Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time
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 } \l...
Saved in:
| Date: | 2022 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7097 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512605430349824 |
|---|---|
| author | Gavrilyuk , I. P. Makarov, V. L. Gavrilyuk , I. P. Makarov, V. L. Makarov, V. L. |
| author_facet | Gavrilyuk , I. P. Makarov, V. L. Gavrilyuk , I. P. Makarov, V. L. Makarov, V. L. |
| author_sort | Gavrilyuk , I. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-10-24T09:23:04Z |
| description | 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 $N$. |
| doi_str_mv | 10.37863/umzh.v74i5.7097 |
| first_indexed | 2026-03-24T03:31:27Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i5.7097
UDC 519.62
I. P. Gavrilyuk (Univ. Cooperative Education Gera-Eisenach, Eisenach, Germany),
V. L. Makarov (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION
WITH EXPONENTIAL ACCURACY AND THEIR APPLICATION TO SOLVING
EVOLUTION EQUATIONS WITH FRACTIONAL DERIVATIVE IN TIME
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 Mittag-Leffler operator function
E1+\alpha ( - At1+\alpha ) =
\sum \infty
k=0
( - At1+\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 presents a solution operator to the evolution problem \partial tu+\partial - \alpha
t Au = 0, t > 0,
u(0) = u0, that includes a spatial operator A and its fractional time-derivative of the order \alpha (in the Riemann – Liouville
sense), i.e., u(t) = E1+\alpha ( - At1+\alpha )u0. We apply the Cayley transform method that allows us to recursively separate the
variables and represent the Mittag-Leffler function in the form of an infinite series of products of the Laguerre – Cayley
functions in time variable (i.e., polynomials in t1+\alpha ) and of powers of the Cayley transform of the spatial operator. The
approximate representation is the truncated series with N terms. We estimate the precision of the N -term approximation
scheme depending on \alpha and N.
Запропоновано та проаналiзовано ефективну дискретизацiю операторної функцiї Мiттаг-Леффлера
E1+\alpha ( - At1+\alpha ) =
\sum \infty
k=0
( - At1+\alpha )k
\Gamma (1 + k(1 + \alpha ))
, де A — самоспряжений додатно визначений оператор. Ця функцiя
має багато застосувань. Наприклад, вона дає операторний розв’язок еволюцiйної задачi \partial tu + \partial - \alpha
t Au = 0, t > 0,
u(0) = u0, з просторовим оператором A та його дробовими похiдними порядку \alpha за часом (у сенсi Рiмана –
Лiувiлля), тобто u(t) = E1+\alpha
\bigl(
- At1+\alpha
\bigr)
u0. Використано метод перетворень Келi, який дозволяє рекурсивно вiд-
окремити змiннi та подати функцiю Мiттаг-Леффлера у виглядi нескiнченного ряду добуткiв функцiй Лагерра – Келi
щодо змiнної часу (тобто полiномiв вiд t1+\alpha ) та степенiв перетворень Келi просторового оператора. Наближення за-
дається скiнченним вiдрiзком ряду, що складається з N доданкiв. Вивчено точнiсть цього наближення в залежностi
вiд \alpha та N.
1. Introduction. During the last two decades the Mittag-Lefler function has come into prominence
after about nine decades of its discovery in 1902 by the Swedish Mathematician Gösta Mittag-Leffler,
due to its success in many areas of science and engineering (see, e.g., [8, 18, 20] and the literature
therein). The (classical) Mittag-Leffler function has been introduced as a power series to give an
answer to a classical question of complex analysis, namely, to describe the procedure of analytic
continuation of power series outside the disc of their convergence.
The importance of the Mittag-Leffler function was re-discovered when its connection to fractional
calculus was fully understood. This function has its applications amongst other things in solving the
problems of physical, biological, engineering, earth and other sciences. Novadays there exist many
modifications of the Mittag-Leffler function in the literature but the literature about methods to
compute the Mittag-Leffler function is rather rare. We mention [23], where the algorithms using
the Taylor series, the exponentially improved asymptotic series, and integral representations to obtain
c\bigcirc I. P. GAVRILYUK, V. L. MAKAROV, 2022
620 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION . . . 621
optimal stability are discussed. Articles about the Mittag-Leffler functions that depend on an operator
are unknown to us.
One of the important and prominent applications of the Mittag-Leffler function is the representa-
tion of the solutions of the fractional differential equations. Diffusion is one of the most prominent
transport mechanisms found in nature. The classical diffusion model \partial tu - Au = f, which employs
a first-order derivative \partial tu in time and the Laplace operator Au = - \Delta u in space, is based on the
assumption that the particle motion is Brownian. One of the distinct features of Brownian motion is a
linear growth of the mean squared particle displacement with the time t. Over the last few decades, a
long list of experimental studies indicates that the Brownian motion assumption may not be adequate
for accurately describing some physical processes, and the mean squared displacement can grow
either sublinearly or superlinearly with time t, which are known as subdiffusion and superdiffusion,
respectively, in the literature (see, e.g., [22]). These experimental studies cover an extremely broad
and diverse range of important practical applications in engineering, physics, biology and finance,
including electron transport in Xerox photocopier, visco-elastic materials, thermal diffusion in fractal
domains, column experiments and protein transport in cell membrane etc. The original equation con-
nects the fractional derivative of the unknown function in time with a spatial operator A. The input
data of a problem and the output ones (the solution) are connected through the so called solution
operator E1+\alpha
\bigl(
- At1+\alpha
\bigr)
which maps the input into output. The connection between a fractional
derivative and fractional powers of operators was studied, e.g., in [3]. An algorithmical representation
of fractional powers of a positive operator A was proposed in [11].
In the present paper we consider a Mittag-Leffler function depending on a self-adjoint positive
definite operator and construct an efficient approximation for it.
