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

  • I. P. Gavrilyuk Univ. Cooperative Education Gera-Eisenach, Eisenach, Germany
  • V. L. Makarov Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev
Keywords: the operator Mittag-Leffler function, the Laguerre-Cayley functions, abstract time-fractional evolution equation, the Cayley transform, $N$-term approximation, exponential accuracy


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 $N$.


N. I. Akhieser, I. M. Glazman, Theory of linear operators in Hilbert space, Pitman Adv. Publ. Program, London (1980).

D. Z. Arov, I. P. Gavrilyuk, A method for solving initial value problems for linear differential equations in Hilbert space based on the Cayley transform, Numer. Funct. Anal. and Optim., 14, № 5-6, 456 – 473 (1993), DOI:

A. Ashyralyev, A note on fractional derivatives and fractional powers of operators, J. Math. Anal. and Appl., 357, 232 – 236 (2009), DOI:

D. Z. Arov, I. P. Gavrilyuk, V. L. Makarov, Representation and approximation of solution of initial value problems for differential equations in Hilbert space based on the Cayley transform, Elliptic and Parabolic Problems, Proc. 2nd Eur. Conf., Pont-a-Mousson, June 1994, Pitman Res. Notes Math. Ser. 325, 40 – 50 (1995).

H. Bateman, A. Erdelyi, Higher transcendental functions, vol. 2, MC Graw-Hill Book Co., New York etc. (1988).

R. Gorenflo, F. Mainardi, S. Rogosin, Mittag-Leffler function: properties and applications, Handbook of Fractional Calculus with Applications, vol. 1, Basic Theory, De Gruyter GmbH, Berlin, Boston, p. 269 – 296 (2019). DOI:

I. P. Gavrilyuk, V. L. Makarov, Explicit and approximate solutions of second order evolution differential equations in Hilbert space, Numer. Methods Partial Different. Equat., 15, 111 – 131 (1999). DOI:<111::AID-NUM6>3.0.CO;2-L

I. Gavrilyuk, V. Makarov, V. Vasylyk, Exponentially convergent algorithms for abstract differential equations, Springer, Basel AG (2011), DOI:

I. P. Gavrilyuk, Strongly $P$ -positive operators and explicit representations of the solutions of initial value problems for second order differential equations in Banach space, J. Math. Anal. and Appl., 236, 327 – 349 (1999), DOI:

I. P. Gavrilyuk, Super exponentially convergent approximation to the solution of the Schrodinger equation in abstract setting, Comput. Methods Appl. Math., 10, № 4, 345 – 358 (2010), DOI:

I. P. Gavrilyuk, An algorithmic representation of fractional powers of positive operators, Numer. Funct. Anal. and Optim., 17, № 3-4, 293 – 305 (1996), DOI:

I. P. Gavrilyuk, W. Hackbusch, B. N. Khoromskij, Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems, Computing, 74, № 2, 131 – 157 (2005), DOI:

I. P. Gavrilyuk, B. N. Khoromskij, Quasi-optimal rank-structured approximation to multidimensional parabolic problems by Cayley transform and Chebyshev interpolation, Comput. Methods Appl. Math., 191, 55 – 71 (2019), DOI:

I. P. Gavrilyuk, V. L. Makarov, Exact and approximate solutions of some operator equations based on the Cayley transform, Linear Algebra and Appl., 282, 97 – 121 (1998), DOI:

I. P. Gavrilyuk, V. L. Makarov, Representation and approximation of the solution of an initial value problem for a first order differential equation in Banach space, Z. Anal. Anwend., 15, № 2, 495 – 527 (1996), DOI:

I. P. Gavrilyuk, V. L. Makarov, V. B. Vasylyk, Exponentially convergent method for abstract integro-differential equation with the fractional Hardy – Titchmarsh integral, Dop. Akad. Nauk Ukr. (to appear).

V. Havu, J. Malinen, The Cayley transform as a time discretization scheme, Numer. Funct. Anal. and Optim., 28, № 7-8, 825 – 851 (2007), DOI:

H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., 2011, Article ID 298628, (2011); DOI:

W. McLean, V. Thomee, Numerical solution via Laplace transform of a fractional order evolution equation, J. Integral Equat. and Appl., 22, № 1, 57 – 94 (2010), DOI:

G. M. Mittag-Leffler, Sur la nouvelle fonction Ealpha (z), C. R. Acad Sci., 137, 554 – 558 (1903).

G. H. Hardy, E. C. Titchmarsh, An integral equation, Proc. Phil. Soc., 28, № 2, 165 – 173 (1932). DOI:

B. Jin, R. Lazarov, Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 62, 1 – 25 (2015), DOI:

H.-J. Seybold, R. Hilfer, Numerical algorithm for calculating the generalized Mittag-Leffler function, SIAM J. Numer. Anal., 47, № 1, 69 – 88 (2008/2009). DOI:

P. K. Suetin, Classical orthogonal polynomials, Nauka, Moscow (1979) (in Russian).

G. Szego, Orthogonal polynomials, Amer. Math. Soc., New York (1959).

How to Cite
Gavrilyuk , I. P., and V. L. Makarov. “Approximations of the Mittag-Leffler Operator Function With Exponential Accuracy and Their Application to Solving of Evolution Equations With Fractional Derivative in Time”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 5, June 2022, pp. 620 -34, doi:10.37863/umzh.v74i5.7097.
Research articles