Abstract
Sequential generalized fractional Riemann–Liouville derivatives are introduced as composites of distributional derivatives on the right half axis and partially defined operators, called Dirac-function removers, that remove the component of singleton support at the origin of distributions that are of order zero on a neighborhood of the origin. The concept of Dirac-function removers allows to formulate generalized initial value problems with less restrictions on the orders and types than previous approaches to sequential fractional derivatives. The well-posedness of these initial value problems and the structure of their solutions are studied.
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1 Introduction
An important issue in applications of fractional differential equations are initial conditions, as emphasized already in [8, 9, p.115] or [10] for fractional relaxation and fractional diffusion. Derivatives \( \mathop { {} {\text {D}}^{ \alpha , \beta }_{0+} } \) of fractional order \( \alpha \) and type \( \beta \) were introduced precisely because their type \( \beta \) parametrizes different types of initial conditions.
Most investigations of fractional initial value problems were until recently concerned with the simplest fractional derivatives \( \mathop { {} {\text {D}}^{ \alpha ,0}_{0+} } \) of type 0 (Riemann–Liouville) or \( \mathop { {} {\text {D}}^{ \alpha ,1}_{0+} } \) of type 1 (Liouville–Caputo) [7, 12, 16, 22, 31]. A number of recent works considered initial value problems for sequential fractional derivatives [3, 24, 25, 30] thereby continuing the classical investigations in [5] and [31, Sec. 42.2]. In [3, 24, 25] generalized fractional derivatives of various types were studied all of which were based on the standard kernel \( t^{ \alpha - 1 }/ {\Gamma } ( \alpha ) \). Other recent works studied generalized fractional derivatives where the kernel \( t^{ \alpha - 1 }/ {\Gamma } ( \alpha ) \) is replaced by a Sonine kernel [19, 20, 26, 27, 29]. Riemann–Liouville derivatives and their generalizations \( \mathop { {} {\text {D}}^{ \alpha , \beta }_{0+} } \) have \( \beta \)-dependent domains and null spaces, but they all coincide on a specific complement of the null space of \( \mathop { {} {\text {D}}^{ \alpha ,1}_{0+} } \) [10, Sec.7]. Exactly on this complement they also coincide with the causal distributional fractional derivative \( {\text {D}}^{ \alpha }_+ \) of order \( \alpha \) defined as [32]
where \( {\mathscr {D}} ' _{0+} \) is the space of distributions supported on the right half axis and \( Y_{ \alpha } \in {\mathscr {D}} ' _{0+} \) are causal power distributions that satisfy \( Y_{ \alpha } ( t ) = t^{ \alpha - 1 }/ {\Gamma } ( \alpha ) \) for \( t > 0 \) (see Equation (2.7)), provided the function spaces for \( \mathop { {} {\text {D}}^{ \alpha , \beta }_{0+} } \) are canonically identified via restriction and zero extrapolation. Motivated by this observation and the recent research activity in sequential fractional derivatives, the main purpose of this work is to generalize and unify the theory of fractional initial value problems that involve several sequential derivatives.
Distributional convolution operators on the full axis modified by certain partially defined operators will be used in this work to reinterpret sequential fractional derivatives
on the right half axis where \(n\in \mathbb {N}_0\). Explicitly, composite operators of the form
with \({\text {D}}^{ \alpha }_+\) from (1.1) are investigated, where each first-order derivative in Eq. (1.2) gives rise to an operator \( {\text {R}} \) in Eq. (1.3). In this representation the operator \( {\text {R}} \) is a partially defined operator on \( {\mathscr {D}} ' _{0+} \) that is called \( {\updelta } \)-eliminator, because it removes the \( {\updelta } \)-part of a distribution at the origin.
Given a generalized sequential fractional derivative of the form (1.3), it can be rewritten in the normal form
where \( \alpha = \alpha _0+ \alpha _1+\ldots + \alpha _n \in \mathbb { R } \) is its fractional order, \((\gamma _1, \dots , \gamma _m )\) is called its sequential type, and \({\text {R}} ^{ \gamma } := {\text {D}}^{ -\gamma }_+ \circ {\text {R}}\circ {\text {D}}^{\gamma }_+ \) is called \(Y_{\gamma }\)-eliminator. Linear combinations of generalized sequential fractional derivatives can then be written in the form
where the operators \( D_l \) are linear combinations of distributional fractional derivatives \( {\text {D}}^{ \alpha }_+ \), \( \alpha \in \mathbb { R } \) (Theorem 2, p. 21). Our Theorem 3 on p. 23 derives a set of basis functions \( K_1, \dots , K_m \) such that every distribution from the null space of D is a linear combination of the \( K_1, \dots , K_m \). Ranges and maximal domains such that D becomes a bijective operator are calculated in Theorem 4, p. 24. If a distribution g belongs to the range of D, then it is an admissible inhomogeneity for the equation \( D f = g \). And a distribution f with \(Df=0\) is a linear combination of the basis functions \( K_1, \dots , K_m \) that solves an initial value problem with initial values obtained by applying the operators \( {\text {V}} \circ {\text {D}}^{ \gamma _l}_+\circ {\text {R}} ^{ \gamma _{l-1} } \circ \cdots \circ {\text {R}} ^{ \gamma _1 } \) to \(K_k\) for \(k,l=1,\dots ,m\), where the \(\delta \)-value operator \( {\text {V}} \) extracts the coefficient multiplying a \(\delta \)-distribution.
While Theorem 3 provides a useful result, a complete characterization of the null space of D requires an analysis of the value and well defined-ness of \( {\text {V}} \circ {\text {D}}^{ \gamma _l}_+\circ {\text {R}} ^{ \gamma _{l-1} } \circ \cdots \circ {\text {R}} ^{ \gamma _1 } (K)\) with K a linear combination of the \( K_1, \dots , K_m \). In the case where D is a linear combination of sequential derivatives with distinct orders Theorem 5, p. 25, gives a helpful simplification for this characterization. This leads to a generalization of the existence and uniqueness results from [3] and [5, Theorem 4]. In the general case however, a complete characterization is quite complicated. For the case with only two sequential types, \( \gamma _1 \) and \( \gamma _2 \), the null space of D is fully characterized here in Section 5.4 below.
Analogous to the generalized fractional derivatives from [19, 20, 29] the operators (1.4) can be generalized to \(\mathrm { C } _K \circ {\text {R}}^{\gamma _n}\circ \cdots \circ {\text {R}}^{\gamma _1 }\) where \( \mathrm { C } _K \) denotes the convolution operator \( f \mapsto K \mathbin {*} f \) on \({\mathscr {D}}'_{0+}\) with convolution kernel \( K \in \mathscr {D}'_{0+} \). Theorems 2, 3 and 4 will be proved for this more general class of operators.
Section 2 summarizes some basic mathematical notations on causal distributional convolution operators, the convolution field generated by causal power distributions [17] and partially defined linear operators. Section 3 studies coefficient operators, projectors and eliminators and their application to certain series of distributions. The generalized sequential fractional derivatives are introduced in Section 4. Their fundamental properties are established and their relation to classical Fractional Calculus operators is elucidated. In the final Section 5, the kernels and maximal injective domains of linear combinations of sequential derivatives are studied.
2 Preliminaries and notations
Subsection 2.1 recalls some basic properties of convolution of distributions with support on the right half axis. Subsection 2.2 summarizes some results from [17] about the field of convolution operators that arises naturally from L. Schwartz’ approach to Fractional Calculus. Subsection 2.3 summarizes definitions for partially defined linear operators and forms.
2.1 Convolution of distributions with support on the right half axis
The following describes some important properties of convolution of distributions with support contained in the right half axis. The details can be found in [32, Chap. IV, §5] or [15]. Convolution of distributions is a bilinear continuous operation on
the space of causal distributions. In other words, \( \mathscr { D } ' _{0+} \) is a convolution algebra with continuous convolution operation. It follows that any distribution \( U \in \mathscr { D } ' _{0+} \) gives rise to a continuous linear translation-invariant endomorphism by means of the definition
Conversely, any continuous linear translation-invariant endomorphism of \( \mathscr { D } ' _{0+} \) is a convolution operator \( \mathrm { C } _U \) for some kernel distribution \( U \in \mathscr { D }'_{0+}\).
The associative law for \((\mathscr {D}'_{0+}, \mathbin {*})\) entails the composition rule
In particular, the convolution operator \( \mathrm { C } _{ {\updelta } } \) coincides with the identity operator \( {\text {E}} \). A convolution operator \( \mathrm { C } _U \) with \( U \in \mathscr { D } ' _{0+} \) is continuously invertible if and only if there exists a distribution \( V \in \mathscr { D } ' _{0+} \) such that \( U \mathbin {*} V = {\updelta } \), where \( {\updelta } \) denotes the “Dirac-function”. Note, that V is unique because the convolution algebra \( \mathscr { D } ' _{0+} \) has no zero divisors.
A sequence \( ( f_n ) \subseteq \mathscr { D } ' _{0+} \) is called absolutely summable if and only if
Summation of sequences is compatible with convolution:
Lemma 1
Let \( ( f_n ), ( g_n ) \subseteq \mathscr { D } ' _{0+} \) be absolutely summable sequences. Then the double sequence \( ( f_n \mathbin {*} g_m ) \) is absolutely summable as well and
Proof
The space \(\mathscr {D}'_{0+}\) is complete. In \( \mathscr { D } ' _{0+} \) sequences converge weakly if and only if they converge strongly. Thus, the lemma follows from continuity of convolution [32, Thm. XIII]. \(\square \)
An immediate consequence of Lemma 1 is the following.
