Elzaki transform

The Elzaki transform is an integral transform established by Tariq Elzaki in 2011. It defined as the set:

$A={f(t)|\ni M,k_{1},k_{2}>0|f(t)<M\exp ^{\frac {|t|}{k_{j}}}ift\epsilon (-1)^{j}X[0,\infty )}$

Where $k_{1}\leq u\leq k_{2}$

Elzaki transform is defined as $E[f(t)]=u\int _{0}^{\infty }f(t)\exp ^{(}-t/u)dt=u^{2}\int _{0}^{\infty }f(ut)\exp ^{-t}dt$

Duality with Laplace transform

$E[f(t)](u)=uL[f(t)]({\frac {1}{u}})$

$L[f(t)]({\frac {1}{v}})=vE[f(t)]({\frac {1}{v}})$ Elzaki transform and Laplace transform

Properties

• Convolution

$E[f(t)*g(t)]={\frac {1}{u}}E[f(t)]E[g(t)]$ where $(f*g)(t)=\int _{0}^{\infty }f(\tau )g(t-\tau )d\tau =(g*f)(t)$ is two convolution function.

• Differentiation

$E[f'(t)](u)={\frac {1}{u}}E[f(t)]-uf(+0)$ $[f''(t)](u)={\frac {1}{u^{2}}}E[f(t)]u-uf(+0)-f(0)$

References

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