### Abstract

A diffusion problem involving a time derivative acting on two time scales represented by two fractional derivatives is investigated. The orders of the fractional derivatives are both between 0 and 1 and therefore the problem corresponds to the subdiffusion case. It is considered on a semi-infinite axis and the forcing term and the initial data are assumed compactly supported. To reduce the problem to that support there is a risk of being lead to an ‘infected’ problem due to the reflected waves on the new settled boundary. To avoid this undesirable effect of reflected waves on the standard boundaries, we establish artificial boundaries and find the appropriate artificial boundary conditions. Then, using the properties of fractional derivatives, a generalized version of the Mittag-Leffler function and some adequate manipulations of inverse Laplace transforms we find the explicit solution of the reduced problem.

Original language | English |
---|---|

Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Boundary Value Problems |

Volume | 2015 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis

### Cite this

*Boundary Value Problems*,

*2015*(1), 1-17. https://doi.org/10.1186/s13661-015-0281-0

**Artificial boundary condition for a modified fractional diffusion problem.** / Awotunde, Abeeb A.; Ghanam, Ryad; Tatar, Nasser eddine.

Research output: Contribution to journal › Article

*Boundary Value Problems*, vol. 2015, no. 1, pp. 1-17. https://doi.org/10.1186/s13661-015-0281-0

}

TY - JOUR

T1 - Artificial boundary condition for a modified fractional diffusion problem

AU - Awotunde, Abeeb A.

AU - Ghanam, Ryad

AU - Tatar, Nasser eddine

PY - 2015

Y1 - 2015

N2 - A diffusion problem involving a time derivative acting on two time scales represented by two fractional derivatives is investigated. The orders of the fractional derivatives are both between 0 and 1 and therefore the problem corresponds to the subdiffusion case. It is considered on a semi-infinite axis and the forcing term and the initial data are assumed compactly supported. To reduce the problem to that support there is a risk of being lead to an ‘infected’ problem due to the reflected waves on the new settled boundary. To avoid this undesirable effect of reflected waves on the standard boundaries, we establish artificial boundaries and find the appropriate artificial boundary conditions. Then, using the properties of fractional derivatives, a generalized version of the Mittag-Leffler function and some adequate manipulations of inverse Laplace transforms we find the explicit solution of the reduced problem.

AB - A diffusion problem involving a time derivative acting on two time scales represented by two fractional derivatives is investigated. The orders of the fractional derivatives are both between 0 and 1 and therefore the problem corresponds to the subdiffusion case. It is considered on a semi-infinite axis and the forcing term and the initial data are assumed compactly supported. To reduce the problem to that support there is a risk of being lead to an ‘infected’ problem due to the reflected waves on the new settled boundary. To avoid this undesirable effect of reflected waves on the standard boundaries, we establish artificial boundaries and find the appropriate artificial boundary conditions. Then, using the properties of fractional derivatives, a generalized version of the Mittag-Leffler function and some adequate manipulations of inverse Laplace transforms we find the explicit solution of the reduced problem.

UR - http://www.scopus.com/inward/record.url?scp=84961348942&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84961348942&partnerID=8YFLogxK

U2 - 10.1186/s13661-015-0281-0

DO - 10.1186/s13661-015-0281-0

M3 - Article

VL - 2015

SP - 1

EP - 17

JO - Boundary Value Problems

JF - Boundary Value Problems

SN - 1687-2762

IS - 1

ER -