The extensive literature is dedicated to various discretization methods for mathematical models
using the Mittag-Leffler functions explicitely or implicitely (see, e.g., [19]). The drawback of some
discretizations, e.g., from [19], is that the constants in accuracy estimates depend exponentially on t.
In [9, 10, 15] the solution operator, which is the operator exponential and the exact solution of the
heat conduction equation in abstract setting was represented by a series using the Laguerre orthogonal
polynomials of t and the Cayley transform of the spatial operator A (the separation of variables).
It was shown that the truncated series with N terms converges exponentially in N in the case of
analytical input data and polynomially in N in the case of input data which belong to the domain
of a power of A. Exponential convergence of approximations to operator valued functions and the
differential equations with unbounded operator coefficients plays a crucial role to obtain algorithms
of optimal or near optimal complexity [12, 13]. Note, that the Cayley transform can be used as a
time discretization scheme (see, e.g., [17] and the literature therein).
The present paper continues the series of works mentioned above. We propose a discretization of
the operator Mittag-Leffler function which is interesting in itself and, besides, represents the solution
operator of the PDE like to the heat conduction equation with the fractional time derivative and with
an abstract operator coefficient A in the “spatial part” of the equation. We propose a representation
of the Mittag-Leffler function as a series with products of the Laguerre – Cayley functions of t and
of powers of the Cayley transform of A. A similar technique for the operator exponential with the
Laguerre polynomials in t was used in [2, 4]. That is the motivation to call these functions as
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
622 I. P. GAVRILYUK, V. L. MAKAROV
the Laguerre – Cayley functionss. Some formal results are inspired by experimentation, conjectures
suggested by experiments and data supporting.
2. The operator-valued functions as the solution operators of differential problems. The
solution operator of a differential problem maps the input data on the problem solution. For example,
in the case of the initial value problem
u\prime (t) +Au = 0, u(0) = u0 (2.1)
with an operator coefficient A in a Hilbert space and the input data u0 the solution operator is the
operator exponential S = S(t, A) = e - At, so that u(t) = S(t, A)u0.
The next example deals with the problem
tD
\alpha +1
\infty u(t) +Au(t) = 0, t \in (0,\infty ), \alpha \in ( - 1, 1),
(2.2)
u(0) = u0,
where tD
1+\alpha
\infty is the (right) Riemann – Liouville derivative given by
tD
\nu
\infty f(t) =
\left\{
1
\Gamma ( - \nu )
\int \infty
t
(s - t) - \nu - 1f(s)ds, \nu < 0,
1
\Gamma
\bigl(
1 - \{ \nu \}
\bigr) \biggl( - d
dt
\biggr) [\nu ]+1 \int \infty
t
f(s)
(s - t)\{ \nu \}
ds, \nu \geq 0,
(2.3)
A is a strongly positive operator with a dense domain D(A) in a Banach space X. Its spectrum lies
in the sector \Sigma (A)
\Sigma (A) =
\Bigl\{
z = a0 + rei\theta : a0 > 0, r \in [0,\infty ), | \theta | < \varphi <
\pi
2
\Bigr\}
and on its boundary \Gamma \Sigma and outside of it the estimate\bigm\| \bigm\| (zI - A) - 1
\bigm\| \bigm\| \leq M
1 + | z|
is valid with some constant M. It was shown in [16] that under the assumptions
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow \infty
\bigl[
(s - t)\alpha +1
sD
\alpha
\infty u(s)
\bigr]
= 0,
(2.4)
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow \infty
\bigl[
(s - t)\alpha sD
\alpha - 1
\infty u(s)
\bigr]
= 0,
the solution is presented by
u(t) = \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- A1/(1+\alpha )t
\bigr)
u(0), (2.5)
i.e., the solution operator is given by
S(t, A) = \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- A1/(1+\alpha )t
\bigr)
. (2.6)
This formula is obtained by applying of the operator (2.3) to the equation (2.2) with taking into
account (2.4). As consequence we obtain the equation
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION . . . 623
- u(t) +A tD
- (\alpha +1)
\infty u(t) = 0, t \in (0,\infty ), \alpha \in ( - 1, 1),
which coindcides with the Hardy – Titchmarsh integral equation [21] (with change \alpha to \alpha +1). Thus,
its solution can be represented by (2.5).
Further, we consider the example of the following abstract initial value problem [19]:
\partial tu+ \partial - \alpha
t Au = f(t), t > 0,
u(0) = u0,
where \partial t =
\partial
\partial t
,
\partial - \alpha
t u =
\left\{
\partial t
\int t
0
(t - s)\alpha
\Gamma (1 + \alpha )
u(s)ds, - 1 < \alpha \leq 0,\int t
0
(t - s)\alpha - 1
\Gamma (\alpha )
u(s)ds, 0 < \alpha < 1,
is the fractional order time-derivative (\alpha < 0) or integral (\alpha > 0) in the Riemann – Liouville sense.
The solution operator S(t, A) of this problem can be represented through the operator
Mittag-Leffler function as [19]
S(t, A) = E1+\alpha ( - At1+\alpha ) =
\infty \sum
k=0
( - At1+\alpha )k
\Gamma
\bigl(
1 + k(1 + \alpha )
\bigr)
and its solution in the form
u(t) = E1+\alpha ( - At1+\alpha )u0 +
t\int
0
E1+\alpha ( - As1+\alpha )f(t - s)ds.
For particular values of \alpha one can obtain the closed explicit representations of the Mittag-Leffler
function. For example, for \alpha = 0 we have
E1( - At) = e - At.
In the case \alpha = - 0.5 this function can be expressed in terms of the complementary error function
E1/2( - A
\surd
t) = eA
2terfc(A
\surd
t).
