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The General Fractional Derivative and Related Fractional Differential Equations

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The General Fractional Derivative and Related Fractional Differential Equations mathematics Review The General Fractional Derivative and Related Fractional Differential Equations Yuri Luchko 1,* and Masahiro Yamamoto 2 1 Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany 2 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan; [email protected] * Correspondence: [email protected] Received: 29 September 2020; Accepted: 23 November 2020; Published: 26 November 2020 Abstract: In this survey paper, we start with a discussion of the general fractional derivative (GFD) introduced by A. Kochubei in his recent publications. In particular, a connection of this derivative to the corresponding fractional integral and the Sonine relation for their kernels are presented. Then we consider some fractional ordinary differential equations (ODEs) with the GFD including the relaxation equation and the growth equation. The main part of the paper is devoted to the fractional partial differential equations (PDEs) with the GFD. We discuss both the Cauchy problems and the initial-boundary-value problems for the time-fractional diffusion equations with the GFD. In the final part of the paper, some results regarding the inverse problems for the differential equations with the GFD are presented. Keywords: general fractional derivative; general fractional integral; Sonine condition; fractional relaxation equation; fractional diffusion equation; Cauchy problem; initial-boundary-value problem; inverse problem MSC: 26A33; 35A05; 35B30; 35B50; 35C05; 35E05; 35L05; 35R30; 45K05; 60E99 1. Introduction In functional analysis, the integral operators with the weakly singular kernels have been an important topic for research for many years. They are defined in the form Z (T f )(t) = K(t, t) f (t) dt, (1) W where W is an open subset of Rn and the kernel K = K(t, t) is a real or complex valued continuous function on W × WnD, D = f(t, t) : t 2 Wg being the diagonal of W × W that satisfies the condition c jK(t, t)j ≤ with a < n. (2) jt − tja In case W is a bounded domain, the operator (1) is called the Schur integral operator. It is compact n 1 1 from C(W) to C(W) [1]. If, additionally, a < q , p + q = 1, 1 ≤ p < +¥, then (1) is a Hilbert-Schmidt operator that is compact from Lp(W) to C(W). In Fractional Calculus (FC), the operators of type (1) with special weakly singular kernels are studied on both bounded and unbounded domains. For instance, the classical left-hand sided Riemann-Liouville fractional integral of order a 2 R+ of a function f on a finite or infinite interval (a, b) is defined as follows: Mathematics 2020, 8, 2115; doi:10.3390/math8122115 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 2115 2 of 20 t 1 Z (Ia f ) (t) = (t − t)a−1 f (t) dt, t 2 (a, b). (3) a+ G(a) a In the case a = 0, this integral is interpreted as the identity operator 0 Ia+ f (t) = f (t) (4) because of the relation ([2]) a lim (Ia+ f ) (t) = f (t), (5) a!0+ that is valid in particular for f 2 L1(a, b) in every Lebesgue point of f , i.e., almost everywhere on (a, b). Evidently, the Riemann-Liouville fractional integral is a generalization of the well-known formula for the n-fold definite integral Z t Z t Z t Z t n 1 n−1 (Ia+ f ) (t) = dt dt··· f (t) dt = (t − t) f (t) dt, n 2 N. a a a (n − 1)! a In a certain sense, this generalization is unique. In the case of a finite interval (without any restriction of generality, we fix the interval (0, 1)), the following result was proved in [3]: Let E be the space Lp(0, 1), 1 ≤ p < +¥, or C[0, 1]. Then there exists precisely one family Ia, a > 0 of operators on E satisfying the following conditions: R t (CM1) (I1 f )(t) = 0 f (t) dt, f 2 E (interpolation condition), (CM2) (Ia Ib f )(t) = (Ia+b f )(t), a, b > 0, f 2 E (index law), (CM3) a ! Ia is a continuous map of R+ into the space L(E) of the linear bounded operators from E to E for some Hausdorff topology on L(E), weaker than the norm topology (continuity), p p (CM4) f 2 E and f (t) ≥ 0 (a.e. for E = L (0, 1)) ) (Ia f )(t) ≥ 0 (a.e. for E = L (0, 1)) for all a > 0 (non-negativity). That family is given by the Riemann-Liouville Formula (3) with a = 0 and b = 1. From the present viewpoint ([4]), the conditions (CM1)–(CM4) are very natural for any definition of the fractional integrals defined on a finite