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)

---

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 in 2019 and applied to both ordinary and partial differential equations.

Formal definition

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

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

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

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.