The substitution s = ty2 in the formula above yields then the representation [19]
u(t) = E1/2( - A
\surd
t) +
1\int
0
E1/2( - A
\surd
ty)f(t - ty2)2tydy.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
624 I. P. GAVRILYUK, V. L. MAKAROV
3. The solution operator and the Cayley transform. The Cayley transform of unbounded
operators allows one to switch from the study of solution operator as a function of an unbounded
operator coefficient to a function (series) of some bounded operators and then to use an appropriate
approximation (e.g., an interpolation formula, a quadrature rule etc.) with a high accuracy.
This idea can be also used to “separate the variables” and in this way to obtain algorithms of low
complexity. In the case when the “spatial” operator involved is of the form B = B1+B2+ . . .+Bd
one can separate the ,,spatial” operators B1, B2, . . . , Bd via the tensor product and so arrive at a fully
separated approximation with a linear dependence on the problem dimension d [12]. Concerning
the time-dependent problems the Cayley transform allows to switch from “continuous time” to the
discrete one [2] as well as to separate the time variable from the spatial ones [2, 4].
In the theory of operators in Hilbert space the Cayley transform T\alpha ,\beta
\gamma ,\delta = (\alpha I +\beta A)(\gamma I - \delta A) - 1,
\alpha , \beta , \gamma , \delta \in \BbbC is frequently used to switch from the study of closed but in general unbounded linear
operator A with the dense domain D(A)(D(A) = H) to that of the bounded operators T\alpha ,\beta
\gamma ,\delta (see,
e.g., [1], where the transform T\gamma = (\gamma I+A)(\gamma I - A) - 1, \gamma = - i converts a self-adjoint (symmetric,
dissipative) operator A into unitary (respectively, isometric, contractive) operator T\gamma ).
In [2, 4] the Cayley transform has been used to obtain explicit and constructive representations of
the solution operator and of solutions of various evolution differential equations with operator coeffi-
cients where, in fact, the solution with continuous time parameter where represented through the ones
with discrete time. In [11, 14] the Cayley transform has been used to obtain explicit representation
of some important operator equations, e.g., the Lyapunov equation, the equation defining fractional
powers of an operator etc. A further important feature of these representation is the fact that they
serve as the basis for algorithms without accuracy saturation, i.e., their accuracy increases automati-
cally and unboundedly together with the smoothness of the input data. In the case of analytical input
data the convergence rate becomes exponential.
In [10] the idea was applied to the Schrödinger differential equation in abstract setting with a
strongly positive “spatial” operator coefficient B in some Banach space (the spectral set of such
operator coefficient lies in some angle in the right half-plane and the resolvent possesses a prescribed
behavior at the infinity). On the basis of an exact representation of the solution with use of the
Cayley transform, an approximation was proposed with the accuracy depending on the smoothness
of this solution. It was shown that for the analytical initial vectors this approximation possesses a
super exponential convergence rate. These ideas together with special tensor-product representation
were developed in [12] for multidimensional (d-dimensional) abstract differential equations in order
to obtain algorithms with linear in d complexity.
In the case when A is a self-adjoint positive definite unbounded operator with the spectrum
\Sigma = \Sigma (A) \in [\lambda 0,\infty ), \lambda 0 > 0, in a Hilbert space H the solution operator for (2.1) can be represented
by the Stieltjes integral
S(t;A) =
\infty \int
\lambda 0
e - t\lambda dE\lambda ,
where E\lambda is the resolution of the identity for A.
We say that the solution operator is generated by the function F (t, \lambda ) = e - t\lambda and by the ope-
rator A. Analogously one can define the so called Schrödinger operator exponential S(t) = eiBt as
the solution operator for the Schrödinger equation which can be represented by the corresponding
Stieltjes integral too.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION . . . 625
In the case when B is a self-adjoint positive definite operator with the spectrum \Sigma = \Sigma (B) \in
\in [\lambda 0,\infty ), \lambda 0 > 0, in a Hilbert space H an arbitrary solution operator generated by B and by a
function F (t, \lambda ) can be represented by the Stieltjes integral
F (t, B) =
\infty \int
\lambda 0
F (t, \lambda )dE\lambda ,
where E\lambda is the resolution of the identity for B.
One can separate the variable t from the operator B by the separation of the variables t and \lambda
in the function F (t, \lambda ).
If B is an unbounded operator, then the more general idea to switch to the study of bounded
operators is the following.
We can use some rational transform \eta = C(w) =
\lambda \alpha - \beta w
- \lambda + w
, w = \lambda
\alpha + \eta
\beta + \eta
=
a+ b\eta
c+ d\eta
, a/c \not =
\not = b/d, \lambda = b/d, \alpha = a/b, \beta = c/d, \alpha \not = \beta , where the variable w can remain in some bounded
domain. The function F (t, \eta ) = F (t, C(w)) = F
\Bigl(
t,
\lambda \alpha - \beta w
- \lambda + w
\Bigr)
can be represented as a power series
in w or approximated by a polynomial of w and we obtain a function of the bounded variable w.
If F (t, z) is analytical with respect to z in the unit disc | z| < 1, then the Taylor expansion
F (t, z) =
\infty \sum
n=0
cn(t)z
n, | z| < 1,
separates the both variables. The Taylor coefficients are given by
cn(t) =
1
n!
F (n)
z (t, z)
\bigm| \bigm|
t=0
=
1
2\pi i
\int
| \xi | =\rho
F (t, z)
\xi n+1
d\xi .
The next example shows the use of the Cayley transform for the representation of the operator
exponential (the solution operator for the heat and the Schrödinger equations).
