In this paper we construct approximations for the Caputo derivative of order 1 − α, 2 − α, 2 and 3 − α. The approximations have weights 0.5 ((k + 1) −α − (k − 1) −α ) /Γ(1 − α) and k −1−α /Γ(−α), and the higher accuracy is achieved by modifying the initial and last weights using the expansion formulas for the left and right endpoints. The approximations are applied for computing the numerical solution of ordinary fractional differential equations. The properties of the weights of the approximations of order 2 − α are similar to the properties of the L1 approximation. In all experiments presented in the paper the accuracy of the numerical solutions using the approximation of order 2 − α which has weights k −1−α /Γ(−α) is higher than the accuracy of the numerical solutions using the L1 approximation for the Caputo derivative.

The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are differentiable functions. In the present paper we propose a method for improving the accuracy of the numerical solutions of ordinary linear FDEs with constant coefficients which uses the fractional Taylor polynomials of the solutions. The numerical solutions of the two-term and three-term FDEs are studied in the paper.

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We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed.

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