Lemma 2
Let \( f, g \in \mathscr { D } ' _{0+} \) such that \( f = {\updelta } - g \). If the expression
is well defined as an absolutely convergent series in \( \mathscr { D } ' _{0+} \), then \( f \mathbin {*} h = {\updelta } \).
2.2 Causal distributional fractional calculus
In his book [32] L. Schwartz considered fractional integrals and derivatives, \( {\text {I}}^{ \alpha }_+ \) and \( {\text {D}}^{ \alpha }_+ \), with orders \( \alpha \in \mathbb { C } \), as convolution operators with distributional kernels operating on the space \( \mathscr { D } ' _+ \) of distributions with support bounded on the left. In this work their restrictions to \( \mathscr { D } ' _{0+} \) are used. That is, one defines \( {\text {I}}^{ \alpha }_+ := \mathrm { C } _{ Y_{ \alpha } } \) and \( {\text {D}}^{ \alpha }_+ := \mathrm { C } _{ Y_{ - \alpha } } \) with the kernels \( Y_{ \alpha } \in \mathscr { D } ' _{0+} \) defined as
Due to Equation (2.3) the semigroup property \( Y_{ \alpha } \mathbin {*} Y_{ \beta } = Y_{ \alpha + \beta } \) automatically translates to the well known index laws
The operators \( {\text {I}}^{ \alpha }_+ \) and \( {\text {D}}^{ \alpha }_+ \) are continuous, linear and inverse to each other.
The distributions \( Y_{ \alpha } \) with orders \( \alpha \in \mathbb { R } \) generate a subalgebra \( \mathscr { A } [ Y_{ \mathbb { R } } ] \) of \( \mathscr { D } ' _{0+} \). The quotient field \( \mathscr { Q } [ Y_{ \mathbb { R } } ] \) of \( \mathscr { A } [ Y_{ \mathbb { R } } ] \) can be realized as a subalgebra of \( \mathscr { D } ' _{0+} \) [17]. The non-zero elements of \( \mathscr { Q } [ Y_{ \mathbb { R } } ] \) were used in [17] to define translation-invariant linear systems that correspond to fractional differential equations. An explicit description of the quotient field \( \mathscr { Q } [ Y_{ \mathbb { R } } ] \) was obtained from Lemma 2. It was found [17, Sec. 3] that the distribution
with \( n \in \mathbb { N } _0 \), \( 0< \alpha _1< \cdots < \alpha _n \) and \( \lambda _1, \dots , \lambda _n \in \mathbb { C } \) possesses a unique convolutional inverse given by
Moreover, using the notation of convolution quotients
the set \( \mathscr { Q } [ Y_{ \mathbb { R } } ] \setminus \{ 0 \} \) coincides with the set of convolution quotients
with uniquely determined numbers \( \alpha _1< \cdots < \alpha _n \), \( 0< \beta _1< \cdots < \beta _m \), \( \lambda _k, \mu _l \in \mathbb { C } ^\times \) and \( n, m \in \mathbb { N } _0 \). Maximal domains for the convolution operators defined by the kernels (2.12) were discussed in [17, Sec. 7]. However, for the purposes of this work it suffices to consider them as convolution operators acting on \( \mathscr { D } ' _{0+} \).
Remark 1
The presented approach to Fractional Calculus resembles operational calculus approaches [13, 23, 28] in the sense that a quotient field construction is used in both cases. However, the restriction to convolution quotients of a certain class of special distributions, whose inverses are well understood distributions (see [17]), has the advantage that the concrete interpretation of the abstract quotients is known very early on.
2.3 Partially defined linear operators and forms
A partially defined linear operator \(A: {\text {dom}} A \subseteq X \rightarrow X\) is a linear operator \({A: {\text {dom}} A\rightarrow X}\) with \( {\text {dom}} A \subseteq X \) a linear subspace. Similarly, a partially defined linear form \( L :{\text {dom}} L \subseteq X \rightarrow \mathbb { K } \) is a linear form \( L :{\text {dom}} L \rightarrow X \) with \( {\text {dom}} L \subseteq X \) a linear subspace. The following recalls basic operations on partially defined linear operators and forms.
Let A, B partially defined linear operators on X. One denotes
The sum \( A + B \) of A and B is defined as
The scalar multiple \( \lambda \cdot A \) with \( \lambda \in \mathbb { K } \) is defined as
The composition \( A \circ B \) of A and B is defined as
Addition (2.14) respectively composition (2.16) defines a semigroup with neutral element given by the zero operator \( {\mathbf {0}} \) with \( {\text {dom}} {\mathbf {0}} = X \) respectively the identity operator \( {\text {E}} \) with \( {\text {dom}} {\text {E}} = X \). The distributive laws read
Equality holds in Equation (2.17b) as well if \( {\text {dom}} C = X \).
Addition of partially defined linear forms is defined analogous to (2.14). The composition \( L \circ A \) of a partially defined linear form L and a partially defined linear operator A is defined analogous to (2.16). This composition defines a semigroup operation of the partially defined linear operators on the partially defined linear forms on X.
The operations for partially defined operators and forms exhibit strange behaviour. For instance, \( A - A = {\mathbf {0}} _{ {\text {dom}} A } \subseteq {\mathbf {0}} \) and \( 0 \cdot A = {\mathbf {0}} _{ {\text {dom}} A } \subseteq {\mathbf {0}} \), but equality holds if and only if \( {\text {dom}} A = X \). For every linear subspace \( Y \subseteq X \) one obtains a linear space of partially defined linear operators \( A :{\text {dom}} A \subseteq X \rightarrow X \) with \( {\text {dom}} A = Y \), but the set of all partially defined linear operators or linear forms does not define a linear space.
3 Eliminators, initial values and series expansions
This section introduces and studies \( Y_{\alpha }\)-coefficient operators, -projectors and -eliminators. These are defined for \( \alpha = 0 \) in Subsection 3.1. Via fractional integration and differentiation the definition is transported to general \( \alpha \) in Subsection 3.2. Then, in Subsection 3.3, the operators are applied to represent the coefficients of a series of distributions \( Y_{ \alpha } \).
3.1 The \({\updelta } \)-limit operator, the \( {\updelta }\)-projector and the \( {\updelta } \)-eliminator
Roughly speaking, the \( {\updelta } \)-elminator, which is denoted by \( {\text {R}} \), removes the “\( {\updelta } \)-component” of a distribution. This “surgery” does not go well without careful preparations. Therefore, the \( {\updelta } \)-eliminator needs to be introduced as a partially defined operator on \( \mathscr { D } ' _{0+} \).
Let \( I \subseteq \mathbb { R } \) be open. The space of distributions of order zero on I, denoted by \( \mathscr { D } ' {}^{ 0 } ( I ) \), is the set of distributions \( \mu \in \mathscr { D } ' ( I ) \) satisfying
where \( \mathscr { D } _J = \left\{ \, \varphi \in \mathscr { D } ( \mathbb { R } ) \,;\, {\text { supp }} \varphi \subseteq J \,\right\} \). Any distribution \( f \in \mathscr { D } ' {}^0( I ) \) can be identified with a Radon measure on I, see [14, Def. 2.1.1] or [15, 4, §4].
Definition 1
Define the space of distributions
The \( {\updelta } \)-value operator is the partially defined linear form \( {\text {V}} \) given by
where the right hand side of (3.3b) is defined for \( \varepsilon > 0 \) small enough by
for any sequence 
such that 
for \( n \rightarrow \infty \).
The \( {\updelta } \)-projector is the partially defined operator
The \( {\updelta } \)-eliminator is the complementary projection of \( {\text {P}} \), that is
The \( {\updelta } \)-residual space of causal distributions is the range of the \( {\updelta } \)-remover
It is immediate from the definition of the space \( \mathscr { D } ' _{ 0+, {\updelta } } \) that, for any \( \varepsilon > 0 \) and any \( f \in \mathscr { D } ' _{ 0+, {\updelta } } \), the distribution f can be written as
The \( {\updelta } \)-coefficient can be defined equivalently as \( {\text {V}} f = \mu ( \{ 0 \} ) \) for \( f = \mu + g \) with \( \mu , g \) as in (3.6), due to the continuity of Radon measures [1, 6]. Here \( \mu ( \{ 0 \} ) \) denotes the \( \mu \)-measure of the Borel set \( \{ 0 \} \subseteq \mathbb { R } \).
Proposition 1
The space \(\mathscr { D }'_{ 0+, {\updelta }}\) is a unitary convolution subalgebra of \(\mathscr {D}'_{0+}\), that is \( \mathscr { D } ' _{ 0+, {\updelta } } \mathbin {*} \mathscr { D } ' _{ 0+, {\updelta } } = \mathscr { D } ' _{ 0+, {\updelta } } \).
