### Abstract

We consider a homogeneous fractional diffusion problem in an infinite reservoir sometimes called a "modified" diffusion equation. The equation involves a (nonlocal in time) memory term in the form of a time fractional derivative (of the Laplacian). For the sake of reducing the computational domain to a bounded one we establish appropriate "artificial" boundary conditions. This is to avoid the effect of reflected waves in case of a "solid" standard boundary. Then, an equivalent problem is studied in this bounded domain. To this end we use the Laplace-Fourier transform, the two-parameter Mittag-Leffler function and some properties of fractional derivatives.

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

Pages (from-to) | 129-152 |

Number of pages | 24 |

Journal | Journal of Mathematical Sciences (Japan) |

Volume | 21 |

Issue number | 1 |

Publication status | Published - 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Artificial boundary condition
- Caputo fractional derivative
- Fractional diffusion problem
- Hilfer fractional derivative
- Mittag-Leffler function
- Reduced equivalent problem.

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Mathematical Sciences (Japan)*,

*21*(1), 129-152.

**Transparent boundary conditions for a diffusion problem modified by hilfer derivative.** / Ghanam, Ryad; Malik, Nadeem A.; Tatar, Nasser Eddine.

Research output: Contribution to journal › Article

*Journal of Mathematical Sciences (Japan)*, vol. 21, no. 1, pp. 129-152.

}

TY - JOUR

T1 - Transparent boundary conditions for a diffusion problem modified by hilfer derivative

AU - Ghanam, Ryad

AU - Malik, Nadeem A.

AU - Tatar, Nasser Eddine

PY - 2014

Y1 - 2014

N2 - We consider a homogeneous fractional diffusion problem in an infinite reservoir sometimes called a "modified" diffusion equation. The equation involves a (nonlocal in time) memory term in the form of a time fractional derivative (of the Laplacian). For the sake of reducing the computational domain to a bounded one we establish appropriate "artificial" boundary conditions. This is to avoid the effect of reflected waves in case of a "solid" standard boundary. Then, an equivalent problem is studied in this bounded domain. To this end we use the Laplace-Fourier transform, the two-parameter Mittag-Leffler function and some properties of fractional derivatives.

AB - We consider a homogeneous fractional diffusion problem in an infinite reservoir sometimes called a "modified" diffusion equation. The equation involves a (nonlocal in time) memory term in the form of a time fractional derivative (of the Laplacian). For the sake of reducing the computational domain to a bounded one we establish appropriate "artificial" boundary conditions. This is to avoid the effect of reflected waves in case of a "solid" standard boundary. Then, an equivalent problem is studied in this bounded domain. To this end we use the Laplace-Fourier transform, the two-parameter Mittag-Leffler function and some properties of fractional derivatives.

KW - Artificial boundary condition

KW - Caputo fractional derivative

KW - Fractional diffusion problem

KW - Hilfer fractional derivative

KW - Mittag-Leffler function

KW - Reduced equivalent problem.

UR - http://www.scopus.com/inward/record.url?scp=84906859027&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906859027&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84906859027

VL - 21

SP - 129

EP - 152

JO - Journal of Mathematical Sciences (Japan)

JF - Journal of Mathematical Sciences (Japan)

SN - 1340-5705

IS - 1

ER -