Example 3.1. Let F (t, z) = F (t, z;\alpha ) = (1 - z) - \alpha - 1e
tz
z - 1 , where for not integer parameter
\alpha we mean the principal value [24]. This is the generating function for the Laguerre polynomials
L
(\alpha )
n (t). For each fixed t the function F (t, z) is analytic in the disc | z| < 1. We have for its Taylor
coefficients
cn(t) = cn(t;\alpha ) =
1
2\pi i
\int
| \xi | =\rho
(1 - \xi ) - \alpha - 1e
t\xi
\xi - 1 \xi - n - 1d\xi , 0 < \rho < 1.
We change the variables by
u =
t
1 - \xi
, \xi = 1 - t
u
, d\xi =
t
u2
du,
where the linear-fractional mapping u =
t
1 - \xi
translates the circle | \xi | = \rho < 1 into a circle \Gamma which
includes the point t > 0 but does not include the point 0. Then we get
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
626 I. P. GAVRILYUK, V. L. MAKAROV
cn(t) = cn(t;\alpha ) =
1
2\pi i
\int
\Gamma
\Bigl( u
t
\Bigr) \alpha +1
et - u
\biggl(
u - t
u
\biggr) - n - 1 t
u2
du =
= t - \alpha et
1
2\pi i
\int
\Gamma
u\alpha +ne - u
(u - t)n+1
du = t - \alpha et
1
n!
\bigl(
t\alpha +ne - t
\bigr) (n)
.
Comparing this equality with the Rodrigue’s formula for the Laguerre polynomials we see that
cn(t) = cn(t;\alpha ) = L
(\alpha )
n (t), i.e., we have the expansion (see also [5], Ch. 10.12)
(1 - z) - \alpha - 1e
tz
z - 1 =
\infty \sum
n=0
L(\alpha )
n (t)zn.
Replacing formally
z
z - 1
by A we obtain the following representation for the operator expo-
nential e - At, generated by the unbounded operator A:
e - At = (E - A) - \alpha - 1
\infty \sum
n=0
L(\alpha )
n (t)
\bigl[
A(A+ E) - 1
\bigr] n
.
Analogously after the substitution z \rightarrow iB(iB - I) - 1 we have formally for the solution operator
eiBt = - (iB - I) - 1
\infty \sum
n=0
L(0)
n (t)Tn,
of the abstract Schrödinger equation
u\prime (t) - iBu = 0, u(0) = u0,
where T = T (B) = iB(iB - I) - 1 is the Cayley transform of the operator B. It was shown in [10]
that truncated series with N summands converge exponentially in N provided that they are applied
to an analytical vector and, thus, represent exponentially convergent algorithms for the corresponding
problems.
An another option for constructing a fractional rational representation of operators is to interpo-
late the function \~F (t, z) = F (t, w) = F
\biggl(
t,
a+ bz
c+ dz
\biggr)
by a polynomial \scrI N \~F (t, z) on the Cheby-
shev, Chebyshev – Radou or Chebyshev – Gauss – Lobatto nodes (see, e.g., [13]), i.e., F (t, w) =
= F
\biggl(
t,
a+ bz
c+ dz
\biggr)
= \scrI N (t, z) + RN (t, z), z \in ( - 1, 1), with the reminder RN (t, z). Then the
operator F (t, B) can be represented by
F (t, B) = \scrI N
\bigl(
t, C(B)
\bigr)
+RN
\bigl(
t, C(B)
\bigr)
,
where C(B) = (\lambda \alpha - \beta B)( - \lambda + B) - 1 is the Cayley transform of B. The corresponding expo-
nentially accurate approximations of such solution operators, e.g., of the operator exponential or the
Schrödinger operator exponential were proposed and justified in [10].
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION . . . 627
4. The Mittag-Leffler operator valued function and the Laguerre – Cayley functions. We
consider the Mittag-Leffler operator function, generating by an operator A:
E1+\alpha
\bigl(
- At1+\alpha
\bigr)
=
\infty \sum
j=0
\bigl(
- At1+\alpha
\bigr) j 1
\Gamma
\bigl(
1 + j(\alpha + 1)
\bigr) .
We replace here A = q(I - q) - 1 with the identity operator I and some operator q and then develop
the Mittag-Leffler operator function into a McLaurin series
E1+\alpha
\bigl(
- q(I - q) - 1t1+\alpha
\bigr)
=
\infty \sum
j=0
\bigl(
- q(I - q) - 1t1+\alpha
\bigr) j 1
\Gamma
\bigl(
1 + j(\alpha + 1)
\bigr) =
\infty \sum
j=0
qjp\alpha j (t
1+\alpha ),
(4.1)
p\alpha j (t
1+\alpha ) =
1
j!
\partial j
\partial qj
E1+\alpha
\bigl(
- q(I - q) - 1t1+\alpha
\bigr) \bigm| \bigm|
q=0
,
and call the functions
p\alpha j (t
1+\alpha ) =
1
j!
\partial j
\partial xj
E1+\alpha
\bigl(
- x(1 - x) - 1t1+\alpha
\bigr) \bigm| \bigm|
x=0
the Laguerre – Cayley functions. From this definition we obtain the following explicit representation
of these functions:
p\alpha k (t
1+\alpha ) = p - ,\alpha
k (t
1+\alpha ) + p+,\alpha
k (t
1+\alpha ) =
k - 1\sum
s=0
Cs
k - 1
( - 1)s+1t(s+1)(1+\alpha )
\Gamma
\bigl(
1 + (\alpha + 1)(s+ 1)
\bigr) ,
p - ,\alpha
k (t
1+\alpha ) = -
[ k+1
2 ]\sum
s=1
C2s - 2
k - 1
t(2s - 1)(1+\alpha )
\Gamma
\bigl(
2s+ (2s - 1)\alpha
\bigr) , (4.2)
p+,\alpha
k (t
1+\alpha ) =
[ k2 ]\sum
s=1
C2s - 1
k - 1
t2s(1+\alpha )
\Gamma (2s+ 1 + 2s\alpha )
.