Proof
The proposition is immediate from Equation (3.6) and the following three facts: It holds \( \inf {\text { supp }} ( f \mathbin {*} g ) \ge \inf {\text { supp }} f + \inf {\text { supp }} g \) for all \( f, g \in \mathscr { D } ' _{0+} \), the set \( \mathscr { D } ' _{0+} \cap \mathscr { D } ' {}^{ 0 } \) is closed with respect to convolution, and \({\updelta }\in \mathscr {D}'_{0+,{\updelta }}\). \(\square \)
Proposition 2
The operators \( {\text {P}} \) and \( {\text {R}} \) are complementary linear projection operators on \( \mathscr { D } ' _{ 0+, {\updelta } } \), that is \( {\text {P}} \circ {\text {P}} = {\text {P}} \), \( {\text {R}} \circ {\text {R}} = {\text {R}} \), \( {\text {R}} \circ {\text {P}} = {\text {P}} \circ {\text {R}} = 0 \) and \( {\text {P}} + {\text {R}} = {\text {E}} _{ \mathscr { D } ' _{ 0+, {\updelta } } } \). Every distribution \( f \in \mathscr { D } ' _{ 0+, {\updelta } } \) has a unique representation
where a and g are given by \( a = {\text {V}} f \) and \( g = {\text {R}} f \). In addition,
Proof
It is immediate from the Definitions (3.3) that \( {\text {P}} { {\updelta } } = {\updelta } \). Thus, \( {\text {P}} \circ {\text {P}} = {\text {P}} \) and the first part of the proposition follows from basic linear algebra. For Equation (3.8), recall the remarks below Equations (3.1) and (3.6). Then use that
for all \( \mu , \nu \in \mathscr { D } ' {}^{ 0 } \cap \mathscr { D } ' _{0+} \).
The latter follows from Equation (1) in [2, Ch. VIII, §1, No. 1]. \(\square \)
Equation (3.8) means, that the operator \({\text {V}} \) defines an “augmentation” of the convolution algebra \( \mathscr { D } ' _{ 0+, {\updelta } } \). That is, the operator \( {\text {V}} \) defines a linear homomorphism \({\text {V}} :\mathscr { D }'_{ 0+, {\updelta } } \rightarrow \mathbb { C }\). Equation (3.8) implies that
In particular, 
is a non-unitary convolution subalgebra of \( \mathscr { D } ' _{ 0+, {\updelta } } \).
3.2 Generalized initial value operators, eliminators and projectors
Composing the operators \( {\text {V}} \), \( {\text {R}} \) and \( {\text {P}} \) with the distributional fractional integrals and derivatives from Subsection 2.2 yields operators that act on the \( Y_{ \gamma } \)-part of a distribution in an analogous way. This section studies these operators and, for the case of real orders, the generated algebra of partially defined linear operators.
Definition 2
Let \( \gamma \in \mathbb { C } \). The \( Y_{ \gamma } \)-coefficient operator, the \( Y_{ \gamma } \)-projector and the \( Y_{ \gamma } \)-eliminator are defined as the composite operators
Further, one defines the distribution spaces
where \( \mathscr { D } ' _{ 0+, {\updelta } } \) and 
are from eqs. (3.2) and (3.5).
It is immediate from the definitions, that the operators \( {\text {I}}^{ \gamma }_+ \) and \( {\text {D}}^{ \gamma }_+ \) induce bijections between spaces \( \mathscr { D } ' _{ \mathrm { Y } , \gamma } \) and 
of different orders. For instance, one has a bijection
The domains of the operators from Definition 2 are
Ranges and kernels of these operators are given by
The brackets \( \left\langle - \right\rangle _{ } \) denote the (complex) linear span.
There hold the composition and commutation rules
Analogous to the operators \( {\text {V}} \), \( {\text {P}} \) and \( {\text {R}} \), one has the relations
The operators \( {\text {P}} ^{ \gamma } \) and \( {\text {R}} ^{ \gamma } \) are complementary projections, just as \( {\text {P}} \) and \( {\text {R}} \), and a statement analogous to (3.7) holds. However, the operator \( {\text {V}} ^{ \gamma } \) does not define an augmentation. From Equation (2.8) and Equation (3.10), one obtains the convolution inclusions
Lemma 3
It holds 
.
Proof
It holds \( { {\text {V}} }\left( Y_{ \gamma } \mathbin {*} \mathscr { M } _{ 0+ } \right) = \{ 0 \} \) for \( \gamma \in \mathbb { H } \) due to \( Y_{ \gamma } \in L^{ 1 } _{ \mathrm {loc} } \). \(\square \)
Corollary 1
Let \( \gamma , \delta \in \mathbb { C } \). If \( \Re \gamma < \Re \delta \), then \( \left. {\text {P}} ^{ \gamma } \right| _{ \mathscr { D } ' _{ \mathrm { Y } , \delta } } = {\mathbf {0}} _{ \mathscr { D } ' _{ \mathrm { Y } , \delta } } \) and \( \left. {\text {R}} ^{ \gamma } \right| _{ \mathscr { D } ' _{ \mathrm { Y } , \delta } } = {\text {E}} _{ \mathscr { D } ' _{ \mathrm { Y } , \delta } } \).
Lemma 4
Every composition of the form
is equal to a composition of the form
where \( \delta _k = \gamma _{ \sigma ( k ) } \) for some function \( \sigma :\{ 1, \dots , m \} \rightarrow \{ 1, \dots , n \} \). For any partially defined linear operator on \( \mathscr { D } ' _{0+} \) of the form (3.20) the parameters \( \delta _1, \dots , \delta _m \), \( m \in \mathbb { N } _0 \) are uniquely determined.
Proof
Corollary 1 implies \( {\text {R}}^{\gamma } \circ {\text {R}}^{\delta } = {\text {R}}^{\delta } \) whenever \( \delta \ge \gamma \). Using this repeatedly, the expression (3.19) can be reduced to an expression of the form (3.20) without changing the resulting operator. Because every set of the form \( \{ Y_{ \delta _1 } , \dots , Y_{ \delta _m } \} \) is linearly independent and the kernel of the operator from Equation (3.20) is equal to \( \left\langle Y_{ \delta _1 } , \dots , Y_{ \delta _m } \right\rangle _{ } \), it follows that the representation (3.20) is unique. \(\square \)
Definition 3
Let \( \gamma _1< \cdots < \gamma _n \) and \( n \in \mathbb { N } \). The \( Y_{ \gamma _1, \dots , \gamma _n } \)-eliminator and the \( Y_{ \gamma _1, \dots , \gamma _n } \)-projector are defined as
The \( Y_{ \gamma _n | \gamma _1, \dots , \gamma _{n-1} } \)-coefficient operator is defined as
The operators \({\text {R}} ^{ \gamma _1, \dots , \gamma _n } \), \({\text {P}} ^{ \gamma _1, \dots , \gamma _n } \) and \( {\text {V}} ^{ \gamma _n | \gamma _1, \dots , \gamma _{n-1} } \) have the joint domain
For kernels and ranges one has
Lemma 5
Let \( \gamma _1< \cdots < \gamma _n \) and \( n \in \mathbb { N } \). Then
In particular
with the convention \({\text {R}}^{\gamma _1, \dots ,\gamma _n}={\text {E}}\) for \(n=0\).
Proof
The distributive law (2.17a) and Eq. (3.17) imply Eq. (3.25b). Let \( k \in \{ 1, \dots , n \} \). Using (3.17) and (2.17a) again one obtains
Equation (3.25a) follows from a repeated application of (3.27). \(\square \)
Proposition 3
Let \( \gamma _1< \cdots < \gamma _n \), \( \delta _1< \cdots < \delta _m \) and \( n, m \in \mathbb { N } \). Then, with \( l = \min \left\{ \, k = 1, \dots , n \,;\, \gamma _k > \delta _m \,\right\} \), it holds
Proof
One applies the cancelation rule \( {\text {R}} ^{ \gamma } \circ {\text {R}} ^{ \delta } = {\text {R}} ^{ \delta } \) for \( \delta \ge \gamma \). \(\square \)
3.3 Series expansions in causal power distributions
Absolutely convergent series over expressions \( c_ \gamma Y_{ \gamma } \), with \( c_ \gamma \in \mathbb { C } \) and \( \gamma \in \mathbb { R } \), are now considered as a kind of fractional Taylor series expansion. Theorem 1 shows how to extract the coefficients \( c_ \gamma \) using coefficient operators \( {\text {V}} ^{ \gamma } \) and composite eliminators \( {\text {R}} ^{ \gamma _1, \dots , \gamma _n } \).
Definition 4
Let \( U \in \mathscr { D } ' _{0+} \) and \( \gamma \in \mathbb { R }\). Then U is said to possess a non-trivial \( Y_{\gamma } \)-part if and only if
The statement \( {\text {O}}\left( U \right) = \gamma \) will be used as equivalent notation for conditions (3.29). The notation \( {\text {O}}\left( U \right) \in \mathbb { R } \) means that there exists \( \gamma \in \mathbb { R } \), such that conditions (3.29) hold true.
Let \( k \in \mathbb { N } _0 \). One defines \( {\text {O}}_{ k }\left( U \right) := {\text {O}}\left( U_k \right) \), where \( U_k \) is recursively defined by the initial condition \( U_0 := U \) and
whenever \( {\text {O}}\left( U_0 \right) , \dots , {\text {O}}\left( U_k \right) \in \mathbb { R } \).
