Venelin Todorov

Yuri Dimitrov, Ivan Dimov, Venelin Todorov

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.

Ivan Dimov, Juri Kandilarov, Venelin Todorov et al.

We present two approaches for enhancing the accuracy of second order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth order compact difference scheme and the fourth order scheme based on Richardson extrapolation. Our interest is concentrated on a system of ten parabolic partial differential equations in air pollution modeling. We analyze numerical experiments to compare the two approaches with respect to accuracy, computational complexity, non-negativity preserving and etc. Sixth-order approximation based on the fourth-order compact difference scheme combined with Richardson extrapolation is also discussed numerically.

Yuri Dimitrov, Radan Miryanov, Venelin Todorov

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.

Yuri Dimitrov, Venelin Todorov, Radan Miryanov

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 Grünwald 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.