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 formula_1 is defined over the set of functions
formula_2
as
formula_3
where formula_4 and formula_5 are the Shehu transform variables. The Shehu transform converges to Laplace transform when the variable formula_6.
Inverse Shehu transform.
The inverse Shehu transform of the function formula_1 is defined as
formula_8
where formula_4 is a complex number and formula_10 is a real number.
Properties and theorems.
Theorems.
Shehu transform of integral.
formula_11
where formula_12 and formula_13
"n"th derivatives of Shehu transform.
If the function formula_14 is the nth derivative of the function formula_15 with respect to formula_16, then formula_17
Convolution theorem of Shehu transform.
Let the functions formula_1 and formula_19 be in set A. If formula_20 and formula_21 are the Shehu transforms of the functions formula_1 and formula_19 respectively. Then
formula_24
Where formula_25 is the convolution of two functions formula_1 and formula_19 which is defined as
formula_28