Lemma 6
Let \( ( \gamma _n ) \) be a sequence of strictly increasing positive numbers and \( ( c_n ) \) a sequence of complex numbers. If the series \( \sum ^ \infty _{n=0} c_n Y_{ \gamma _n } \) converges absolutely in \( \mathscr { D } ' \), then it converges absolutely in .
Proof
Let \( K \subseteq \mathbb { R } \) compact, \( \chi \in \mathscr { D } \), \( \chi \ge 0 \) and \( \chi ( K ) \subseteq \{ 1 \} \). Then
Let 
be compact. There exists \( \varepsilon > 0 \) such that 
and a function 
, \( \theta \ge 0 \) and \( \theta ( L ) \subseteq \{ 1 \} \). It holds
for all \( \gamma > 0 \) and \( m \in \mathbb { N } \) that satisfy \( m - \gamma \notin \mathbb { N } _0 \). (If \( m - \gamma \in \mathbb { N } _0 \), then \( {\text {D}} ^m Y_{ \gamma } \) vanishes on 
.) \(\square \)
Corollary 2
If the series from Lemma 6 converges absolutely in \( \mathscr { D } ' \), then its limit belongs to 
.
Theorem 1
Let \( ( \gamma _k ) \subseteq \mathbb { R } \) and \( ( c_k ) \subseteq \mathbb { C } \) such that \( ( \gamma _k ) \) is strictly increasing and the series \( \sum ^ \infty _{k=0} c_k Y_{ \gamma _k } \) converges absolutely to the limit U. Then \( {\text {O}}_{ k }\left( U \right) = \gamma _k \) and \( { {\text {V}} ^{ \gamma _k } \circ {\text {R}} ^{ \gamma _1, \dots , \gamma _{k-1} } }\left( U \right) = c_k \) for all \( k \in \mathbb { N } _0 \).
Proof
For all \( k \in \mathbb { N } _0 \) it holds \( U_k = \sum ^ \infty _{n=k} c_k Y_{ \gamma _n } = c_k Y_{ \gamma _k } + U_{k+1} \) with 
due to Corollary 2. \(\square \)
Proposition 4
Let \( U, V \in \mathscr { D } ' _{0+} \) such that \( {\text {O}}\left( U \right) \) and \( {\text {O}}\left( V \right) \) exist. Then \( {\text {O}}\left( U \mathbin {*} V \right) \) exists as well and \( {\text {O}}\left( U \mathbin {*} V \right) = {\text {O}}\left( U \right) + {\text {O}}\left( V \right) \).
Proof
Equation (3.18) implies
for all \( \gamma , \delta , c, d \in \mathbb { C } \), which proves the proposition. \(\square \)
Lemma 7
Let \( \alpha _1< \cdots < \alpha _n \), \( 0< \beta _1< \cdots < \beta _m \), \( \lambda _1, \dots , \lambda _n {\in } \mathbb { C } ^\times \), \( \mu _1, \dots , \mu _m {\in } \mathbb { C } ^\times \), \( m \in \mathbb { N } _0 \) and \( n \in \mathbb { N } \). Then
Proof
The limit \( U^{ \mathbin {*} -1} \) of the series in (2.10) satisfies \({\text {O}}\left( U^{ \mathbin {*} -1} \right) =0\) and one has \({\text {O}}\left( \lambda _1 Y_{ \alpha _1 } + \dots + \lambda _n Y_{ \alpha _n } \right) = \alpha _1 \) according to Theorem 1. Thus, Equation (3.33) follows follows from Proposition 4. \(\square \)
4 Generalized sequential fractional derivatives
The generalized sequential fractional derivatives are introduced and their basic properties summarized in Subsection 4.1. Then, in Subsection 4.2, restrictions and extrapolations of measures and distributions are defined. Subsection 4.3 discusses examples of generalized sequential fractional derivatives and their relations to other Fractional Calculus operators on the right half axis from the literature.
4.1 Definition and fundamental properties
Generalized sequential fractional derivatives are introduced as catenations of a distributional fractional derivative and a composite of eliminators.
Definition 5
Let \( \alpha \in \mathbb { R } \), \( \gamma _1< \cdots < \gamma _n \) and \( n\in \mathbb { N } \). The sequential fractional derivative of order \( \alpha \) and sequential type \( ( \gamma _1, \dots , \gamma _n ) \) is defined as
where \( {\text {D}}^{ \alpha | }_{0+} :={\text {D}}^{ \alpha }_+\) is used to link the notation for sequential fractional derivatives with that for simple fractional derivatives.
The operators defined in Equation (4.1) coincide with alternating compositions of eliminators and fractional derivatives with real orders. This means operators
Note, that \({\text {R}}\circ {\text {D}}^{\alpha }_+ \circ {\text {R}}={\text {D}}^{\alpha }_+ \circ {\text {R}}\) holds for all \( \alpha < 0 \), so that \( \alpha _1, \dots , \alpha _{n-1} > 0 \) may be assumed without loss of generality. The expression on the right hand side of Equation (4.1) constitutes a normal form of the operators from Equation (4.2). The advantage of the normal form from Equation (4.1) is that domain and kernel are given by
because the operator \( {\text {D}}^{ \alpha }_+ \) is bijective. Thus, domain and kernel are given by Equation (3.23) and (3.24a).
Another normal form is obtained when the operators \( {\text {D}}^{ \alpha }_+ \) and \( {\text {R}} ^{ \gamma _1, \dots , \gamma _n } \) are commuted. Let \( \alpha _0, \dots , \alpha _n \in \mathbb { R } \) with \( \alpha _1, \dots , \alpha _{n-1} > 0 \). Using the commutation rules (3.16) the relations between different normal forms can be derived. They read
with the parameters
for \( k = 1, \dots , n \). Using Equations (3.17) and (3.25a) one obtains
for \( \alpha \in \mathbb { R } \), \( \gamma _1< \cdots < \gamma _n \), \( n \in \mathbb { N } \).
Proposition 5
Let \( \alpha , \beta \in \mathbb { R } \), \( \gamma _1< \cdots < \gamma _n \), \( \delta _1< \cdots < \delta _m \) and \( m, n \in \mathbb { N } _0 \). Then the composition law
holds with \( l = \min \left\{ \, k \in 1, \dots , n \,;\, \gamma _k + \beta > \delta _m \,\right\} \).
Proof
The commutation rules from Eq. (3.16) give
and an application of Proposition 3 yields Equation (4.5). \(\square \)
Remark 2
The normal form (4.4c) exists as well for composite operators of the form \( {\text {R}} ^{ \gamma } \circ \mathop { \mathrm { C } _K } \) with \( \gamma \in \mathbb { R } \) and \( K \in \mathscr { Q } [ Y_{ \mathbb { R } } ] \). It holds
Equation (4.7) holds also when \( K \in \mathscr { D } ' _{ \mathrm { Y } , {\text {O}}\left( K \right) } \). However, even for \( K \in \mathscr { A } [ Y_{ \mathbb { R } } ] \) a similar description of composites of the form \( {\text {R}} ^{ \gamma _1, \dots , \gamma _n } \circ \mathop { \mathrm { C } _K } \) leads to a more complicated expression that involves multiple case distinctions. Let us remark, that for future studies of the operators \( \mathrm { C } _K \circ {\text {R}}^{ \beta _1, \dots , \beta _n} \), it seems plausible to generalize the eliminators as well.
4.2 Restriction and extrapolation of distributions
Let \( I \subseteq \mathbb { R } \) be an interval and \( J \subseteq \mathbb { R } \) an open interval. The restriction of a distribution \( f :\mathscr { D } ( \mathbb { R } ) \rightarrow \mathbb { C } \) to J is defined as \( \left. f \right| _{ J } := \left. f \right| _{ \mathscr { D } _J } :\mathscr { D } _J \rightarrow \mathbb { C } \) with \( \mathscr { D } _J = \left\{ \, \varphi \in \mathscr { D } \,;\, {\text { supp }} \varphi \subseteq J \,\right\} \). The restriction of a Radon measure \( \mu :\mathscr { K } ( \mathbb { R } ) \rightarrow \mathbb { C } \) to I is defined by \( \left. \mu \right| _{ I } ( \varphi ) := \mu ( \left. \varphi \right| ^{ \text {zero} }_{ \mathbb { R } } ) \) with \( \left. \varphi \right| ^{ \text {zero} }_{ \mathbb { R } } ( x ) \) equal to 0 for \( x \in I \) and equal to \( \varphi ( x ) \) for \( x \notin I \). Here, the Borel measurable function \( \left. \varphi \right| ^{ \text {zero} }_{ \mathbb { R } } \) is the zero extrapolation of \( \varphi \in \mathscr { K } ( I ) \).
The restriction of a distribution \( f \in \mathscr { D } ' _{ 0+, {\updelta } } \) to \( \mathbb { R } _{0+} \) is defined as the linear form \( f :\mathscr { D } ( \mathbb { R } _{0+} ) \rightarrow \mathbb { C } \) such that
where \( \mu \) and g are defined as in (3.6), \( \mathscr { D } ( \mathbb { R } _{0+} ) = \{\, \left. \varphi \right| _{ \mathbb { R } _{0+} } \,;\, \varphi \in \mathscr { D } ( \mathbb { R } ) \,\} \) and \( \chi \in \mathscr { E } ( \mathbb { R } _{0+} ) \) is such that \( \chi ( {\text { supp }} g ) \subseteq \{ 1 \} \) and for some \( \varepsilon > 0 \). This is well defined due to \( 0 \notin {\text { supp }} g \).
