Sumudu transform

The Sumudu transform is an integral transform introduced in 1990 by G K Watagala.^[1][2][3] It is defined over the set of functions ^[4][5][6]

$${\displaystyle A=\{f(t):\ni M,p,q>0,|f(t)|=M\exp(1/u)\}}$$

where ${\displaystyle p\leq u\leq q}$, the Sumudu transform is defined as

$${\displaystyle S[f(t)]={\frac {1}{u}}\int _{0}^{\infty }f(t)\exp \left(-{\frac {1}{u}}\right)\,dt.}$$

Relationship with other transforms

Sumudu transform is 1/u Laplace transform ${\displaystyle S[f(t)](u)={\frac {1}{u}}L[f(t)]({\frac {1}{u}})}$ And with u² Elzaki transform ${\displaystyle S[f(t)](u)=u^{2}E[f(t)](u)}$