N-transform

In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan¹ in 2008. It converges to both Laplace and Sumudu transform just by changing variables. Given the convergence to the Laplace and Sumudu transforms, the N-transform inherits all the applied aspects of the both transforms. Most recently, F. B. M. Belgacem² has renamed it the natural transform and has proposed a detail theory and applications.³⁴

The natural transform of a function f(t), defined for all real numbers t ≥ 0, is the function R(u, s), defined by:

${\displaystyle R(u,s)={\mathcal {N}}\{f(t)\}=\int _{0}^{\infty }f(ut)e^{-st}\,dt.\qquad (1)}$

Khan¹ showed that the above integral converges to Laplace transform when u = 1, and into Sumudu transform for s = 1.