Lemma 4.1. For the Laguerre – Cayley functions the formula (4.2) is valid iff
p\alpha k+1(t
1+\alpha ) = p\alpha k (t
1+\alpha ) - 1
\Gamma (\alpha + 1)
t\int
0
(t - s)\alpha p\alpha k (s
1+\alpha )ds, \alpha \in ( - 1, 1), k = 1, 2, . . . ,
(4.3)
p\alpha k (t
1+\alpha ) = - t1+\alpha
\Gamma (\alpha + 2)
.
Proof. Let (4.2) holds true, then using the relation
1
\Gamma (\alpha + 1)
t\int
0
(t - s)\alpha sk(1+\alpha )ds =
tk(1+\alpha )+\alpha +1
\Gamma (\alpha + 1)
1\int
0
(1 - s)\alpha sk(1+\alpha )ds =
=
t(k+1)(1+\alpha )\Gamma (k\alpha + k + 1)
\Gamma
\bigl(
(k + 1)\alpha + k + 2
\bigr)
one can see that the both parts of (4.3) are equal.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
628 I. P. GAVRILYUK, V. L. MAKAROV
The sufficiency of (4.3) can be proven by induction.
Lemma is proved.
Let us introduce the generating function
f\alpha (q, t) =
\infty \sum
k=0
qkp\alpha k (t
1+\alpha ), p\alpha 0 (t
1+\alpha ) = 1.
Multiplying the both parts of (4.3) by zk and summing over k from 0 to \infty , we come to the integral
equation
(1 - q)f (\alpha )(q, t) +
z
\Gamma (1 + \alpha )
t\int
0
(t - s)\alpha f (\alpha )(q, s)ds = 1.
Using the Laplace transform one can obtain the explicite solutions of this integral equation. In
particular, for \alpha = - 1/2, 0, 1/2, 1, we obtain
f ( - 1/2)(q, t) = eq
2t/(1 - q)2
\biggl[
erf
\biggl(
- q
\surd
t
1 - q
\biggr)
+ 1
\biggr]
, f (0)(q, t) = e
qt
q - 1 ,
f (1)(q, t) = \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
t
\sqrt{}
q
1 - q
\biggr)
,
f (1/2)(q, t) =
1
3
\surd
\pi ( - 1 + q)
\Biggl[
4q t3/2hypergeom
\biggl(
[1],
\biggl[
5
6
,
7
6
,
9
6
\biggr]
,
q2t3
27( - 1 + q)2
\biggr)
+
+
\surd
\pi ( - 1 + q)2 \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl(
- q2/3t
2( - 1 + q)2/3
\Biggr)
\mathrm{c}\mathrm{o}\mathrm{s}
\Biggl( \surd
3q2/3t
2( - 1 + q)2/3
\Biggr)
-
\surd
\pi \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl(
q2/3t
( - 1 + q)2/3
\Biggr) \Biggr]
.
More simple way to obtain solutions for different \alpha is the use of the left part of formula (4.1).
The following proposition holds true.
Proposition 4.1. It holds
1
k!
\infty \sum
s=k
p\alpha s (1)
k!
(k - s)!
= ( - 1)k
k - 1\sum
s=0
( - 1)sCs
k - 1
\Gamma
\bigl(
- \alpha - s(1 + \alpha )
\bigr) =
= \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 1 - 0
1
k!
\partial k
\partial xk
f (\alpha )(x, 1). (4.4)
Proof performed by using the Abel theorem with the support of computer algebra programm
Maple.
In cases when for concrete \alpha the use of formula (4.4) causes difficulties it is resonable to go an
other way.
One can choosethe Taylor series
E1+\alpha
\bigl(
- q(I - q) - 1t1+\alpha
\bigr)
=
=
\infty \sum
j=0
\bigl(
- q(I - q) - 1t1+\alpha
\bigr) j 1
\Gamma (1 + j(\alpha + 1))
=
\infty \sum
j=0
(q + I)jS\alpha
j (t
1+\alpha ),
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION . . . 629
where
S\alpha
j (t
1+\alpha ) =
1
j!
\partial j
\partial xj
E1+\alpha
\biggl(
- x
1 - x
t1+\alpha
\biggr) \bigm| \bigm| \bigm| \bigm|
q= - 1
=
1
j!
\infty \sum
s=j
( - 1)sp\alpha s (t
1+\alpha )
s!
(s - j)!
. (4.5)
Proposition 4.2. Let \alpha \in ( - 1, 0), then for each natural j the formula (4.5) is valid.
The calculation of the left-hand side of formula (4.5) is technically much more simple then the
calculation of the right-hand side of formula (4.4), since the function E1+\alpha
\bigl(
- x(1 - x) - 1t1+\alpha
\bigr)
and
all its partial derivatives with respect to x do not have singularities at x - 1.
The propositions above imply that for all \alpha \in ( - 1, 0) and for each non negative j the limit value
holds true
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow \infty
p\alpha s (t
1+\alpha )
s!
(s - j)!
= 0
provided the series (4.5) is convergent. This equality means that there exists a constant c independent
of s such that \bigm| \bigm| p\alpha s (t1+\alpha )
\bigm| \bigm| \leq c
(s - j)!
s!
=
1
(s - j + 1)(s - j + 2) . . . s
\leq s - s.
In the case \alpha = 0 it follows from the generating function f (0)(q, t) that the Laguerre – Cayley
functions can be represented through the Laguerre polynomials
p0k(t) = Lk(t) - Lk - 1(t),
for which the inequality \bigm| \bigm| p0k(t)\bigm| \bigm| \leq 2e
t
2
holds true.
