The decay rate for a fractional differential equation

Nasser Eddine Tatar

doi:10.1016/j.jmaa.2004.01.047

The decay rate for a fractional differential equation

Nasser Eddine Tatar^*

^*Corresponding author for this work

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

18 Scopus citations

Abstract

We consider the fractional differential equation u_tt(t,x)=∫₀^t k(t-s)u_sxx(s,x) ds+u_xx(t,x), 0, x∈(0,1), with Dirichlet boundary conditions and initial values. This problem, with a particular kernel, may be looked at as an internally damped wave equation with (a strong) damping of order less than one. It is proved that the solution of this problem with a weakly singular kernel decays exponentially to zero.

Original language	English
Pages (from-to)	303-314
Number of pages	12
Journal	Journal of Mathematical Analysis and Applications
Volume	295
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2004.01.047
State	Published - 15 Jul 2004

Bibliographical note

Funding Information:
✩ This work has been supported by a FAST TRACK (Saudi Arabia) grant. E-mail address: [email protected].

Keywords

Exponential decay
Fractional derivative
Positive definite function
Weakly singular kernel

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2004.01.047

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title = "The decay rate for a fractional differential equation",

abstract = "We consider the fractional differential equation utt(t,x)=∫0t k(t-s)usxx(s,x) ds+uxx(t,x), 0, x∈(0,1), with Dirichlet boundary conditions and initial values. This problem, with a particular kernel, may be looked at as an internally damped wave equation with (a strong) damping of order less than one. It is proved that the solution of this problem with a weakly singular kernel decays exponentially to zero.",

keywords = "Exponential decay, Fractional derivative, Positive definite function, Weakly singular kernel",

author = "Tatar, \{Nasser Eddine\}",

note = "Funding Information: ✩ This work has been supported by a FAST TRACK (Saudi Arabia) grant. E-mail address: [email protected].",

year = "2004",

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doi = "10.1016/j.jmaa.2004.01.047",

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T1 - The decay rate for a fractional differential equation

AU - Tatar, Nasser Eddine

N1 - Funding Information: ✩ This work has been supported by a FAST TRACK (Saudi Arabia) grant. E-mail address: [email protected].

PY - 2004/7/15

Y1 - 2004/7/15

N2 - We consider the fractional differential equation utt(t,x)=∫0t k(t-s)usxx(s,x) ds+uxx(t,x), 0, x∈(0,1), with Dirichlet boundary conditions and initial values. This problem, with a particular kernel, may be looked at as an internally damped wave equation with (a strong) damping of order less than one. It is proved that the solution of this problem with a weakly singular kernel decays exponentially to zero.

AB - We consider the fractional differential equation utt(t,x)=∫0t k(t-s)usxx(s,x) ds+uxx(t,x), 0, x∈(0,1), with Dirichlet boundary conditions and initial values. This problem, with a particular kernel, may be looked at as an internally damped wave equation with (a strong) damping of order less than one. It is proved that the solution of this problem with a weakly singular kernel decays exponentially to zero.

KW - Exponential decay

KW - Fractional derivative

KW - Positive definite function

KW - Weakly singular kernel

UR - https://www.scopus.com/pages/publications/3242668227

U2 - 10.1016/j.jmaa.2004.01.047

DO - 10.1016/j.jmaa.2004.01.047

M3 - Article

AN - SCOPUS:3242668227

SN - 0022-247X

VL - 295

SP - 303

EP - 314

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -

The decay rate for a fractional differential equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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