Transparent boundary conditions for a diffusion problem modified by hilfer derivative

Ryad Ghanam, Nadeem A. Malik, Nasser Eddine Tatar

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3 Citations (Scopus)


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 languageEnglish
Pages (from-to)129-152
Number of pages24
JournalJournal of Mathematical Sciences (Japan)
Issue number1
Publication statusPublished - 2014
Externally publishedYes



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

ASJC Scopus subject areas

  • Mathematics(all)

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