2 research outputs found
Asymptotic expansions and approximations for the Caputo derivative
In this paper we use the asymptotic expansions of the binomial coefficients
and the weights of the L1 approximation to obtain approximations of order
and second-order approximations of the Caputo derivative by
modifying the weights of the shifted Gr\"unwald-Letnikov difference
approximation and the L1 approximation of the Caputo derivative. A modification
of the shifted Gr\"unwald-Letnikov approximation is obtained which allows
second-order numerical solutions of fractional differential equations with
arbitrary values of the solutions and their first derivatives at the initial
point
Approximations for the Caputo derivative (II)
In the present paper we use the expansion formula of the polylogarithm
function to construct approximations of the Caputo derivative which are related
to the midpoint approximation of the integral in the definition of the Caputo
derivative. The asymptotic expansion formula of the Riemann sum approximation
of the beta function and the first terms of the expansion formulas of the
approximations of the Caputo derivative of the power function are obtained in
the paper. The induced shifted approximations of the Gr\"unwald formula and the
approximations of the Caputo derivative studied in the first part of the paper
are constructed and applied for numerical solution of fractional differential
equations