A Numerical Method for Solving Fractional Differential Equations

Abstract

In this paper, we solve the fractional differential equations (FDEs) with boundary value conditions in Sobolev space $H^n\left[\mathrm{0,1}\right]$ . The strategy is constructing multiscale orthonormal basis for $H^n\left[\mathrm{0,1}\right]$ to get the approximation for the problems. The convergence of the method is proved, and it is tested on some numerical experiments; the tests show that our method is more efficient and accurate. The notion of numerical stability with respect to the condition number is introduced proving that the proposed method is numerically stable in this sense.