Shehu transform
Integral transform generalizing both Laplace and Sumudu transforms
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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[1][2][3] in 2019 and applied to both ordinary and partial differential equations.[4][3][5][6][7][8]
Formal definition
The Shehu transform of a function is defined over the set of functions
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)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c9d11474472081a0d33c5b889b64b275626cd5f1)
where and are the Shehu transform variables.[1] The Shehu transform converges to Laplace transform when the variable .
Inverse Shehu transform
The inverse Shehu transform of the function is defined as
![{\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)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b70b23e94dc05e2dcd718e8680341a8787967681)
where is a complex number and is a real number.[1]
Properties and theorems
...| Property | Explanation |
|---|---|
| Linearity | Let the functions and be in set A. Then |
| Change of scale | Let the function be in set A, where in an arbitrary constant. Then |
| Exponential shifting | Let the function be in set A and is an arbitrary constant. Then |
| Multiple shift | Let and . Then |
![{\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].}](http://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).}](http://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).}](http://wikimedia.org/api/rest_v1/media/math/render/svg/353e6d8f2c29fd6552b1c7d1496200b13bed1a95)
![{\displaystyle {\mathbb {S} }\left[f(t)\right]=F(s,u)}](http://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).}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f94e9a84a6c3b1314ea1a97c449d8a323dfd3dcc)
Theorems
Shehu transform of integral
![{\displaystyle {\mathbb {S} }\left[\int _{0}^{t}f(\zeta )d\zeta \right]={\frac {u}{s}}F(s,u),}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2f58d02683509e844765d1871d5da3561ffc33e8)
![{\displaystyle {\mathbb {S} }\left[f(\zeta )\right]=F(s,u)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/23a0f8c37f360aceda702a45f9ef7ad4c89a96bb)
nth derivatives of Shehu transform
If the function is the nth derivative of the function with respect to , then [1][3]
![{\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).}](http://wikimedia.org/api/rest_v1/media/math/render/svg/655ccce6429e815dcd6c96f2a30ebcdb0ffb9b30)
Convolution theorem of Shehu transform
Let the functions and be in set A. If and are the Shehu transforms of the functions and respectively. Then
![{\displaystyle {\mathbb {S} }\left[(f*g)(t)\right]=F(s,u)G(s,u).}](http://wikimedia.org/api/rest_v1/media/math/render/svg/fa5434e6a68095a990a521e1f868497f18b1cc74)
Where is the convolution of two functions and which is defined as
