Draft:Shehu transform

• Comment: In accordance with Wikipedia's Conflict of interest policy, I disclose that I have a conflict of interest regarding the subject of this article. Smaitama (talk) 17:09, 3 May 2025 (UTC)

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In mathematics, the Shehu transform is an integral transform  which generalized both the Laplace transform and the Sumudu integral transform. It was  introduced by Maitama and Zhao^[1][2] in 2019 and applied to both ordinary and partial differential equations^{[3][4][5][6][7]}.

Formal definition

The Shehu transform of a function f(t) is defined over the set of functions

${\displaystyle A=\{f(t):\exists M,p_{1},p_{2}>0,|f(t)|<M\exp(|t|/p_{i}),\,\,\,{\text{if}}\,\,\,t\in (-1)^{i}\times [0,\,\infty )\}}$

as${\displaystyle \mathbb {S} [f(t)]=F(s,u)=\int _{0}^{\infty }\exp \left(-{\frac {st}{u}}\right)f(t)\,dt=\lim _{\alpha \rightarrow \infty }\int _{0}^{\alpha }\exp \left(-{\frac {st}{u}}\right)f(t)\,dt,\,s>0,\,u>0.\,\,\,\,\,\,(1)}$

Inverse Shehu transform

The inverse Shehu transform of the function f(t) is defined as

${\displaystyle f(t)=\mathbb {S} ^{-1}[F(s,u)]=\lim _{\beta \rightarrow \infty }{\frac {1}{2\pi i}}\int _{\alpha -i\beta }^{\alpha +i\beta }{\frac {1}{u}}\exp \left({\frac {st}{u}}\right)F(s,u)ds.}$

References

1. Maitama, Shehu; Zhao, Weidong (2019-02-24). "New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace Transform for Solving Differential Equations". International Journal of Analysis and Applications. 17 (2): 167–190. ISSN 2291-8639.
2. Maitama, Shehu; Zhao, Weidong (2021). "New Laplace-type integral transform for solving steady heat-transfer problem". Thermal Science. 25 (1 Part A): 1–12. doi:10.2298/TSCI180110160M.
3. Akinyemi, Lanre; Iyiola, Olaniyi S. (2020). "Exact and approximate solutions of time-fractional models arising from physics via Shehu transform". Mathematical Methods in the Applied Sciences. 43 (12): 7442–7464. Bibcode:2020MMAS...43.7442A. doi:10.1002/mma.6484. ISSN 1099-1476.
4. Maitama, Shehu; Zhao, Weidong (2021-03-16). "Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives". Computational and Applied Mathematics. 40 (3): 86. doi:10.1007/s40314-021-01476-9. ISSN 1807-0302.
5. Yadav, L. K.; Agarwal, G.; Gour, M. M.; Akgül, A.; Misro, Md Yushalify; Purohit, S. D. (2024-04-01). "A hybrid approach for non-linear fractional Newell-Whitehead-Segel model". Ain Shams Engineering Journal. 15 (4): 102645. doi:10.1016/j.asej.2024.102645. ISSN 2090-4479.
6. Sartanpara, Parthkumar P.; Meher, Ramakanta (2023-01-01). "A robust computational approach for Zakharov-Kuznetsov equations of ion-acoustic waves in a magnetized plasma via the Shehu transform". Journal of Ocean Engineering and Science. 8 (1): 79–90. Bibcode:2023JOES....8...79S. doi:10.1016/j.joes.2021.11.006. ISSN 2468-0133.
7. Abujarad, Eman S.; Jarad, Fahd; Abujarad, Mohammed H.; Baleanu, Dumitru (August 2022). "APPLICATION OF q-SHEHU TRANSFORM ON q-FRACTIONAL KINETIC EQUATION INVOLVING THE GENERALIZED HYPER-BESSEL FUNCTION". Fractals. 30 (5): 2240179–2240240. Bibcode:2022Fract..3040179A. doi:10.1142/S0218348X2240179X. ISSN 0218-348X.