2018/06/09 by Dimitrov, Yuri, Miryanov, Radan, Todorov, Venelin
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1806.03421
In this paper we use the asymptotic expansions of the binomial coefficients and the weights of the L1 approximation to obtain approximations of order 2-α and second-order approximations of the Caputo derivative by modifying the weights of the shifted Grünwald-Letnikov difference approximation and the L1 approximation of the Caputo derivative. A modification of the shifted Grünwald-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.