arXiv subjects

Yuri Dimitrov

Publications and source records attributed to Yuri Dimitrov.

9 recordsLinked to original sources

arXiv2018

Asymptotic expansions and approximations for the Caputo derivative

In this paper we use the asymptotic expansions of the binomial coefficients and the weights of the L1 approximation to obtain approximations of order $2-\alpha$ and second-order approximations of the Caputo derivative by modifying the weights of the shifted Gr\"unwald-Letnikov difference approximation and the L1 approximation of the Caputo derivative. A modification of the shifted Gr\"unwald-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.

math.NA↗

arXiv2018

Numerical solutions of ordinary fractional differential equations with singularities

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.

math.NA↗

arXiv2016

Approximations for the Caputo Derivative (I)

In this paper we construct approximations for the Caputo derivative of order $1-\alpha,2-\alpha,2$ and $3-\alpha$. The approximations have weights $0.5\left((k+1)^{-\alpha}-(k-1)^{-\alpha}\right)/\Gamma(1-\alpha)$ and $k^{-1-\alpha}/\Gamma(-\alpha)$, 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 $2-\alpha$ 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 $2-\alpha$ which has weights $k^{-1-\alpha}/\Gamma(-\alpha)$ is higher than the accuracy of the numerical solutions using the $L1$ approximation for the Caputo derivative.

math.NA↗

arXiv2016

Approximations for the Caputo derivative (II)

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 Gr\"unwald 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.

math.NA↗

arXiv2016

Higher-Order Numerical Solutions of the Fractional Relaxation-Oscillation Equation using Fractional Integration

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 $\alpha,1+\alpha,2+\alpha,3+\alpha$ and $4+\alpha$. The approximations are applied for computing the numerical solutions of the fractional relaxation-oscillation equation.

math.NA↗

arXiv2015

Three-Point Compact Approximation for the Caputo Fractional Derivative

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}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^{n} \delta_k^{(\alpha)} y_{n-k}=\dfrac{13}{12}y^{(\alpha)}_n-\dfrac{1}{6}y^{(\alpha)}_{n-1}+\dfrac{1}{12}y^{(\alpha)}_{n-2}+O\left(h^{3-\alpha}\right), \end{equation*} with weights $\delta_0^{(\alpha)}=1-\zeta(\alpha-1),\; \delta_n^{(\alpha)}=(n-1)^{1 -\alpha}-n^{1-\alpha},$ $$ \delta_1^{(\alpha)}=2^{1-\alpha}-2+2\zeta(\alpha-1),\; \delta_2^{(\alpha)}=1-2^{2-\alpha}+3^{1-\alpha}-\zeta(\alpha-1),$$ $$\delta_k^{(\alpha)}=(k-1)^{1-\alpha}-2k^{1-a}+(k+1)^{1-\alpha},\quad (k=3\cdots,n-1),$$ 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.

math.NA↗

arXiv2015

A New Method for Numerical Solution of the Fractional Relaxation and Subdiffusion Equations Using Fractional Taylor Polynomials

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.

math.NA↗

arXiv2015

A Second Order Approximation for the Caputo Fractional Derivative

When $0<\alpha<1$, the approximation for the Caputo derivative $$y^{(\alpha)}(x) = \frac{1}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^n \sigma_k^{(\alpha)} y(x-kh)+O\bigl(h^{2-\alpha}\bigr),$$ where $\sigma_0^{(\alpha)} = 1, \sigma_n^{(\alpha)} = (n-1)^{1-a}-n^{1-a}$ and $$\sigma_k^{(\alpha)} = (k-1)^{1-\alpha}-2k^{1-a}+(k+1)^{1-\alpha},\quad (k=1...,n-1),$$ has accuracy $O\bigl(h^{2-\alpha}\bigr)$. We use the expansion of $\sum_{k=0}^n k^\alpha$ to determine an approximation for the fractional integral of order $2-\alpha$ and the second order approximation for the Caputo derivative $$y^{(\alpha)}(x) = \frac{1}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^n \delta_k^{(\alpha)} y(x-kh)+O\bigl(h^{2}\bigr),$$ where $\delta_k^{(\alpha)} = \sigma_k^{(\alpha)}$ for $2\leq k\leq n$, $$\delta_0^{(\alpha)} = \sigma_0^{(\alpha)}-\zeta(\alpha-1), \delta_1^{(\alpha)} = \sigma_1^{(\alpha)}+2\zeta(\alpha-1),\delta_2^{(\alpha)} = \sigma_2^{(\alpha)}-\zeta(\alpha-1),$$ and $\zeta(s)$ is the Riemann zeta function. The numerical solutions of the fractional relaxation and subdiffusion equations are computed.

math.NA↗

arXiv2013

Numerical Approximations for Fractional Differential Equations

The Gr\"unwald and shifted Gr\"unwald formulas for the function $y(x)-y(b)$ are first order approximations for the Caputo fractional derivative of the function $y(x)$ with lower limit at the point $b$. We obtain second and third order approximations for the Gr\"unwald and shifted Gr\"unwald formulas with weighted averages of Caputo derivatives when sufficient number of derivatives of the function $y(x) $ are equal to zero at $b$, using the estimate for the error of the shifted Gr\"unwald formulas. We use the approximations to determine implicit difference approximations for the sub-diffusion equation which have second order accuracy with respect to the space and time variables, and second and third order numerical approximations for ordinary fractional differential equations.

math.NA↗