For \( I = \mathbb { R } _+ \), \( \alpha \in \mathbb { R } \), or, \( I = \mathbb { R } _{0+} \), \( \alpha \ge 0 \), define the space
with 
defined in Eq. (3.12).
Define the zero extrapolation of a Radon measure \( \mu \in \mathscr { K } ( I ) \) to \(\mathbb { R } \) as
The zero extrapolation \( \left. f \right| ^{ \text {zero} }_{ \mathbb { R } } \) of \( f \in \mathscr { D } ' _{ 0+, {\updelta } } ( \mathbb { R } _+ ) \) is defined as
where \( \mu \) and g are defined as in (3.6) and \( \theta \in \mathscr { E } ( \mathbb { R } _+ ) \) is such that \( \theta ^{-1}( 1 ) \supseteq {\text { supp }} \varphi \) and \( {\text { supp }} \theta \subseteq \mathbb { R } _+ \). Define the zero extrapolation of a distribution \( f \in \mathscr { D } ' _{ 0+, {\updelta } } ( \mathbb { R } _{0+} ) \) to \( \mathbb { R } \) as
with \( \mu \) resp. g as in the representation formula (3.6) and their zero extrapolations defined by Equations (4.10) resp. (4.11). The continuous constant extrapolation \( \left. f \right| ^{ \text {c.c.} }_{ \mathbb { R } } \) of f is defined as
where \( 1_{ \mathbb { R } _- } \) denotes the indicator function for the open left half axis.
Restricting a distribution from \(\mathscr { D }'_{ \mathrm { Y } , 0 } \) to \( \mathbb { R }_+ \) and extending to \( \mathbb { R } \) by zero afterwards has the same effect as applying the eliminator. Restricting to \( \mathbb { R } _{0+} \) and zero-extrapolating to \( \mathbb { R } \) has no effect. This can be expressed as
Because \( {\text {R}} \) reduces to the identity on 
, the map**s
define mutually inverse isomorphisms that restrict to isomorphmisms between \( \mathscr { D } ' _{ \mathrm { Y } , \alpha } \) and \( \mathscr { D } ' _{ \mathrm { Y } , \alpha } \) for \( \alpha > 0 \).
4.3 Examples
Numerous operators can be reinterpreted as generalized sequential fractional derivatives on subspaces of \( \mathscr { D } ' _{0+} \) in the context of restriction to the right half axis and extrapolation to the full real axis. This section collects some examples.
4.3.1 Distributional derivatives on the right half axis
The distributional derivative is well defined as an operator acting on distributions defined on the open right half axis \( \mathbb { R } _+ \). Using restriction and zero extrapolation operators the sequential derivative \( {\text {D}}^{ 1 | 1 }_{0+} \) can be related to the operator \( {\text {D}} :\mathscr { D } ' ( \mathbb { R } _+ ) \rightarrow \mathscr { D } ' ( \mathbb { R } _+ ) \) via
The sequential derivative \( {\text {D}}^{ 1 | 1 }_{0+} f \) of \( f \in \mathscr { D } ' _{ \mathrm { Y } , 1 } ( \mathbb { R } ) \) is a modification of the distributional derivative of f that ignores jumps at the origin.
According to (4.16b), this is equivalent to interpreting f as a distribution on the open right half axis and extrapolating by zero after the calculation of the derivative.
Unfortunately, the case of the closed right half axis is more involved, despite the existence of the isomorphism \( \mathscr { D } ' _{ \mathrm { Y } , 1 } ( \mathbb { R } _+ ) \rightarrow \mathscr { D } ' _{ \mathrm { Y } , 1 } ( \mathbb { R } _{0+} ) \). The reason is, that the definition of the derivative of a function \( f \in \mathscr { D } ' _{ \mathrm { Y } , 1 } ( \mathbb { R } _{0+} ) \) at the origin depends on its extrapolation to a neighborhood of the origin. In particular, it holds
for all \( f \in \mathscr { D } ' _{ \mathrm { Y } , 1 } ( \mathbb { R } _{0+} ) \). These relations reflect the fact that it is always necessary give a precise interpretation of the derivatives when a boundary is involved.
4.3.2 Riemann–Liouville fractional integrals
Riemann–Liouville fractional integrals with orders \( \alpha \in \mathbb { H } \), that act on functions on the closed or open right half axis, can be represented using the sequential derivatives \( {\text {D}}^{ - \alpha | }_{0+} = {\text {I}}^{ \alpha }_+ \). Note, that the Riemann–Liouville integral \( \mathop { {}_ { \mathrm { RL } } }{\text {I}}^{ \alpha }_{0+} \mu ( t ) \) of a measure \( \mu \in \mathscr { M } ( \mathbb { R } ) \) or \( \mu \in \mathscr { M } ( \mathbb { R } _+ ) \) can be defined, in the almost everywhere sense for the variable t, by applying the classical formula to \( \mu \). (In the latter case, the domains of integration must include the point zero.) This extends most common definitions of the Riemann–Liouville integral (compare [24, Remark 4.4]).
With these definitions one obtains the relations
However, the action of the sequential derivative \( {\text {D}}^{ - \alpha | }_{0+} = {\text {I}}^{ \alpha }_+ \) on functions \( f \in \mathscr { D } ' _{ \mathrm { Y } , - \alpha } \setminus \mathscr { D } ' _{ 0+, {\updelta } } \) can not be described using Riemann–Liouville integrals defined on the closed or open right half axis, because neither the corresponding restriction operators nor the corresponding extrapolation operators are well defined for such functions f.
4.3.3 Generalized Riemann–Liouville fractional derivatives [7]
The fractional derivatives of Riemann–Liouville and their generalizations can be reinterpreted by replacing the first order derivatives with \( {\text {D}}^{ 1 | 1 }_{0+} \) and \( \mathop { {}_ { \mathrm { RL } } }{\text {I}}^{1- \alpha }_{0+} \) with \( {\text {D}}^{ \alpha -1 | }_{0+} \). Specifically, the generalized fractional derivative \( \mathop { {} {\text {D}}^{ \alpha , \mu }_{0+} } \) of order \( \alpha \) and type \(\mu \) with 
, can be interpreted as a generalized sequential derivative
of order \( \alpha \) and sequential type \( \alpha +\mu - \alpha \mu \). The sequential reinterpretation \( \mathop { {} {\text {D}}^{ \alpha ;\mu }_{0+} } \) is distinguished notationally by a semicolon instead of a comma from the original operator \( \mathop { {} {\text {D}}^{ \alpha ,\mu }_{0+} } \). The domain of \( \mathop { {} {\text {D}}^{ \alpha ; \mu }_{0+} } \) is
With the notations from Equation (4.19) the well-known relation [7, p.434] between two derivatives of same order 
, but distinct types \( 0 \le \mu _1 < \mu _2 \le 1 \) reads
The relation is immediate from the fact that
according to Equation (4.4c), because the partially defined linear form \( {\text {V}} ^{ \alpha + \mu _1 ( 1 - \alpha ) } \) reduces to zero on the domain of partially defined linear form \( {\text {V}} ^{ \alpha + \mu _2 ( 1 - \alpha ) } \).
4.3.4 \(\varvec{n}\)th-level fractional derivatives [5, 24]
The sequential fractional derivatives from Definition 5 extend earlier definitions as studied in [5], or later ones, in [24]. In the following, the operators from Definition 5 are compared with the “nth-level derivatives” from Definition 3.6 in [24]. Similar remarks apply to the sequential derivatives given in [31]. For the purpose of a more convenient comparison the domains \( X^1_{nL} \) of nth-level derivatives are defined as spaces of functions on the whole real line. This definition becomes equal to the definition in Equation (3.41) from [24] when the functions are restricted to 
.
More precisely, let \( 0 < \alpha \le 1 \), \( \gamma = ( \gamma _1, \dots , \gamma _n ) \) such that \( \gamma _k \ge 0 \) and \( \alpha + s_k \le k \) for all \( k = 1, \dots , n \), with the notation \( s_k := \sum ^k_{i=1} \gamma _i \). The indices for the corresponding generalized sequential fractional derivatives are defined as \( \delta _{n-k+1} := \alpha + s_k - k + 1 \). These satisfy \( \delta _1< \dots < \delta _n \) if and only if \( \gamma _k < 1 \) for \( k = 2, \dots , n - 1 \). Further, it holds \( 0 < \delta _1 \) if and only if \( \alpha + s_n > n + 1 \).
Consider now, the space \( X^1_{nL,+}(\mathbb {R})\), defined as
The space \( X^1_{nL} \) from Definition 3.6 in [24] can be characterized as the set of restrictions 
with \( f \in X^1_{nL,+}( \mathbb { R } ) \) by using the following lemma.