For \alpha \in (0, 1) with assistance of Maple it can be shown that there exist such natural \mu (\alpha ), that
the series
\infty \sum
k=1
p\alpha k (1) k
- \mu (\alpha )
are convergent (Maple provides the exact sums). Thus, the estimate\bigm| \bigm| p\alpha k (1)\bigm| \bigm| \leq Ck\mu (\alpha ), \alpha \in (0, 1),
is valid. In particular, \mu (\alpha ) = 6 for \alpha =
1
m
, m = 4, . . . , 13, \mu (\alpha ) = 7 for \alpha =
m
m+ 1
, m = 2, 3, 4,
and \mu
\biggl(
1
2
\biggr)
= 8. Further analogously one can get that
\infty \sum
k=1
\bigm| \bigm| p1k(1)\bigm| \bigm| k - 4 = 0.53631821 . . . ,
i.e., the estimate \bigm| \bigm| p1k(1)\bigm| \bigm| \leq C k4
holds true.
Let us summarize the estimates for p\alpha k (t
1+\alpha ) in the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
630 I. P. GAVRILYUK, V. L. MAKAROV
Table 4.1
N
N
\delta (2)
20 7.72 \cdot 10 - 11
40 3.42 \cdot 10 - 19
80 6.49 \cdot 10 - 34
160 5.02 \cdot 10 - 64
Lemma 4.2. The following estimates hold true:
\bigm| \bigm| p\alpha k (t1+\alpha )
\bigm| \bigm| \leq c
\left\{
k - k, if \alpha \in ( - 1, 0),
1, if \alpha = 0,
k\mu (\alpha ), if \alpha \in (0, 1).
(4.6)
Besides we have \bigm| \bigm| p1k(1)\bigm| \bigm| \leq c1k
4, (4.7)
where constants c, c1 do not depend on k.
Remark 4.1. Such technique demands to proof simultaniously whether the general summand of
the series tends to zero or whether the series is absolutely convergent.
Example 4.1. Let us consider the example (2.2) with \alpha = - 1/2, A = 1, t = 2 which was
considered also in [19] and with the exact solution
u(2) = e2erfc(
\surd
2).
Note that in the case of the differential operator A = - \partial 2
\partial x2
this example deals with a parabolic
problem in the language of PDEs.
The Cayley transform method leads to the approximate N -terms representation
N
u(2) = 1 +
N\sum
k=1
2 - kp
- 1/2
k (
\surd
2), q = A(1 +A) - 1 =
1
2
.
The numerical values of the absolute error \delta N (2) =
\bigm| \bigm| \bigm| u(2) - N
u(2)
\bigm| \bigm| \bigm| for Example 4.1 are given in
Table 4.1.
Example 4.2. Now let us consider problem (4.1) with u0 = 1, \alpha = 1/2, A = 1, q = 1/2. The
solution is the Mittag-Leffler function
u(t) = E3/2
\bigl(
- t3/2
\bigr)
=
\infty \sum
j=0
\bigl(
- t3/2
\bigr) j 1
\Gamma (1 + 3j/2)
. (4.8)
For t = 1 we have
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION . . . 631
Table 4.2
N
N
\delta (1)
8 1.22 \cdot 10 - 9
16 8.460928 \cdot 10 - 6
32 3.814396 \cdot 10 - 11
64 1.856974 \cdot 10 - 20
Table 4.3
N
N
\delta (1)
8 0.7683484749
16 0.2696535165
32 0.930343601 \cdot 10 - 1
64 0.84225987 \cdot 10 - 4
u(1) = E3/2 ( - 1) =
e
3
+
2e - 1/2
3
\mathrm{c}\mathrm{o}\mathrm{s}
\Biggl( \surd
3
2
\Biggr)
- 4
3
\surd
\pi
hypergeom
\biggl(
[1],
\biggl[
5
6
,
7
6
,
9
6
\biggr]
,
1
27
\biggr)
=
= .396629365318088084491614 . . . .
The exact representation due to the Cayley transform method (the power series in q) is
u(1) = E3/2 ( - 1) =
\infty \sum
k=0
\biggl(
1
2
\biggr) k
p
1/2
k (1)
and the N -term approximation is
N
u(1) =
N\sum
k=0
\biggl(
1
2
\biggr) k
p
1/2
k (1).
The numerical values of the absolute error
N
\delta (1) =
\bigm| \bigm| \bigm| u(1) - N
u(1)
\bigm| \bigm| \bigm|
are given by Table 4.2.
Example 4.3. Now let us consider the two-dimensional problem (4.1) with the unbounded opera-
tor A = - d2
dx2
and t = 1, \alpha = 1/2, u0 = \mathrm{s}\mathrm{i}\mathrm{n}(\pi x), x =
1
2
and the homogeneous Dirichlet boundary
conditions. In this case formula (4.8) takes the form
E3/2
\biggl(
d2
dx2
\biggr)
\mathrm{s}\mathrm{i}\mathrm{n}(\pi x)
\bigm| \bigm| \bigm| \bigm|
x=1/2
=
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
632 I. P. GAVRILYUK, V. L. MAKAROV
=
e\pi
4/3
3
+
2e - \pi 4/3/2
3
\mathrm{c}\mathrm{o}\mathrm{s}
\Biggl( \surd
3\pi 4/3
2
\Biggr)
- 4\pi 3/2
3
hypergeom
\biggl(
[1],
\biggl[
5
6
,
7
6
,
9
6
\biggr]
,
\pi 4
27
\biggr)
=
= - .1152743484427076753630 . . . .
Our algorithm for t = 1, x = 1/2 provides numerical results given by Table 4.3.
5. Accuracy estimates of the Cayley transform method. We consider the case of the Hilbert
space and of the positive definite operator A and let us estimate the accuracy of the application of
the Cayley transform approximation of the Mittag-Leffler function to an vector u0 from the domain
of an operator A\sigma .
