In mathematics, the Shehu transform is an integral transform which generalizes both the Laplace transformhttps://link.springer.com/article/10.1007/BF00377050 and the Sumudu transformhttps://www.tandfonline.com/doi/abs/10.1080/0020739930240105. 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 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 .
Inverse Shehu transform The inverse Shehu transform of the function is defined as
where is a complex number and is a real number.
Properties and theorems + Properties of the Shehu transform 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
Theorems Shehu transform of integral
where and
nth derivatives of Shehu transform
If the function is the nth derivative of the function with respect to , then
Convolution theorem of Shehu transform 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
Shehu transform of Caputo fractional order derivative If for and of exponential order. Then
,
where
Shehu transform of Riemann-Liouville fractional order derivative If and Then
,
where
Shehu transform of Caputo-Fabrizio fractional derivative The Shehu transform of Caputo-Fabrizio fractional derivative in Caputo sense is defined as
Shehu transform of Atangana-Baleanu fractional derivative The Shehu transform of Atangana-Baleanu fractional derivative in Caputo sense is defined as
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
Shehu transform of regularized Prabhakar fractional derivative The Shehu transform of regularized Prabhakar fractional derivative in Caputo sense is defined as
References