Yuri Dimitrov

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.

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

When $0<α<1$, the approximation for the Caputo derivative $$y^{(α)}(x) = \frac{1}{Γ(2-α)h^α}\sum_{k=0}^n σ_k^{(α)} y(x-kh)+O\bigl(h^{2-α}\bigr),$$ where $σ_0^{(α)} = 1, σ_n^{(α)} = (n-1)^{1-a}-n^{1-a}$ and $$σ_k^{(α)} = (k-1)^{1-α}-2k^{1-a}+(k+1)^{1-α},\quad (k=1...,n-1),$$ has accuracy $O\bigl(h^{2-α}\bigr)$. We use the expansion of $\sum_{k=0}^n k^α$ to determine an approximation for the fractional integral of order $2-α$ and the second order approximation for the Caputo derivative $$y^{(α)}(x) = \frac{1}{Γ(2-α)h^α}\sum_{k=0}^n δ_k^{(α)} y(x-kh)+O\bigl(h^{2}\bigr),$$ where $δ_k^{(α)} = σ_k^{(α)}$ for $2\leq k\leq n$, $$δ_0^{(α)} = σ_0^{(α)}-ζ(α-1), δ_1^{(α)} = σ_1^{(α)}+2ζ(α-1),δ_2^{(α)} = σ_2^{(α)}-ζ(α-1),$$ and $ζ(s)$ is the Riemann zeta function. The numerical solutions of the fractional relaxation and subdiffusion equations are computed.

Yuri Dimitrov

In this paper we construct approximations for the Caputo derivative of order $1-α,2-α,2$ and $3-α$. The approximations have weights $0.5\left((k+1)^{-α}-(k-1)^{-α}\right)/Γ(1-α)$ 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.

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.

Yuri Dimitrov

In the present paper we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral. We use the expansion formula to obtain approximations for the fractional integral of order $α,1+α,2+α,3+α$ and $4+α$. The approximations are applied for computing the numerical solutions of the fractional relaxation-oscillation equation.

Yuri Dimitrov

The accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution. The derivatives of the solutions of fractional differential equations often have a singularity at the initial point, which may result in a lower accuracy of the numerical solutions. We propose a method for improving the accuracy of the numerical solutions of the fractional relaxation and subdiffusion equations based on the fractional Taylor polynomials of the solution at the initial point.

Yuri Dimitrov

In this paper we derive the fourth-order asymptotic expansions of the trapezoidal approximation for the fractional integral and the $L1$ approximation for the Caputo derivative. We use the expansion of the $L1$ approximation to obtain the three point compact approximation for the Caputo derivative \begin{equation*} \dfrac{1}{Γ(2-α)h^α}\sum_{k=0}^{n} δ_k^{(α)} y_{n-k}=\dfrac{13}{12}y^{(α)}_n-\dfrac{1}{6}y^{(α)}_{n-1}+\dfrac{1}{12}y^{(α)}_{n-2}+O\left(h^{3-α}\right), \end{equation*} with weights $δ_0^{(α)}=1-ζ(α-1),\; δ_n^{(α)}=(n-1)^{1 -α}-n^{1-α},$ $$ δ_1^{(α)}=2^{1-α}-2+2ζ(α-1),\; δ_2^{(α)}=1-2^{2-α}+3^{1-α}-ζ(α-1),$$ $$δ_k^{(α)}=(k-1)^{1-α}-2k^{1-a}+(k+1)^{1-α},\quad (k=3\cdots,n-1),$$ where $y$ is a differentiable function which satisfies $y'(0)=0$. The numerical solutions of the fractional relaxation and the time-fractional subdiffusion equations are discussed.