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.
Formal definition The Shehu transform of a function f ( t ) {\displaystyle f(t)} is defined over the set of functions
A = { f ( t ) : ∃ M , p 1 , p 2 > 0 , | f ( t ) | < M exp ( | t | / p i ) , if t ∈ ( − 1 ) i × [ 0 , ∞ ) } {\displaystyle 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
S [ f ( t ) ] = F ( s , u ) = ∫ 0 ∞ exp ( − s t u ) f ( t ) d t = lim α → ∞ ∫ 0 α exp ( − s t u ) f ( t ) d t , s > 0 , u > 0 , ( 1 ) {\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)}
where s {\displaystyle s} and u {\displaystyle u} are the Shehu transform variables. The Shehu transform converges to Laplace transform when the variable u = 1 {\displaystyle u=1} .
Inverse Shehu transform The inverse Shehu transform of the function f ( t ) {\displaystyle f(t)} is defined as
f ( t ) = S − 1 [ F ( s , u ) ] = lim β → ∞ 1 2 π i ∫ α − i β α + i β 1 u exp ( s t u ) F ( s , u ) d s , ( 2 ) {\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)}
where s {\displaystyle s} is a complex number and α {\displaystyle \alpha } is a real number.
Properties and theorems
Theorems
Shehu transform of integral
S [ ∫ 0 t f ( ζ ) d ζ ] = u s F ( s , u ) , {\displaystyle {\mathbb {S} }\left[\int _{0}^{t}f(\zeta )d\zeta \right]={\frac {u}{s}}F(s,u),}
where S [ f ( ζ ) ] = F ( s , u ) {\displaystyle {\mathbb {S} }\left[f(\zeta )\right]=F(s,u)} and f ( ζ ) ∈ A . {\displaystyle f(\zeta )\in A.}
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