An explicit form for higher order approximations of fractional derivatives

W. A. Gunarathna, H. M. Nasir*, W. B. Daundasekera

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


An explicit form for coefficients of shifted Grünwald type approximations of fractional derivatives is presented. This form directly gives approximations for any order of accuracy with any desired shift leading to efficient and automatic computations of the coefficients. To achieve this, we consider generating functions in the form of power of a polynomial. Then, an equivalent characterization for consistency and order of accuracy established on a general generating function is used to form a linear system of equations with Vandermonde matrix. This linear system is solved for the coefficients of the polynomial in the generating function. These generating functions completely characterize Grünwald type approximations with shifts and order of accuracy. Incidentally, the constructed generating functions happen to be a generalization of the previously known Lubich forms of generating functions without shift. We also present a formula to compute leading and some successive error terms from the coefficients. We further show that finite difference formulas for integer-order derivatives with desired shift and order of accuracy are some special cases of our explicit form.

Original languageEnglish
Pages (from-to)51-60
Number of pages10
JournalApplied Numerical Mathematics
Publication statusPublished - Sept 2019


  • Finite difference formula
  • Fractional derivative
  • Generating function
  • Grünwald approximation
  • Vandermonde system

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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