veeresha, P., Baskonus, H. M., & Gao, W. (2021). Strong interacting internal waves in rotating ocean: novel fractional approach. Axioms, 10(2), 123. https://doi.org/10.3390/axioms10020123
[26] Veeresha, P., Ilhan, E., & Baskonus, H. M. (2021). Fractional approach for analysis of the model describing wind-in
uenced projectile motion. Physica Scripta, 96(7), 075209. https://doi.org/10.1088/1402-4896/abf868
[27] Watugala, G. K. (1993). Sumudu transform: a new integral transform to solve di erential equations and control engineering problems. Integrated Education, 24(1), 35-43. https://doi.org/10.1080/0020739930240105
[28] Yang, J., Sarkar, T. K., & Antonik, P. (2007). Applying the Fourier{modi ed Mellin transform (FMMT) to Doppler-distorted waveforms. Digital Signal Processing, 17(6), 1030-1039. 17(6), 1030-1039. https://doi.org/10.1016/.dsp.2007.01.003
[29] Yao, S. W., Ilhan, E., Veeresha, P., & Baskonus, H. M. (2021). A powerful iterative approach for quintic complex Ginzburg{Landau equation within the frame of fractional operator. Fractals, 29(05), 2140023. https://doi.org/10.1142/S0218348X21400235
[30] Zolotarev, V. M. (1957). Mellin-Stieltjes transforms in probability theory. Theory of Probability & Its Applications, 2(4), 433-460. https://doi.org/10.1137/1102031 
Journal of Mahani Mathematical Research

Articles in Press, Accepted Manuscript
Available Online from 02 July 2026

Files

History

  • Receive Date: 25 November 2025
  • Revise Date: 10 April 2026
  • Accept Date: 28 June 2026

Share

How to cite

Statistics