# N-transform

In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan[1] 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. Belgacem[2] has renamed it the natural transform and has proposed a detail theory and applications.[3][4]

## Formal definition

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

$R(u, s) = \mathcal{N}\{f(t)\} = \int_0^\infty f(ut)e^{-st}\,dt.\qquad(1)$

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

## References

1. ^ a b Khan, Z.H., Khan, W.A., "N-Transform-Properties and Applications" NUST Journal of Engineering Sciences 1(2008), 127–133.
2. ^ Belgacem, F. B. M and Silambarasan, R. Theory of the Natural transform. Mathematics in Engg Sci and Aerospace (MESA) journal. Vol. 3. No. 1. pp 99–124. 2012.
3. ^ Belgacem, F. B. M., and R. Silambarasan. "Advances in the Natural transform." AIP Conference Proceedings. Vol. 1493. 2012.
4. ^ Silambarasan, R., and F. B. M. Belgacem. "Applications of the Natural transform to Maxwell’s Equations." 12–16.