interval. As proved in [3], they are also sufficient for uniqueness of the family of the Riemann-Liouville fractional integrals. Thus, in this sense, the Riemann-Liouville fractional integrals are the only “right” one-parameter fractional integrals defined on a finite one-dimensional interval. The problem regarding the “right” fractional derivatives is more delicate and has no unique solution. Presently, the main approach for introducing the fractional derivatives is to define them as the left-inverse operators to the fractional integrals ([4–6]). However, even for the Riemann-Liouville fractional integral, there exist infinitely many different families of operators that fulfill this property ([6]). In particular, for 0 < a ≤ 1, the Riemann-Liouville fractional derivative d (Da f )(t) = (I1−a f )(t), (6) RL dt 0+ the Caputo fractional derivative d f (Da f )(t) = (I1−a )(t), (7) C 0+ dt and the Hilfer fractional derivative b(1−a) d (1−a)(1−b) (Da f )(t) = (I I f )(t), 0 ≤ b ≤ 1 (8) H 0+ dt 0+ are the left-inverse operators to the Riemann-Liouville fractional integral on the suitable nontrivial spaces of functions including the space of the absolutely continuous functions on [0, 1], i.e., the Fundamental Theorem of FC holds true ([6]): Mathematics 2020, 8, 2115 3 of 20 a a (DX I0+ f )(t) = f (t), t 2 [0, 1], X 2 fRL, C, Hg. (9) Moreover, in [6], infinitely many other families of the fractional derivatives in the sense of Formula (9) called the nth level fractional derivatives were introduced. Let the parameters g1, g2,..., gn 2 R satisfy the conditions k 0 ≤ gk and a + sk ≤ k with sk := ∑ gi, k = 1, 2, . , n. (10) i=1 The nth level fractional derivative of order a, 0 < a ≤ 1 and type g = (g1, g2, ... , gn) is defined as follows: n ! a,(g) gk d n−a−sn (DnL f )(t) = ∏(I0+ ) (I0+ f )(t). (11) k=1 dt This derivative satisfies the Fundamental Theorem of FC, i.e., the relation a,(g) a (DnL I0+ f )(t) = f (t), t 2 [0, 1] (12) holds true on a nontrivial space of functions (see [6] for details). To keep an overview of these and many other fractional derivatives, it is very natural to consider some general integro-differential operators of convolution type and to clarify the question under what conditions can they be interpreted as a kind of the fractional derivatives. In particular, one expects that for these derivatives and the appropriate defined fractional integrals the Fundamental Theorem of FC holds true. Moreover, for the sake of possible applications, one wants to keep some fundamental properties of solutions to the differential equations with these derivatives. In particular, the property of complete monotonicity of solutions to the appropriate relaxation equation or the positivity of the fundamental solution to the Cauchy problem for the fractional diffusion equation with the time-derivatives of this type. In the theory of the abstract Volterra integral equations in the Banach spaces, the evolution equations including the integro-differential operators of convolution type d Z t (Dk u)(t) = k(t − t)u(t) dt, t 2 [0, T], 0 < T ≤ +¥ (13) dt 0 have been a subject for research for more than a half century. In particular, in [7] (see also the references therein), the abstract Volterra integral equations including the operators (13) with the completely positive kernels k 2 L1(0, T) have been studied. This class of the kernels can be characterized as follows: A function k 2 L1(0, T) is completely positive on [0, T] if and only if there exist a ≥ 0 and l 2 L1(0, T), non-negative and non-increasing, satisfying the relation Z t a k(t) + k(t − t)l(t) dt = 1, t 2 (0, T]. (14) 0 In particular, the completely monotone kernels are completely positive. The notion of completely positive kernels originated from the “positivity preserving property” that is valid for the corresponding Volterra integral equations in the case of the Banach space X = R with the usual norm. In [8,9], the properties of the appropriate defined weak solutions to the linear and quasi-linear evolutionary partial integro-differential equations of second order with the time-operators of type (13) in the form d Z t (Dk u)(t) = k(t − t)(u(t) − u0) dt, t 2 R+ (15) dt 0 were addressed in the case of the kernels k 2 Lloc(R+) that satisfy the following conditions: Mathematics 2020, 8, 2115 4 of 20 (Z1) The kernel k is non-negative and non-increasing on R+, R t (Z2) There exists a kernel l 2 Lloc(R+) such that 0 k(t − t)l(t) dt = 1, t 2 R+.
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