Van der Corput lemmas for Mittag-Leffler functions. II. α–directions
Ruzhansky M. Torebek B.T.
October 2021Elsevier Masson s.r.l.
Bulletin des Sciences Mathematiques
2021#171
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 Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) 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 Schrödinger equations are considered.
Asymptotic estimate , Mittag-Leffler function , Van der Corput lemma
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Department of Mathematics: Analysis, Logic and Discrete, Mathematics, Ghent University, Krijgslaan 281, Ghent, Belgium
School of Mathematical Sciences, Queen Mary University of London, United Kingdom
Al–Farabi Kazakh National University, Al–Farabi ave. 71, Almaty, 050040, Kazakhstan
Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., Almaty, 050010, Kazakhstan
Department of Mathematics: Analysis
School of Mathematical Sciences
Al–Farabi Kazakh National University
Institute of Mathematics and Mathematical Modeling
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