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Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions

  • Michael Ruzhansky and Berikbol T. Torebek EMAIL logo
Published/Copyright: December 31, 2020
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 23 Issue 6

Abstract

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions Eα, β(i λ ϕ(x)), x ∈ ℝN and Eα, β(iαλ ϕ(x)), x ∈ ℝN for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

MSC 2010: Primary 42B20; Secondary 26D10; 33E12
Key Words and Phrases: van der Corput-type estimates; Mittag-Leffler function; time-fractional Schrödinger equation; time-fractional Klein-Gordon equation

Editorial Note

This paper has been presented at the online international conference “WFC 2020: Workshop on Fractional Calculus”, Ghent University, Belgium, 9–10 June 2020.

Acknowledgements

The authors were supported in parts by the FWO Odysseus Project 1 grant G.0H94.18N: Analysis and Partial Differential Equations. The first author was supported in parts by the EPSRC grant EP/R003025/1. The second author was supported by the Ministry of Education and Science of the Republic of Kazakhstan Grant AP08052046. No new data was collected or generated during the course of research.

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Received: 2020-06-09
Published Online: 2020-12-31
Published in Print: 2020-12-16

© 2020 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA special 2020 conferences' issue (FCAA–Volume 23–6–2020)
  4. Survey Paper
  5. Wave propagation dynamics in a fractional Zener model with stochastic excitation
  6. A survey on numerical methods for spectral Space-Fractional diffusion problems
  7. Research Paper
  8. Determination of the order of fractional derivative for subdiffusion equations
  9. Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions
  10. Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data
  11. Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation
  12. Nakhushev extremum principle for a class of integro-differential operators
  13. Bounded solutions of second order of accuracy difference schemes for semilinear fractional schrödinger equations
  14. Fractional nonlinear stochastic heat equation with variable thermal conductivity
  15. Implementation of fractional optimal control problems in real-world applications
  16. Historical Survey
  17. Mkhitar Djrbashian and his contribution to Fractional Calculus
  18. Archive Paper
  19. Fractional derivatives and cauchy problem for differential equations of fractional order
https://doi.org/10.1515/fca-2020-0082
Keywords for this article
van der Corput-type estimates; Mittag-Leffler function; time-fractional Schrödinger equation; time-fractional Klein-Gordon equation

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA special 2020 conferences' issue (FCAA–Volume 23–6–2020)
  4. Survey Paper
  5. Wave propagation dynamics in a fractional Zener model with stochastic excitation
  6. A survey on numerical methods for spectral Space-Fractional diffusion problems
  7. Research Paper
  8. Determination of the order of fractional derivative for subdiffusion equations
  9. Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions
  10. Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data
  11. Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation
  12. Nakhushev extremum principle for a class of integro-differential operators
  13. Bounded solutions of second order of accuracy difference schemes for semilinear fractional schrödinger equations
  14. Fractional nonlinear stochastic heat equation with variable thermal conductivity
  15. Implementation of fractional optimal control problems in real-world applications
  16. Historical Survey
  17. Mkhitar Djrbashian and his contribution to Fractional Calculus
  18. Archive Paper
  19. Fractional derivatives and cauchy problem for differential equations of fractional order
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