Shehu transform

{{Short description|Integral transform generalizing both Laplace and Sumudu transforms}}

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{{Cite journal |last1=Maitama |first1=Shehu |last2=Zhao |first2=Weidong |date=2019-02-24 |title=New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace Transform for Solving Differential Equations |url=https://www.etamaths.com/index.php/ijaa/article/view/1771 |journal=International Journal of Analysis and Applications |language=en |volume=17 |issue=2 |pages=167–190 |issn=2291-8639}}{{Cite journal |last1=Maitama |first1=Shehu |last2=Zhao |first2=Weidong |date=2021 |title=New Laplace-type integral transform for solving steady heat-transfer problem |url=https://doiserbia.nb.rs/Article.aspx?id=0354-98361900160M |journal=Thermal Science |volume=25 |issue=1 Part A |pages=1–12|doi=10.2298/TSCI180110160M }} in 2019 and applied to both ordinary and partial differential equations.{{Cite journal |last1=Akinyemi |first1=Lanre |last2=Iyiola |first2=Olaniyi S. |date=2020 |title=Exact and approximate solutions of time-fractional models arising from physics via Shehu transform |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.6484 |journal=Mathematical Methods in the Applied Sciences |language=en |volume=43 |issue=12 |pages=7442–7464 |doi=10.1002/mma.6484 |bibcode=2020MMAS...43.7442A |issn=1099-1476}}{{Cite journal |last1=Maitama |first1=Shehu |last2=Zhao |first2=Weidong |date=2021-03-16 |title=Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives |url=https://link.springer.com/article/10.1007/s40314-021-01476-9 |journal=Computational and Applied Mathematics |language=en |volume=40 |issue=3 |pages=86 |doi=10.1007/s40314-021-01476-9 |issn=1807-0302}}{{Cite journal |last1=Yadav |first1=L. K. |last2=Agarwal |first2=G. |last3=Gour |first3=M. M. |last4=Akgül |first4=A. |last5=Misro |first5=Md Yushalify |last6=Purohit |first6=S. D. |date=2024-04-01 |title=A hybrid approach for non-linear fractional Newell-Whitehead-Segel model |url=https://www.sciencedirect.com/science/article/pii/S2090447924000200 |journal=Ain Shams Engineering Journal |volume=15 |issue=4 |pages=102645 |doi=10.1016/j.asej.2024.102645 |issn=2090-4479}}{{Cite journal |last1=Sartanpara |first1=Parthkumar P. |last2=Meher |first2=Ramakanta |date=2023-01-01 |title=A robust computational approach for Zakharov-Kuznetsov equations of ion-acoustic waves in a magnetized plasma via the Shehu transform |url=https://www.sciencedirect.com/science/article/pii/S2468013321001339 |journal=Journal of Ocean Engineering and Science |volume=8 |issue=1 |pages=79–90 |doi=10.1016/j.joes.2021.11.006 |bibcode=2023JOES....8...79S |issn=2468-0133}}{{Cite journal |last1=Abujarad |first1=Eman S. |last2=Jarad |first2=Fahd |last3=Abujarad |first3=Mohammed H. |last4=Baleanu |first4=Dumitru |date=August 2022 |title=APPLICATION OF q-SHEHU TRANSFORM ON q-FRACTIONAL KINETIC EQUATION INVOLVING THE GENERALIZED HYPER-BESSEL FUNCTION |url=https://www.worldscientific.com/doi/abs/10.1142/S0218348X2240179X |journal=Fractals |volume=30 |issue=5 |pages=2240179–2240240 |doi=10.1142/S0218348X2240179X |bibcode=2022Fract..3040179A |issn=0218-348X}}{{Cite journal |last1=Mlaiki |first1=Nabil |last2=Jamal |first2=Noor |last3=Sarwar |first3=Muhammad |last4=Hleili |first4=Manel |last5=Ansari |first5=Khursheed J. |date=2025-04-29 |title=Duality of Shehu transform with other well known transforms and application to fractional order differential equations |journal=PLOS ONE |language=en |volume=20 |issue=4 |pages=e0318157 |doi=10.1371/journal.pone.0318157 |doi-access=free |issn=1932-6203 |pmc=12040285 |pmid=40299951|bibcode=2025PLoSO..2018157M }}

Formal definition

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

$A = \{f(t) : \exists M, p_1 ,p_2> 0 , |f(t)|< M \exp(|t|/p_i),\,\,\,\text{if}\,\,\,t\in(-1)^i\times[0,\,\infty) \}$

$\mathbb S[f(t)]=F(s,u)= \int_0^\infty\exp\left(-\frac{st}{u}\right)f(t) \, dt=\lim_{\alpha\rightarrow\infty}\int_0^\alpha\exp\left(-\frac{st}{u}\right)f(t) \, dt,\,s>0,\,u>0,\,\,\,\,\,\,$

where $s$ and $u$ are the Shehu transform variables. The Shehu transform converges to Laplace transform when the variable $u = 1$ .

Inverse Shehu transform

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

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

where $s$ is a complex number and $\alpha$ is a real number.

Properties and theorems

class="wikitable" \|+ Properties of the Shehu transform ! Property !! Explanation
Linearity	Let the functions $\alpha f(t)$ and $\beta w(t)$ be in set A. Then ${\mathbb S}\left[\alpha f(t)+\beta w(t)\right]= \alpha{\mathbb S}\left[f(t)\right]+\beta{\mathbb S}\left[w(t)\right].$
Change of scale	Let the function $f(\beta t)$ be in set A, where $\beta$ in an arbitrary constant. Then ${\mathbb S}\left[f(\beta t)\right]=\frac{1}{\beta}F\left(\frac{s}{\beta},u\right).$
Exponential shifting	Let the function $\exp\left(\alpha t\right)f(t)$ be in set A and $\alpha$ is an arbitrary constant. Then ${\mathbb S}\left[\exp\left(\alpha t\right)f(t)\right]=F(s-\alpha u, u).$
Multiple shift	Let ${\mathbb S}\left[f(t)\right]=F(s,u)$ and $f(t)\in A$ . Then ${\mathbb S}\left[t^nf(t)\right]=(-u)^n\frac{d^n}{ds^n}F(s, u).$

= Theorems =

==Shehu transform of integral==

${\mathbb S}\left[\int_{0}^{t}f(\zeta)d\zeta\right]=\frac{u}{s}F(s,u),$

where ${\mathbb S}\left[f(\zeta)\right]=F(s,u)$ and $f(\zeta)\in
A.$

==''n''th derivatives of Shehu transform==

If the function $f^{(n)}(t)$ is the nth derivative of the function $f(t)\in A$ with respect to $t$ , then ${\mathbb S}
\left[f^{(n)}(t)\right]
=\left(\frac{s}{u}\right)^{n}F(s,u)-
\sum_{k=0}^{n-1}\left(\frac{s}{u}\right)^{n-(k+1)}f^{(k)}(0).$

References

Category:Integral transforms

Category:Mathematics