Lemma 8
For every \( f \in \mathscr { D } ' _{ \mathrm { Y } , 1 } ( \mathbb { R } ) \) there holds the equivalence
The operator \( \mathop { {}_ { n \mathrm { L } } {\text {D}}^{\alpha , (\gamma )}_{0+} } \) with \({\text {dom}}\mathop { {}_ {n \mathrm { L }} {\text {D}}^{\alpha , (\gamma )}_{0+} } = X^1_{nL,+}(\mathbb {R})\) and the parameters \(\gamma = ( \gamma _1, \dots , \gamma _n ) \) can now be defined as in [24, Def. 3.6], but for the whole right half axis. With the comparison of partially defined operators from (2.13), one has
The advantage of the different parameterization from Definition 5 is, that the domain and the kernel of the operators \( {\text {D}}^{ \alpha | \delta _1, \dots , \delta _n }_{0+} \) can be deducted directly from the indices \( \delta _1, \dots , \delta _n \). Compare Equations (3.23), (3.24) and (4.3) with Equation (3.45) from [24]. Further, Definition 5 allows more general indices because the only condition is \( \delta _1< \dots < \delta _n \) and it is always guaranteed that the derivative is “truly nth-level”.
Let \( 0 < \gamma _k \le 1 \) and define \( \sigma _k := \sum _{k=1}^n \gamma _k - 1 \), \( k = 1, \dots , n \). Similar to the nth-level derivatives, the Dzherbashian-Nersesian derivatives \( {}_{ \mathrm { D } \mathrm { N } } \mathrm { D } ^{ \sigma _n}_{0+} \), defined in Equation (2.5) from [5], satisfy
4.3.5 Generalized fundamental theorem of fractional calculus
The composition law from Proposition 5 implies:
Corollary 3
Let \( \alpha \in \mathbb { R } \) and \( \gamma _1< \cdots < \gamma _n \). Then
Equations (4.26) correspond to the Fundamental Theorem of Fractional Calculus. More specifically, Equation (4.26b) implies the projector formula [25, Theorem 2]. The fact that the right-hand side of Equation (4.26a) is not the identity operator on a function space does not contradict the Fundamental Theorem of Fractional Calculus from [24, Theorem 3.4]. Because, under suitable assumptions on the parameters \( \alpha , \gamma _1, \dots , \gamma _n \) and when the distributional translation-invariant Riemann–Liouville operator \( {\text {I}}^{ \alpha }_+ \) in Equation (4.26a) is restricted to a classical domain of the Riemann–Liouville integral on the half axis, then the right hand side becomes an identity operator on this restricted domain.
This can be expressed by the slightly more general relations
The domain 
contains all domains that are commonly used to define Riemann–Liouville operators on the right half axis, as discussed on page 17. In particular, it holds 
, where \( X_{FT,+}( \mathbb { R } ) \) is defined as
Using Lemma 8 it follows that the space \( X_{FT} \) from [24, Eq. (3.18)] is characterized by the restrictions 
with \( f \in X_{FT,+}( \mathbb { R } ) \). Thus, (4.27b) implies Theorem 3.4 from [24].
4.3.6 Generalized derivatives with Sonine kernels
Generalizations of fractional derivatives and integrals where the kernels \( Y_{ \alpha } \) are replaced by pairs of kernels K and L from \( L^{ 1 } _{ \mathrm {loc} } \cap \mathscr { D } ' _{0+} \) that satisfy the Sonine relations \( K \mathbin {*} L = Y_{ 1 } \) were introduced in [18] and recently discussed in [21, 29]. The Sonine relations imply
The convolution operator \( \mathrm { C } _K \) and the composite
correspond to the generalized integral and the generalized derivative of Caputo type in [29, Eq. (26), (20)]. Equations (4.29) and (4.7) imply
From similar considerations as for Equations (4.27) it is found that the Equations (4.31) imply Theorem 1 from [29].
4.3.7 Two counterexamples
The operators of Caputo-Fabrizio [4, Eq.(1)] and Atangana-Baleanu-Caputo [4, Eq. (2)] \( \mathop { {}_ { \mathrm { CF } } {\text {D}}^{ \alpha }_{0+} } \) and \( \mathop { {}_ { \mathrm { ABC } } {\text {D}}^{ \alpha }_{0+} } \), can be represented as composites of convolution operators with kernels from \( \mathscr { Q } [ Y_{ \mathbb { R } } ] \) and eliminators. Explicitly, one obtains
with \({}_ { \mathrm { CF } } K_ \alpha =( \alpha \cdot {\updelta } +(1- \alpha ) \cdot Y_{ 1 } )/M( \alpha )\), and
with \({}_ { \mathrm { ABC } } K_ \alpha =( \alpha \cdot {\updelta } +(1- \alpha ) \cdot Y_{ \alpha } )/B( \alpha )\) and normalization constants \( M( \alpha ), B( \alpha ) \in \mathbb { R } \). The formulas follow immediately from the form of the derivative operators in Equations (1) and (3) in [4] and the formulas for the corresponding operators, \( \mathop { {}_ { \mathrm { CF } } {\text {J}}^{ \alpha }_{0+} } = \mathrm { C } _{ {}_ { \mathrm { CF } } K_ \alpha } \) and \( \mathop { {}_ { \mathrm { ABC } } {\text {J}}^{ \alpha }_{0+} } = \mathrm { C } _{ {}_ { \mathrm { ABC } } K_ \alpha } \), from Equations (5) and (9) in [4]. An application of Equation (4.7) yields the relations
that correspond to Equations (6) and (10) from [4]. This shows, once again, that these operators do not satisfy a Fundamental Theorem such as (4.31).
5 Generalized sequential fractional initial value problems
The main purpose of this section is to study the kernels of generalized sequential fractional differential operators. Subsection 5.1 provides an algorithm (Theorem 2) that can be used to simplify linear combinations of sequential derivatives to an eventually simpler normal form. The result is used in Subsection 5.2 to derive a representation formula for kernels (Theorem 3) and a characterization of the maximal injective domains (Theorem 4). The case of sequential derivatives with distinct order is considered in Subsection 5.3 in more detail. This yields a simplified criterion for kernel functions in general (Theorem 5) and a full characterization for important special cases (Theorem 6). Last, in Subsection 5.4, all possible cases with two types are described.
5.1 Simplifying linear combinations of sequential derivatives
Due to annihilation effects evaluating linear combinations of partially defined linear operators can be cumbersome. The theorem to be established below provides an algorithm that reduces sums of composites \( \mathrm { C } _K \circ {\text {R}} ^{ {\varGamma } } \) of convolution operators \( \mathrm { C } _K \) and composite eliminators \( {\text {R}} ^{ {\varGamma } } \) to a simple expression. Due to the theorem, the operators to be investigated can be assumed to be given in the form (5.3).
Note, that every finite subset \( {\varGamma } \subseteq \mathbb { R } \) can be represented as \( {\varGamma } = \{ \gamma _1, \dots , \gamma _n \} \) with unique \( \gamma _1< \dots < \gamma _n \) and \( n \in \mathbb { N } _0 \). Thus, the notations
are well defined with the conventions \( {\text {R}} ^{ \emptyset } = {\text {E}} \) and \( {\text {P}} ^{ \emptyset } = {\mathbf {0}} \).
Theorem 2
Every partially defined linear operator C of the form
with \( U_1, \dots , U_n \in \mathscr { D } ' _{0+} \), \( {\varGamma } _1, \dots , {\varGamma } _n \subseteq \mathbb { R } \) finite and \( n \in \mathbb { N } _0 \), can be written
with unique convolution kernels \( V_0, \dots , V_m \in \mathscr { D } ' _{0+} \), types \( {\varGamma } = \{ \gamma _1, \dots , \gamma _m \} \) and a step number \( m \in \mathbb { N } _0 \), given by
for \( l = 1, \dots , m \).
Proof
Reducibility to the normal form: Let \( n \in \mathbb { N } _0 \). Define the total number of eliminators arising in expression (5.2) as \( {\varSigma } := \sum ^n_{k=1} \# {\varGamma } _k \). The reducibility will be proved via of induction over \( {\varSigma } \). If \( {\varSigma } = 0 \) or \( n = 0 \) the expression (5.2) obviously has the form (5.3a). Thus, the Lemma holds for \( {\varSigma } = 0 \) or \( n = 0 \).
Assume now, that \( {\varSigma } > 0 \) and \( n > 0 \). The induction hypothesis is, that the statement of the Lemma holds for all expressions of the form (5.2) that have a total number of eliminators \( {\varSigma } ' < {\varSigma } \). Consider the order parameter
In the expression (5.2), all operators \( {\text {R}} ^{ \gamma } \) with \( \gamma < \gamma _1 \) can be canceled from the right due to (2.14), (3.14) and Corollary 1. A new expression of the form (5.2) emerges with a reduced total number of eliminators \( {\varSigma } ' < {\varSigma } \). Thus, the induction hypothesis applies to the new expression and it thus reduces to the form (5.3a).
If no eliminator can be canceled in this way, then for all \( k = 1, \dots , n \) either \( \gamma _1 = \max {\varGamma } _k \) or \( {\varGamma } _k = \emptyset \). Then, the operator \( {\text {R}} ^{ \gamma _1 } \) can be factored out as
The expression inside the brackets has the form (5.2) and satisfies \( {\varSigma } ' < {\varSigma } \). Thus, the induction hypothesis applies and the whole expression (5.5) can be written in the form (5.3a) by inserting an expression of the form (5.3a) into the brackets in Equation (5.5) and factoring out all composed eliminators.
