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Applications of the Sumudu transform to fractional differential equations. (English) Zbl 1223.44001

The Sumudu transform was introduced by G. F. Watugala in an effort to improve Laplace transform techniques (see for instance [Int. J. Math. Educ. Sci. Technol. 24, No. 1, 35–43 (1993; Zbl 0768.44003)]). The Sumudu transform of a function $f\left(x\right)$ is defined by the formula

$G\left(u\right)=S\left[f\left(t\right)\right]\left(u\right)={\int }_{0}^{\infty }f\left(ut\right)\phantom{\rule{0.166667em}{0ex}}{e}^{-t}\phantom{\rule{0.166667em}{0ex}}dt,$

and is connected to the Laplace transform

$f\left(t\right)\to F\left(s\right)={\int }_{0}^{\infty }f\left(t\right){e}^{-st}\phantom{\rule{0.166667em}{0ex}}dt$

in a natural way, $G\left(u\right)=\frac{1}{u}F\left(\frac{1}{u}\right)$. Using this formula one can translate properties of the Laplace transform into properties of the Sumudu transform and vice versa. This transform is used practically for the same purpose the Laplace transform is used – for solving ordinary and partial-differential equations. Some formulas turn out to be very convenient.

In the present paper, the authors apply the Sumudu transform to fractional calculus, computing the transforms of fractional derivatives and integrals. For instance, if ${D}^{-\alpha }f$ is the Riemann-Liouville fractional integral, it is shown that $S\left[{D}^{-\alpha }f\left(t\right)\right]={u}^{\alpha }G\left(u\right)$. The authors also demonstrate how to solve fractional-differential equations with this transform. As an added bonus for the reader, the paper is accompanied by a representative table of transforms $S\left[{D}^{-\alpha }f\left(t\right)\right]$.

##### MSC:
 44A15 Special transforms (Legendre, Hilbert, etc.) 45A05 Linear integral equations 44A35 Convolution (integral transforms) 44A99 Miscellaneous topics of integral transforms and operational calculus