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Van der Corput lemmas for Mittag-Leffler functions. II. $alpha$-directions | | Michael Ruzhansky
; Berikbol T. Torebek
; | | Date: |
10 May 2020 | | Abstract: | The paper is devoted to study analogues of the van der Corput lemmas
involving Mittag-Leffler functions. The generalisation is that we replace the
exponential function with the Mittag-Leffler-type function, to study
oscillatory integrals appearing in the analysis of time-fractional partial
differential equations. More specifically, we study integral of the form
$I_{alpha,eta}(lambda)=int_mathbb{R}E_{alpha,eta}left(i^alphalambda
phi(x)
ight)psi(x)dx,$ for the range $0<alphaleq 2,,eta>0$. This
extends the variety of estimates obtained in the first part, where integrals
with functions $E_{alpha,eta}left(i lambda phi(x)
ight)$ have been
studied. Several generalisations of the van der Corput lemmas are proved. As an
application of the above results, the generalised Riemann-Lebesgue lemma, the
Cauchy problem for the time-fractional Klein-Gordon and time-fractional
Schr"{o}dinger equations are considered. | | Source: | arXiv, 2005.4546 | | Services: | Forum | Review | PDF | Favorites | |
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