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Shehu transform

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In mathematics, the Shehu transform is an integral transform which generalizes both the Laplace transform and the Sumudu integral transform. It was introduced by Shehu Maitama and Zhao Weidong[1][2] in 2019 and applied to both ordinary and partial differential equations.[3][4][5][6][7][8]

Formal definition

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The Shehu transform of a function is defined over the set of functions

as

where and are the Shehu transform variables.[1] The Shehu transform converges to Laplace transform when the variable .

Inverse Shehu transform

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The inverse Shehu transform of the function is defined as

where is a complex number and is a real number.[1]

Properties and theorems

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Properties of the Shehu transform[1]
Property Explanation
Linearity Let the functions and be in set A. Then
Change of scale Let the function be in set A, where in an arbitrary constant. Then
Exponential shifting Let the function be in set A and is an arbitrary constant. Then
Multiple shift Let and . Then

Theorems

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Shehu transform of integral

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where and [1]

nth derivatives of Shehu transform

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If the function is the nth derivative of the function with respect to , then [1]

References

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  1. ^ a b c d e f 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.
  8. ^ Mlaiki, Nabil; Jamal, Noor; Sarwar, Muhammad; Hleili, Manel; Ansari, Khursheed J. (2025-04-29). "Duality of Shehu transform with other well known transforms and application to fractional order differential equations". PLOS ONE. 20 (4): e0318157. Bibcode:2025PLoSO..2018157M. doi:10.1371/journal.pone.0318157. ISSN 1932-6203. PMC 12040285. PMID 40299951.