Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations

Kyeong Hun Kim; Sungbin Lim

doi:10.4134/JKMS.j150343

Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations

Kyeong Hun Kim, Sungbin Lim

Research output: Contribution to journal › Article › peer-review

38 Citations (Scopus)

Abstract

Let p(t; x) be the fundamental solution to the problem (Formula Presented), then the kernel p(t; x) becomes the transition density of a Lévy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t; x) and its space and time fractional derivatives (Formula Presented); where Dⁿ_x is a partial derivative of order n with respect to x, (-Δ_x)^γ is a fractional Laplace operator and D^σ_t and I^σ_t are Riemann-Liouville fractional derivative and integral respectively.

Original language	English
Pages (from-to)	929-967
Number of pages	39
Journal	Journal of the Korean Mathematical Society
Volume	53
Issue number	4
DOIs	https://doi.org/10.4134/JKMS.j150343
Publication status	Published - 2016

Bibliographical note

Publisher Copyright:
© 2016 Korean Mathematical Society.

Keywords

Asymptotic behavior
Fractional diffusion
Fundamental solution
Lévy process
Space-time fractional differential equation

ASJC Scopus subject areas

General Mathematics

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10.4134/JKMS.j150343

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abstract = "Let p(t; x) be the fundamental solution to the problem (Formula Presented), then the kernel p(t; x) becomes the transition density of a L{\'e}vy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t; x) and its space and time fractional derivatives (Formula Presented); where Dnx is a partial derivative of order n with respect to x, (-Δx)γ is a fractional Laplace operator and Dσt and Iσt are Riemann-Liouville fractional derivative and integral respectively.",

keywords = "Asymptotic behavior, Fractional diffusion, Fundamental solution, L{\'e}vy process, Space-time fractional differential equation",

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AB - Let p(t; x) be the fundamental solution to the problem (Formula Presented), then the kernel p(t; x) becomes the transition density of a Lévy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t; x) and its space and time fractional derivatives (Formula Presented); where Dnx is a partial derivative of order n with respect to x, (-Δx)γ is a fractional Laplace operator and Dσt and Iσt are Riemann-Liouville fractional derivative and integral respectively.

KW - Asymptotic behavior

KW - Fractional diffusion

KW - Fundamental solution

KW - Lévy process

KW - Space-time fractional differential equation

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ER -

Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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