Uniqueness of the normal form: According to equations (3.23) and (4.3) it holds
and therefore \( Y_{ \gamma } \in {\text {dom}} C \) for all \( \gamma \in \mathbb { R } \) with \( \gamma \ge \gamma _m \). Further, (3.23) and (4.3) imply
Thus, the orders \( \gamma _1< \cdots < \gamma _m \) and the number m are uniquely determined by C. For \( \gamma \in \mathbb { R }\) one calculates
From this equations it is clear how to represent the convolution kernels \( U_l \), \( l = 0, \dots , m \) as linear combinations of expressions \( Y_{ - \gamma } \mathbin {*} C( Y_{ \gamma } ) \). Therefore, the \( U_l \) are uniquely determined by the operator C. \(\square \)
Example 1
Corresponding to the problem from [3] consider
with \( \lambda _1, \dots , \lambda _{n-1} \in \mathbb { C } ^\times \), \( \lambda _n = 1 \), \( 0< \alpha _1< \dots < \alpha _n \) and \( n \in \mathbb { N } \). An application of Theorem 2 yields
where \( n^* \) is the largest k with \( \alpha _n - \alpha _k \notin \mathbb { N } \) if it exists and \( n^* = n \) otherwise. Thus, the proof of Theorem 2 generalizes Lemma 2 from [3].
5.2 The structure of the kernel
Consider an operator C in the form of Equation (5.3a). Changing to a different normal form makes it easy to obtain certain structural results on its kernel \( {\text {ker}} C \). Using Lemma 5 one calculates directly that
for all \( V_0, \dots , V_m, W_0, \dots , W_m \in \mathscr { D } ' _{0+} \), \( \gamma _1< \cdots < \gamma _m \) and \( m \in \mathbb { N } _0 \) with
where the convention \( W_{m+1} = 0 \) applies. Equation (5.10b) is equivalent to Equation (5.10c). The described transformation works as well under the restrictions that the distributions \( V_0, \dots , V_m \) and \( W_0, \dots , W_m \) all belong to \( \mathscr { Q } [ Y_{ \mathbb { R } } ] \), or belong to \( \mathscr { A } [ Y_{ \mathbb { R } } ] \).
For the remainder of this subsection let C be an operator of the form of Equation (5.10) such that the distributions \( V_0, \dots , V_m \), or, equivalently, the distributions \( W_0, \dots , W_m \), belong to the convolution field \( \mathscr { Q} [ Y_{\mathbb { R }}] \). Define the number
With these notations, the structure of \({\text {ker}} C \) is characterized by the following.
Theorem 3
Every distribution \( K \in {\text {dom}} C \) satisfies
where \( K_l \) and \( L_l \) are defined as
In particular, the kernel satisfies the inclusions
Proof
The right hand side of (5.10) can be rewritten as
Let \( K \in {\text {dom}} C \) and set \( {\tilde{K}} = {\text {R}} ^{ \gamma _1, \dots , \gamma _{l_0} } ( K ) \). Applying the operator from (5.14) to K and setting the result to zero yields
Solving for \( {\tilde{K}} \) in Eq. (5.15) and noting that the distributions \( Y_{ \gamma _1 } , \dots , Y_{ \gamma _{l_0} } \) belong to \( {\text {ker}} C \), due to (5.14), proves the proposition. \(\square \)
The following theorem provides the maximal domains for admissible inhomogeneities g of the equation \( C f = g\).
Theorem 4
The operator C from (5.10) restricts to a bijection
Proof
This is immediate from the formula (4.3) for kernels of sequential derivatives and Proposition 4. \(\square \)
5.3 Sums of sequential derivatives with distinct orders
In the following, those generalized fractional sequential differential operators are considered that are sums of generalized fractional sequential derivatives with distinct orders. That means, an operator D of the form
is considered with parameters \( n, m \in \mathbb { N } _0 \), \( 0 = \alpha _0< \alpha _1 \cdots < \alpha _n \), \( 1 = \lambda _0, \lambda _1, \dots , \lambda _n \in \mathbb { C } ^\times \), \( \gamma _1< \dots < \gamma _m \) and a function \( \sigma :\{ 0, \dots , n \} \rightarrow \{ 0, \dots , m \} \) such that \( m = \max \{\, \sigma ( k ) \,;\, k = 0, \dots , n \,\} \). In order to exclude trivial cases, \( \min \{\, \sigma ( k ) \,;\, k = 0, \dots , n \,\} = 0 \) is assumed as well.
Using Theorem 3 the kernel of the operator D can be expressed in terms of the functions \( K_l \) and the composed coefficient operators \( L_l \). The distributions \( K_l \) are given by
and the distributions \( W_l \) are given by
with the notation \( Z_k := \mu _k Y_{ \beta _k } \) and the parameters \( \beta _k := \alpha _n - \alpha _{n-k} \) and \( \mu _k := \lambda _{n-k} / \lambda _n \) for \( k = 0, \dots , n \).
In the following, elementary properties of the kernel distributions \( K_l \) are summarized. That will help to shed more light on the structure of \( {\text {ker}} D \). After an examination of Equation (5.18b), one concludes that
Let \( l, l' \in \{ 1, \dots , m \} \). Equation (5.19) yields the implications
and the equation
Note, that \({\text {O}}\left( K_l \right) < {\text {O}}\left( K_{l'}\right) \) holds for all \( l \in \{ 1, \dots , \sigma ( n ) \} \), \( l' \in \{ 1, \dots , m \} \) with \( l < l' \) according to Eq. (5.19). However, if \( l \in \{ \sigma ( n ) + 1, \dots , m - 1 \} \), then \( {\text {O}}\left( K_l \right) < {\text {O}}\left( K_{l'} \right) \) does not necessarily hold. Nevertheless, the functions \( K_1, \dots , K_l \) are linearly independent. This is because in the expression for \( W_l \) from Equation (5.18b), as l increases the number of terms \( Z_k \) reduces or \( {\text {O}}\left( W_l \right) \) increases.
Theorem 5
Let \( K = \sum ^m_{l=1} a_l K_l \) with \( a_1, \dots , a_m \in \mathbb { C } \). Then
-
1.
It holds \( K \in {\text {dom}} D \) if and only if
$$\begin{aligned} \sum ^{l-1}_{i=1} a_i \cdot K_i&\in {\text {dom}} L_l&\text {for all} ~ l = 1, \dots , m. \end{aligned}$$(5.22) -
2.
It holds \( K \in {\text {ker}} D \) if and only if \( K \in {\text {dom}} D \) and
$$\begin{aligned} { L_l }\left( \sum ^{l-1}_{i=1} a_i \cdot K_i \right) = {\left\{ \begin{array}{ll} 0 &{} \text {if} ~ l \le \sigma ( n ) ,\\ a_l &{} \text {if} ~ l > \sigma ( n ) , \end{array}\right. }&\text {for all} ~ l = 1, \dots , m . \end{aligned}$$(5.23)
Proof
Due to Equation (5.20a), and because \( {\text {dom}} D \) is linear, \( K \in {\text {dom}} D \) is equivalent to (5.22). Inserting \( K \in {\text {dom}} D \) into Equation (5.12a) yields Equation (5.23) after canceling all terms that are zero according to Equation (5.21). \(\square \)
Corollary 4
The estimate \( {\text {dim}} {\text {ker}} D \le \sigma ( n ) \) holds.
Corollary 5
If \( l' \in \{ 1, \dots , m \} \) is such that \( K_l \notin {\text {ker}} L_{l+1} \) for all \( l = 1, \dots , l' \), then \( {\text {ker}} D \subseteq \left\langle K_{l'+1}, \dots , K_m \right\rangle _{ } \).
Theorem 6
The kernel of D satisfies \( {\text {dim}} {\text {ker}} D = m = n \) if the conditions \( \sigma ( n ) = m \) and \( \beta _{ \varphi ( l, 1 ) } > \gamma _m - \gamma _l \), \( l = 1, \dots , m - 1 \) hold, where \( \varphi ( l, 1 ) := \min \left\{ \, k \in \{ 1, \dots , n \} \,;\, \sigma ( n - k ) < l \,\right\} \).
Proof
Introducing a term \( W_0 \mathbin {*} ( W_0 )^{ \mathbin {*} -1} \) one calculates
Therefore, \( {\text {O}}_{ 1 }\left( K_l - Y_{ \gamma _l } \right) {=} {\text {O}}\left( W_l - W_0 \right) {=} \beta _{ \varphi ( l, 1 ) } \) and the distributions \( K_1, \dots ,\) \( K_m \) satisfy the condition (5.23) from Theorem 5. \(\square \)
Example 2
The operator D from Equation (5.9b) satisfies the requirements of Theorem 6. Thus, for linear combinations of Riemann–Liouville operators the cancellation of the dimension of the kernel is already explained by the cancellation effects described in Theorem 3.
