Shehu transform

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 Weidong 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

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 ${\mathbb {S} }\left[t^{n}f(t)\right]=(-u)^{n}{\frac {d^{n}}{ds^{n}}}F(s,u).$

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 $f(\zeta )\in A.$

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 ${\mathbb {S} }\left[f^{(n)}(t)\right]=\left({\frac {s}{u}}\right)^{n}F(s,u)-\sum _{k=0}^{n-1}\left({\frac {s}{u}}\right)^{n-(k+1)}f^{(k)}(0).$

Convolution theorem of Shehu transform

Let the functions $f(t)$ and $g(t)$ be in set $A$. If $F(s,u)$ and $G(s,u)$ are the Shehu transforms of the functions $f(t)$ and $g(t)$ respectively. Then

${\mathbb {S} }\left[(f*g)(t)\right]=F(s,u)G(s,u).$

Where $f*g$ is the convolution of two functions $f(t)$ and $g(t)$ which is defined as

$\int _{0}^{t}f(\tau )g(t-\tau )d\tau =\int _{0}^{t}f(t-\tau )g(\tau )d\tau .$

Shehu transform of Caputo fractional order derivative

If $f\in AC^{n}(a,b)$ for $b>a$ and of exponential order. Then

$\mathbb {S} \left[^{C}D_{t}^{\alpha }f(t)\right]=\left({\frac {s}{u}}\right)^{\alpha }{\mathbb {S} }\left[f(t)\right]-\sum _{k=0}^{m-1}\left({\frac {s}{u}}\right)^{\alpha -k-1}f^{(k)}(0+)$,

where $\alpha >0,\,m=1+[\alpha ].$

Shehu transform of Riemann-Liouville fractional order derivative

If $\alpha >0,\,\,m=1+[\alpha ],$ and $f(t),\,I^{m-\alpha }f(t),\,{\frac {d}{dt}}I^{m-\alpha }f(t),\cdots ,\,^{RL}Df(t)\in A.$ Then

$\mathbb {S} \left[^{RL}D_{t}^{\alpha }f(t)\right]=\left({\frac {s}{u}}\right)^{\alpha }{\mathbb {S} }\left[f(t)\right]-\sum _{k=0}^{m-1}\left({\frac {s}{u}}\right)^{m-k-1}{\frac {d^{k-1}}{dt^{k-1}}}I^{m-\alpha }f(0+)$,

where $\alpha >0,\,m=1+[\alpha ].$

Shehu transform of Caputo-Fabrizio fractional derivative

The Shehu transform of Caputo-Fabrizio fractional derivative in Caputo sense is defined as

$\mathbb {S} \left[^{CFC}{_{0}}D_{t}^{\alpha }f(t)\right]={\frac {{\mathcal {B}}(\alpha )}{s+\alpha (u-s)}}\left(sF(s,u)-uf(0)\right).$

Shehu transform of Atangana-Baleanu fractional derivative

The Shehu transform of Atangana-Baleanu fractional derivative in Caputo sense is defined as

$\mathbb {S} \left[^{ABC}{_{0}}D_{t}^{\alpha }f(t)\right]={\frac {{\mathcal {B}}(\alpha )}{1-\alpha +\alpha \left({\frac {u}{s}}\right)^{\alpha }}}\left(F(s,u)-{\frac {u}{s}}f(0)\right).$

Shehu transform of Atangana-Baleanu fractional derivative in Riemann-Liouville sense

The Shehu transform of Atangana-Baleanu fractional derivative in Riemann-Liouville sense is defined as

$\mathbb {S} \left[^{ABR}{_{0}}D_{t}^{\alpha }f(t)\right]={\frac {{\mathcal {B}}(\alpha )}{1-\alpha +\alpha \left({\frac {u}{s}}\right)^{\alpha }}}F(s,u).$

Shehu transform of regularized Prabhakar fractional derivative

The Shehu transform of regularized Prabhakar fractional derivative in Caputo sense is defined as

$\mathbb {S} \left[^{C}D_{\mu ,\beta ,\omega ,0}^{\gamma }f(t)\right]=\left({\frac {u}{s}}\right)^{-\beta }\left(1-\omega \left({\frac {u}{s}}\right)^{\mu }\right)^{\gamma }F(s,u).$