It means that we assume the initial vector u0 to be such that\bigm\| \bigm\| A\sigma u0
\bigm\| \bigm\| < \infty .
Then we obtain: 1) in the case \alpha \in ( - 1, 0)
\bigm\| \bigm\| \bigm\| E1+\alpha
\bigl(
- At1+\alpha
\bigr)
-
N
E1+\alpha
\bigl(
- At1+\alpha
\bigr) \bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
qkp\alpha k (t
1+\alpha )u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
\infty \int
\gamma
F (\lambda )p\alpha k (t
1+\alpha )dE\lambda A
\sigma u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ,
where F (\lambda ) = \lambda - \sigma
\biggl(
\lambda
1 + \lambda
\biggr) k
. The function F (\lambda ) arrives its maximum Fmax = \scrO (k - \sigma ) at the
point \lambda max =
k - \sigma
\sigma
:
F \prime (\lambda ) = - \sigma \lambda - \sigma - 1
\biggl(
\lambda
1 + \lambda
\biggr) k
+ k\lambda - \sigma
\biggl(
\lambda
1 + \lambda
\biggr) k - 1 1
(1 + \lambda )2
,
\lambda - \sigma - 1
\biggl(
\lambda
1 + \lambda
\biggr) k - 1\biggl(
- \sigma
\lambda
1 + \lambda
+ \lambda k
1
(1 + \lambda )2
\biggr)
= 0.
Taking into account Lemma 4.2, we obtain
\bigm\| \bigm\| \bigm\| E1+\alpha ( - At1+\alpha )u0 -
N
E1+\alpha ( - At1+\alpha )u0
\bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
qkp\alpha k (t
1+\alpha )u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
\infty \int
\gamma
\lambda - \sigma
\biggl(
\lambda
1 + \lambda
\biggr) k
p\alpha k (t
1+\alpha )dE\lambda A
\sigma u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq c
\infty \sum
k=N+1
k - k - \sigma +\varepsilon +1k - 1 - \varepsilon \| A\sigma u0\| \leq cN - N - \sigma +\varepsilon \| A\sigma u0\| , (5.1)
where c is a constant independent of N and \varepsilon is an arbitrarily small positive number.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
APPROXIMATIONS OF THE MITTAG-LEFFLER OPERATOR FUNCTION . . . 633
2) Analogously for \alpha = 0 we have\bigm\| \bigm\| \bigm\| E1( - At)u0 -
N
E1( - At)u0
\bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
qkp\alpha k (t)u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
\infty \int
\gamma
\lambda - \sigma
\biggl(
\lambda
1 + \lambda
\biggr) k
p0k(t)dE\lambda A
\sigma u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq c
\infty \sum
k=N+1
k - \sigma +\varepsilon +1k - 1 - \varepsilon \| A\sigma u0\| \leq cN - \sigma +1+\varepsilon \| A\sigma u0\| . (5.2)
3) If \alpha \in (0, 1), then using estimate (4.6) we get\bigm\| \bigm\| \bigm\| E1+\alpha ( - At1+\alpha )u0 -
N
E1+\alpha ( - At1+\alpha )u0
\bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
qkp\alpha k (t
1+\alpha )u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
\infty \int
\gamma
\lambda - \sigma
\biggl(
\lambda
1 + \lambda
\biggr) k
p\alpha k (t
1+\alpha )dE\lambda A
\sigma u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq c
\infty \sum
k=N+1
k - \sigma +\mu (\alpha )+\varepsilon +1k - 1 - \varepsilon \| A\sigma u0\| \leq cN - \sigma +\mu (\alpha )+\varepsilon \| A\sigma u0\| . (5.3)
By using the estimate (4.7), we obtain analogously the following accuracy estimate for \alpha = 1 at
t = 1: \bigm\| \bigm\| \bigm\| E2( - At2)u0 -
N
E2( - At2)
\bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\infty \sum
k=N+1
qkp1k(t
2)u0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq c
\infty \sum
k=N+1
k - \sigma +4+\varepsilon +1k - 1 - \varepsilon \| A\sigma u0\| \leq cN - \sigma +5+\varepsilon \| A\sigma u0\| . (5.4)
Thus, we have proven the following assertion.
Theorem 5.1. Let A be a self-adjoint positive definite operator in a Hilbert space H and
q = A(I + A) - 1. Then for the accuracy of the Cayley transform method on a vector u0 \in D(A\sigma )
the estimates (5.1) – (5.3) or (5.4) hold true provided the corresponding assumptions from above are
fulfilled and \sigma is such that the powers of N are negative.
Remark 5.1. With \alpha = 1 and with the additional initial condition
du(0)
dt
= 0 one can see that
the Cayley transform method (without accuracy saturation) improves our algorithm from [7, 9] for
the corresponding initial value problem for an abstract hyperbolic equation with the exact solution
u(t) = \mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
A)u0.
References
1. N. I. Akhieser, I. M. Glazman, Theory of linear operators in Hilbert space, Pitman Adv. Publ. Program, London
(1980).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
634 I. P. GAVRILYUK, V. L. MAKAROV
2. 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).
3. A. Ashyralyev, A note on fractional derivatives and fractional powers of operators, J. Math. Anal. and Appl., 357,
232 – 236 (2009).
4. 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).
5. H. Bateman, A. Erdelyi, Higher transcendental functions, vol. 2, MC Graw-Hill Book Co., New York etc. (1988).
6. 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 (2019), p. 269 – 296.
7. 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).
8. I. Gavrilyuk, V. Makarov, V. Vasylyk, Exponentially convergent algorithms for abstract differential equations,
Springer, Basel AG (2011).
9. 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).
