Shehu transform

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 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

Theorems

Shehu transform of integral

where and

''n''th 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 A. If and are the Shehu transforms of the functions and respectively. Then
Where is the convolution of two functions and which is defined as