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In mathematics, the Shehu transform is an integral transform which generalizes both the Laplace transform and the Sumudu transform. It was introduced by Shehu Maitama and Weidong Zhao in 2019 and applied to both ordinary differential equation and partial differential equation.
The Shehu transform of a function is defined over the set of functions
as
where and are the Shehu transform variables. The Shehu transform converges to Laplace transform when the variable .
The inverse Shehu transform of the function is defined as
where is a complex number and is a real number.
| Property | Explanation |
|---|---|
| Linearity | For any functions , where , are arbitrary constants. Then, the Shehu transform satisfies
|
| Change of scale | Let the function be in set , where is an arbitrary constant. Then |
| Exponential shifting | Let the function be in set and is an arbitrary constant. Then |
| Multiple shift | Let and . Then |
where and
If the function is the nth derivative of the function with respect to , then
Let the functions and be in set . If and are the Shehu transforms of the functions and respectively. Then
Where is the convolution of two functions and which is defined as
If for and of exponential order. Then
,
where
If and Then
,
where
The Shehu transform of Caputo-Fabrizio fractional derivative in Caputo sense is defined as
The Shehu transform of Atangana-Baleanu fractional derivative in Caputo sense is defined as
The Shehu transform of Atangana-Baleanu fractional derivative in Riemann-Liouville sense is defined as
The Shehu transform of regularized Prabhakar fractional derivative in Caputo sense is defined as
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d11474472081a0d33c5b889b64b275626cd5f1)
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b70b23e94dc05e2dcd718e8680341a8787967681)
![{\displaystyle {\mathbb {S} }\left[\alpha f(t)+\beta w(t)\right]=\alpha {\mathbb {S} }\left[f(t)\right]+\beta {\mathbb {S} }\left[w(t)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7b5350f85bd47e2135d180a36b2deebcad543d)
![{\displaystyle {\mathbb {S} }\left[f(\beta t)\right]={\frac {1}{\beta }}F\left({\frac {s}{\beta }},u\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/308a5364784c1af11da351edccb06858a0e0bea4)
![{\displaystyle {\mathbb {S} }\left[\exp \left(\alpha t\right)f(t)\right]=F(s-\alpha u,u).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/353e6d8f2c29fd6552b1c7d1496200b13bed1a95)
![{\displaystyle {\mathbb {S} }\left[f(t)\right]=F(s,u)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/070293adea5d728e8afddf6008cb9ad06437b373)
![{\displaystyle {\mathbb {S} }\left[t^{n}f(t)\right]=(-u)^{n}{\frac {d^{n}}{ds^{n}}}F(s,u).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f94e9a84a6c3b1314ea1a97c449d8a323dfd3dcc)
![{\displaystyle {\mathbb {S} }\left[\int _{0}^{t}f(\zeta )d\zeta \right]={\frac {u}{s}}F(s,u),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f58d02683509e844765d1871d5da3561ffc33e8)
![{\displaystyle {\mathbb {S} }\left[f(\zeta )\right]=F(s,u)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23a0f8c37f360aceda702a45f9ef7ad4c89a96bb)
![{\displaystyle {\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/655ccce6429e815dcd6c96f2a30ebcdb0ffb9b30)
![{\displaystyle {\mathbb {S} }\left[(f*g)(t)\right]=F(s,u)G(s,u).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5434e6a68095a990a521e1f868497f18b1cc74)
![{\displaystyle \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+)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a542eebeddbe098f8f85e4fce918660aee633f4)
![{\displaystyle \alpha >0,\,m=1+[\alpha ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f42653aedbe3ad2d1943f291307b58e748f0f28b)
![{\displaystyle \alpha >0,\,\,m=1+[\alpha ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cac76cd70302453a22708a12d8b498acd689675)
![{\displaystyle \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+)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9363955e93df6ffd7485df72477cfda8bad156)
![{\displaystyle \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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/521afca6ed07d95685894b2ffd755cbe1d12f7aa)
![{\displaystyle \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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6de27da711a6bacba985fbe4c980662a755bdd9)
![{\displaystyle \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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77682effcf8e04fa3054be03f4f5b9992c426cd6)
![{\displaystyle \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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d6b80ed84e427870ff5314c9a42c457a673a70)