Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time

Abstract

UDC 519.62

Наближення операторної функцiї Мiттаг-Леффлера з експоненцiальною точнiстю та їх застосування до розв’язування еволюцiйних рiвнянь з дробовою похiдною за часом

In the present paper we propose and analyse an efficient discretization of the operator Mittag-Leffler function $E_{1+\alpha } \left(-At^{1+\alpha } \right)=\sum _{k=0}^{\infty }\frac{(-At^{1+\alpha } )^{k} }{\Gamma (1+k(1+\alpha ))}$, where $A$ is a self-adjoint positive definite operator. This function possesses a broad field of applications, for example, it represents the solution operator for an evolution problem $\partial_t u +\partial_t^{-\alpha}A u=0, t>0, u(0)=u_0$ with a spatial operator $A$ and with the fractional time-derivative of the order $\alpha$ (in the Riemann-Liouville sense), i.e. $u(t)=E_{1+\alpha} \left(-At^{1+\alpha } \right) u_{0}$ .
We apply the Cayley transform method \cite{ag, agm} that allows to recursively separate the variables and to represent the Mittag-Leffler function as an infinite series of products of the Laguerre-Cayley functions of the time variable (polynomials of $t^{1+\alpha}$) and of the powers of the Cayley transform of the spatial operator. The approximate representation is the truncated series with $N$ terms. We study the accuracy of the $N$-term approximation scheme depending on $\alpha$ and $N$.