Example 3
Let \( {\tilde{L}} \) be the operator from Equation (2.16) in [5] for the special case of constant coefficients \( {\tilde{p}}_0, \dots , {\tilde{p}}_n \). For simplicity, assume that the \( {\tilde{p}}_k \) are non-zero. (Within this remark, symbols x from [5] are decorated as \( {\tilde{x}} \).) Recalling Equation (4.25) the operator \( {\text {D}}^{ -{\tilde{\sigma }}_0 }_+ \circ {\tilde{L}} \) can be identified with an operator of the form D as in Equation (5.17) above with the definitions \( n = \tilde{n} + 1 \), \( m = \tilde{ n } \), \( \alpha _1 = -{\tilde{\sigma }}_0 \), \( \alpha _k = {\tilde{\sigma }}_{k-1} - {\tilde{\sigma }}_0 \) for \( k = 2, \dots , n \), \( \gamma _l = {\tilde{\sigma }}_{l-1} + 1 \) for \( l = 1, \dots , m \) and \( \lambda _1 = 1 \), \( \lambda _k = \tilde{p}_{\tilde{n}+1-k} \) for \( k = 2, \dots , n - 2 \), \( \lambda _{n-1} = \tilde{ p }_{\tilde{n}-1} \), \( \lambda _n = \tilde{ p }_{\tilde{n}} \).
One verifies, that \( \varphi ( l, 1 ) = n + 1 - l \) and thus \( \beta _{ \varphi (l,1)} = {\tilde{\sigma }}_{\tilde{n}} - {\tilde{\sigma }}_l > {\tilde{\sigma }}_{\tilde{n}-1} - {\tilde{\sigma }}_l \) for all \( l = 1, \dots , m \). Therefore, Theorem 6 and Theorem 3 imply the existence result [5, Thm. 4] for a trivial inhomogeniety \( \tilde{ f } = 0 \). From \( {\text {O}}\left( W_0 \right) = - \alpha _n \), the relation
and Theorem 4 one finds that \( {\text {D}}^{ -{\tilde{\sigma }}_0 }_+ \circ \tilde{ L } \) restricts to a bijection
This extend the result from [5, Thm. 4] to inhomogeneities .
The restriction \( {\tilde{\gamma }}_0 > 1 - {\tilde{\gamma }}_n \) is required in [5, Thm. 4] in order to ensure that \( Y_{ {\tilde{\sigma }}_k + 1 } = {\text {I}}^{ 1 - {\tilde{\gamma }}_n }_+( Y_{ {\tilde{\sigma }}_k + {\tilde{\gamma }}_n } ) \) is well defined as a classical Riemann–Liouville integral. The condition becomes superflous in the generalized sequential setting because it is not required that \( Y_{ {\tilde{\sigma }}_k + {\tilde{\gamma }}_n } \) is locally integrable.
5.4 Characterization of the kernel for two primary orders
In the following analysis the representation formula
is used with the notations
for \( i = 1, \dots , {\varPhi } ( l ) \) and \( l = 1, \dots , m \). Further, let
and define the notation
Lemma 9
Let \( U_1, \dots , U_n, V_1, \dots , V_m \in \mathscr { Q } [ Y_{ \mathbb { R } } ] \setminus \{ 0 \} \) with \( m \le n \), \( m, n \in \mathbb { N } \) and assume, that \( 0< {\text {O}}\left( U_1 \right)< \cdots < {\text {O}}\left( U_n \right) \) and \( {\text {O}}\left( V_1 \right)< \cdots < {\text {O}}\left( V_m \right) \). Then
where \( l' \) is the largest \( k' \) such that \( V_{k+1} - V_1 \mathbin {*} U_k = 0 \) for all \( k < k' \).
Proof
The assumptions imply \( {\updelta } + U_1 + \cdots + U_n \ne 0 \) and one obtains
The assumptions and Proposition 4 imply that
With these considerations Equation (5.30) is immediate from (5.31). \(\square \)
Lemma 10
For all \( l = 1, \dots , m \) one has
with the order parameter
where \(\beta _k\) was defined just after Eq. (5.18b).
Proof
One applies Lemma 9 to Equation (5.27a) with \( U_k = Z_k \), \( k = 1, \dots , n \) and \( V_k = Z_{ \varphi (l,k)} \), \( k = 1, \dots , {\varPhi } ( l ) \). \(\square \)
An application of Lemma 10 yields:
Lemma 11
Let \( l \in \{ 1, \dots , m - 1 \} \). Then
If \( K_l \in {\text {dom}} L_{l+1} \), then
In particular, it holds
Example 4
Combining Theorem 5 and Lemma 11 a complete characterization of \( {\text {ker}} D \) can be obtained. The possible cases are given by
A minimal example for the first, and trivial, case is the operator
The cases with \( \sigma ( n ) = 1 \) are covered by the minimal examples
where \( 0< \alpha _1 < \alpha _2 \), \( \gamma _1 < \gamma _2 \) and \( \lambda _1, \lambda _2 \in \mathbb { C } ^\times \). The cases with \( \sigma ( n ) = 2 \) are covered by the examples
where \( 0 < \alpha \), \( \gamma _1 < \gamma _2 \) and \( \lambda \in \mathbb { C } ^\times \).
Example 5
Linear combinations of a first order time derivative and a generalized Riemann–Liouville fractional derivative were used in [10] as infinitesimal generators for composite fractional time evolutions. Reinterpreting the operator \( \mathop { {} {\text {D}}^{ \alpha ,\mu }_{0+} } \) in [10] along the lines of eq. (4.19) as \( \mathop { {} {\text {D}}^{ \alpha ; \mu }_{0+} } \) the solution f(t) , \( t \ge 0 \) of the initial value problem (32) in [10] can be represented as the restriction \( f := \left. K \right| _{ \mathbb { R } _{0+} } \) of a distribution \( K \in \mathscr { D } ' _{0+} \), where the distribution K is the solution to the following generalized initial value problem: Define the operator
where \( \lambda _1 = ( \tau _ \alpha )^ \alpha \), \( \lambda _2 = \tau _1 \) and the parameters \( \tau _ \alpha , \tau _1 > 0 \), \( 0< \alpha < 1 \) and \( 0 \le \mu \le 1 \). According to Theorem 2 the equality
holds if and only if \( \mu \ne 1 \) (corresponding to Equation (34) in [10]). In any case, there exists a unique distribution \( K \in \mathscr { D } ' _{0+} \) such that
The relation \( R = S \mathbin {*} {\check{Y}} _{ 1 } \) between the normalized relaxation function \( R = K + {\check{Y}} _{ 1 } \) and the time-domain representation of the corresponding normalized susceptibility S was verified using convolutional calculus in [17]. Here, \( {\check{Y}} _{ 1 } \) denotes the reflection of \( Y_{ 1 } \). The Laplace-transform of S (in the sense of [33]) defines the susceptibility function \( \varepsilon ( u ) \) in the frequency domain.
Another interpretation of the infinitesimal generator D from [10] was recently given in [11]. It is obtained from observing \( \mathop { {} {\text {D}}^{ \alpha ;\mu }_{0+} } = {\text {R}} ^{ \mu -\mu \alpha } \circ {\text {D}}^{ \alpha | }_{0+} \) and shifting \( {\text {R}} ^{ \mu -\mu \alpha } \) from \( {\text {D}}^{ \alpha | }_{0+} \) to the infinitesimal generator \( {\text {D}}^{ 1 | 1 }_{0+} \) of translations in eq. (5.41). The physical motivation for this are relaxation processes that are too fast to be resolved [11]. The modified interpretation leads to a sequential first order derivative
with \( \delta =1+\mu (1- \alpha )\). Instead of D and problem (5.41) the modified fractional initial value problem involves the modified operator
where \( \lambda _1=( \tau _ \alpha )^ \alpha \), \( \lambda _2= \tau _1\), \( \delta =1+\mu (1- \alpha )\) with the parameters \( \tau _ \alpha , \tau _1 > 0 \), \( 0< \alpha < 1 \) and \( 0< \mu < 1 \). Note, that \( 1 - \delta < 1 - \alpha \). Therefore, Theorem 6 guarantees the existence of a unique distribution \(\tilde{K}\in \mathscr { D } ' _{0+} \) such that
with \( \nu = ( \tau _{ \mathrm { a } \mathrm { b } } )^ \delta \). Theorem 3 and Equations (5.18) yield the solution
As above, the relaxation motion \( \tilde{ R } \) satisfies \( \tilde{ R } = \tilde{ K } + {\check{Y}} _{ 1 } = \tilde{ S } \mathbin {*} {\check{Y}} _{ 1 } \), with the normalized susceptibility \( \tilde{ S } \) given by the convolution quotient
The normalized susceptibility function \( \varepsilon (u)\) from Equation (3) in [11] coincides with the Laplace-transform of the distribution \( \tilde{ S } \). Thus, \( \tilde{ S } \) is the time-domain representation of \( {\tilde{\varepsilon }}( u ) \) and it follows that the function \( t \mapsto \tilde{ K }( t ) \) for \( t \ge 0 \) is the normalized relaxation motion corresponding to \( {\tilde{\varepsilon }}( u ) \). As shown in [11, 17] the solution (5.45) agrees over a range of 12 decades in time or frequency with a physical relaxation experiment.
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Kleiner, T., Hilfer, R. Sequential generalized Riemann–Liouville derivatives based on distributional convolution. Fract Calc Appl Anal 25, 267–298 (2022). https://doi.org/10.1007/s13540-021-00012-0
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DOI: https://doi.org/10.1007/s13540-021-00012-0