Self-similar cauchy problems and generalized Mittag-Leffler functions
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Pierre Patie
Abstract
By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15].
Acknowledgements
This paper is dedicated to the memory of Mark Meerschaert who sadly passed away on September 29th 2020. We decided to leave the acknowledgements from the first version of the paper, as: The authors are indebted to Mark Meerschaert for providing them many interesting references on the spectral approach in the context of the fractional Cauchy problem and also for his invaluable encouragements.
The authors are grateful to the referees for careful reading, constructive comments and providing several interesting references on different aspects of the paper.
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© 2021 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial Note
- Anniversary of Prof. S.G. Samko, FC Events (FCAA–Volume 24–2–2021)
- Research Paper
- Operational calculus for the general fractional derivative and its applications
- Riesz potentials and orthogonal radon transforms on affine Grassmannians
- Characterizations of variable martingale Hardy spaces via maximal functions
- Survey Paper
- Contributions on artificial potential field method for effective obstacle avoidance
- Research Paper
- Self-similar cauchy problems and generalized Mittag-Leffler functions
- Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives
- A boundary value problem for a partial differential equation with fractional derivative
- Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications
- Duality theory of fractional resolvents and applications to backward fractional control systems
- Kinetic solutions for nonlocal stochastic conservation laws
- A fractional analysis in higher dimensions for the Sturm-Liouville problem
- Averaging theory for fractional differential equations
Articles in the same Issue
- Frontmatter
- Editorial Note
- Anniversary of Prof. S.G. Samko, FC Events (FCAA–Volume 24–2–2021)
- Research Paper
- Operational calculus for the general fractional derivative and its applications
- Riesz potentials and orthogonal radon transforms on affine Grassmannians
- Characterizations of variable martingale Hardy spaces via maximal functions
- Survey Paper
- Contributions on artificial potential field method for effective obstacle avoidance
- Research Paper
- Self-similar cauchy problems and generalized Mittag-Leffler functions
- Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives
- A boundary value problem for a partial differential equation with fractional derivative
- Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications
- Duality theory of fractional resolvents and applications to backward fractional control systems
- Kinetic solutions for nonlocal stochastic conservation laws
- A fractional analysis in higher dimensions for the Sturm-Liouville problem
- Averaging theory for fractional differential equations