10. I. P. Gavrilyuk, Super exponentially convergent approximation to the solution of the Schrödinger equation in abstract
setting, Comput. Methods Appl. Math., 10, № 4, 345 – 358 (2010).
11. I. P. Gavrilyuk, An algorithmic representation of fractional powers of positive operators, Numer. Funct. Anal. and
Optim., 17, № 3-4, 293 – 305 (1996).
12. 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).
13. 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).
14. 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).
15. 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).
16. 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).
17. V. Havu, J. Malinen, The Cayley transform as a time discretization scheme, Numer. Funct. Anal. and Optim., 28,
№ 7-8, 825 – 851 (2007).
18. H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., 2011,
Article ID 298628, (2011); DOI:10.1155/2011/298628.
19. W. McLean, V. Thomèe, Numerical solution via Laplace transform of a fractional order evolution equation, J. Integral
Equat. and Appl., 22, № 1, 57 – 94 (2010).
20. G. M. Mittag-Leffler, Sur la nouvelle fonction E\alpha (z), C. R. Acad Sci., 137, 554 – 558 (1903).
21. G. H. Hardy, E. C. Titchmarsh, An integral equation, Proc. Phil. Soc., 28, № 2, 165 – 173 (1932).
22. 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).
23. H.-J. Seybold, R. Hilfer, Numerical algorithm for calculating the generalized Mittag-Leffler function, SIAM J. Numer.
Anal., 47, № 1, 69 – 88 (2008/2009).
24. P. K. Suetin, Classical orthogonal polynomials, Nauka, Moscow (1979) (in Russian).
25. G. Szegö, Orthogonal polynomials, Amer. Math. Soc., New York (1959).
Received 11.01.22
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
|
| id | umjimathkievua-article-7097 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:27Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a5/64a92ee51af37fd60fec4389da5d64a5.pdf |
| spelling | umjimathkievua-article-70972022-10-24T09:23:04Z Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time Gavrilyuk , I. P. Makarov, V. L. Gavrilyuk , I. P. Makarov, V. L. Makarov, V. L. the operator Mittag-Leffler function, the Laguerre-Cayley functions, abstract time-fractional evolution equation, the Cayley transform, $N$-term approximation, exponential accuracy 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 $\partial_t u +\partial_t^{-\alpha}A u=0, t&gt;0, u(0)=u_0$ 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 $N$. УДК 519.62 Запропоновано та проаналізовано ефективну дискретизацію операторної функції Міттаг-Леффлера&nbsp;&nbsp;$E_{1+\alpha } (-At^{1+\alpha })=\displaystyle \sum\nolimits _{k=0}^{\infty }\frac{(-At^{1+\alpha } )^{k} }{\Gamma (1+k(1+\alpha ))},$&nbsp;&nbsp;де $A$ – самоспряжений додатно визначений оператор.&nbsp;&nbsp;Ця функція має багато застосувань.&nbsp;&nbsp;Наприклад, вона дає операторний розв'язок еволюційної задачі $\partial_t u +\partial_t^{-\alpha}A u =0,$ $t&gt;0,$ $u(0)=u_0,$ з просторовим оператором $A$ та його дробовими похідними порядку $\alpha$ за часом (у сенсі Рімана–Ліувілля), тобто $u(t)=E_{1+\alpha} \left(-At^{1+\alpha } \right) u_{0}.$&nbsp;&nbsp;Використано метод перетворень Келі, який дозволяє рекурсивно відокремити змінні та подати функцію Міттаг-Леффлера у вигляді нескінченного ряду добутків функцій Лагерра–Келі щодо змінної часу (тобто поліномів від $t^{1+\alpha}$) та степенів перетворень Келі просторового оператора.&nbsp;&nbsp;Наближення задається скінченним відрізком ряду, що складається з $N$ доданків.&nbsp;&nbsp;Вивчено точність цього наближення в залежності від $\alpha$ та $N.$ Institute of Mathematics, NAS of Ukraine 2022-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7097 10.37863/umzh.v74i5.7097 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 5 (2022); 620 - 634 Український математичний журнал; Том 74 № 5 (2022); 620 - 634 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7097/9245 Copyright (c) 2022 Володимир Леонідович Макаров |
| spellingShingle | Gavrilyuk , I. P. Makarov, V. L. Gavrilyuk , I. P. Makarov, V. L. Makarov, V. L. Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| title | Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| title_alt | Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| title_full | Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| title_fullStr | Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| title_full_unstemmed | Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| title_short | Approximations of the Mittag-Leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| title_sort | approximations of the mittag-leffler operator function with exponential accuracy and their application to solving of evolution equations with fractional derivative in time |
| topic_facet | the operator Mittag-Leffler function the Laguerre-Cayley functions abstract time-fractional evolution equation the Cayley transform $N$-term approximation exponential accuracy the operator Mittag-Leffler function the Laguerre-Cayley functions abstract time-fractional evolution equation the Cayley transform $N$-term approximation exponential accuracy |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7097 |
| work_keys_str_mv | AT gavrilyukip approximationsofthemittagleffleroperatorfunctionwithexponentialaccuracyandtheirapplicationtosolvingofevolutionequationswithfractionalderivativeintime AT makarovvl approximationsofthemittagleffleroperatorfunctionwithexponentialaccuracyandtheirapplicationtosolvingofevolutionequationswithfractionalderivativeintime AT gavrilyukip approximationsofthemittagleffleroperatorfunctionwithexponentialaccuracyandtheirapplicationtosolvingofevolutionequationswithfractionalderivativeintime AT makarovvl approximationsofthemittagleffleroperatorfunctionwithexponentialaccuracyandtheirapplicationtosolvingofevolutionequationswithfractionalderivativeintime AT makarovvl approximationsofthemittagleffleroperatorfunctionwithexponentialaccuracyandtheirapplicationtosolvingofevolutionequationswithfractionalderivativeintime |