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 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. 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 of Shehu transform

1. Linearity Property: Let the functions $\alpha f(t)$ and $\beta w(t)$ be in set A. Then ${\mathbb {S} }\left[\alpha f(t)+\beta w(t)\right]=\alpha {\mathbb {S} }\left[f(t)\right]+\beta {\mathbb {S} }\left[w(t)\right].$
2. Change of Scale Property: Let the function $f(\beta t)$ be in set A, where $\beta$ in an arbitrary constant. Then ${\mathbb {S} }\left[f(\beta t)\right]={\frac {1}{\beta }}F\left({\frac {s}{\beta }},u\right).$
3. Exponential Shifting Property: Let the function $\exp \left(\alpha t\right)f(t)$ be in set A and $\alpha$ is an arbitrary constant. Then ${\mathbb {S} }\left[\exp \left(\alpha t\right)f(t)\right]=F(s-\alpha u,u).$
4. Multiple Shift Property: Let ${\mathbb {S} }\left[f(t)\right]=F(s,u)$ and $f(t)\in A$. Then ${\mathbb {S} }\left[t^{n}f(t)\right]=(-u)^{n}{\frac {d^{n}}{ds^{n}}}F(s,u).$
5. Theorem 1: Shehu transform of integral: Let ${\mathbb {S} }\left[f(\zeta )\right]=F(s,u)$ and $f(\zeta )\in A$. Then ${\mathbb {S} }\left[\int _{0}^{t}f(\zeta )d\zeta \right]={\frac {u}{s}}F(s,u).$
6. Theorem 2: nth-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 ${\mathbb {S} }\left[f^{(n)}(t)\right]=\left({\frac {s}{u}}\right)^{n}F(s,u)-\sum _{k=0}^{n-1}\left({\frac {s}{u}}\right)^{n-(k+1)}f^{(k)}(0).$