TY - JOUR

T1 - An explicit form for higher order approximations of fractional derivatives

AU - Gunarathna, W. A.

AU - Nasir, H. M.

AU - Daundasekera, W. B.

N1 - Funding Information:
This research was supported by Sultan Qaboos University Internal Grant No. IG/SCI/DOMAS/16/10.☆ This research was supported by Sultan Qaboos University Internal Grant No. IG/SCI/DOMAS/16/10.
Publisher Copyright:
© 2019 IMACS

PY - 2019/9

Y1 - 2019/9

N2 - 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.

AB - 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.

KW - Finite difference formula

KW - Fractional derivative

KW - Generating function

KW - Grünwald approximation

KW - Vandermonde system

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U2 - 10.1016/j.apnum.2019.03.017

DO - 10.1016/j.apnum.2019.03.017

M3 - Article

AN - SCOPUS:85063897281

SN - 0168-9274

VL - 143

SP - 51

EP - 60

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -