An interpolation between the wave and diffusion equations through the fractional evolution equations Dirac like

Abstract The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-fractional evolution equations. In this work we establish basic properties of the Prabhakar type generalization of this function with the main emphasis on complete monotonicity. As particular examples, the relaxation functions for equations with multiple time-derivatives in the so-called “natural” and “modified” forms are studied in detail and useful estimates are derived. The obtained results extend known properties of the classical Mittag-Leffler function. The main tools used in this work are Laplace transform and Bernstein functions’ technique.

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Uncertain Fractional Evolution Equations with Non-Lipschitz Conditions Using the Condensing Mapping Approach

Acta Mathematica Vietnamica ◽

10.1007/s40306-020-00405-y ◽

2021 ◽

Author(s):

Nguyen Thi Kim Son

Keyword(s):

Evolution Equations ◽

Fractional Evolution Equations ◽

Mapping Approach ◽

Condensing Mapping ◽

Lipschitz Conditions

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

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0082 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1663-1677

Author(s):

Michael Ruzhansky ◽

Berikbol T. Torebek

Keyword(s):

Cauchy Problem ◽

Exponential Function ◽

Evolution Equations ◽

Oscillatory Integrals ◽

Schrödinger Equations ◽

Fractional Evolution Equations ◽

Type Function ◽

Fractional Schrödinger Equations ◽

The Cauchy Problem ◽

Klein Gordon

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 E α, β (i λ ϕ(x)), x ∈ ℝ N and E α, β (i α λ ϕ(x)), x ∈ ℝ N 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.

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Study of Hilfer fractional evolution equations by the properties of controllability and stability

Alexandria Engineering Journal ◽

10.1016/j.aej.2021.02.014 ◽

2021 ◽

Vol 60 (4) ◽

pp. 3741-3749

Author(s):

Pallavi Bedi ◽

Anoop Kumar ◽

Thabet Abdeljawad ◽

Aziz Khan

Keyword(s):

Evolution Equations ◽

Fractional Evolution Equations

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A study on controllability of impulsive fractional evolution equations via resolvent operators

Boundary Value Problems ◽

10.1186/s13661-021-01499-5 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Haide Gou ◽

Yongxiang Li

Keyword(s):

Banach Space ◽

Fixed Point ◽

Fixed Point Theorem ◽

Resolvent Operator ◽

Evolution Equations ◽

Validity Study ◽

Resolvent Operators ◽

Fractional Evolution Equations ◽

Mönch Fixed Point Theorem ◽

Alpha Beta

AbstractIn this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the $(\alpha ,\beta )$ ( α , β ) -resolvent operator, we concern with the term $u'(\cdot )$ u ′ ( ⋅ ) and finding a control v such that the mild solution satisfies $u(b)=u_{b}$ u ( b ) = u b and $u'(b)=u'_{b}$ u ′ ( b ) = u b ′ . Finally, we present an application to support the validity study.

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