Fractional Differential Equations: Change of Variables and Nonlocal Symmetries

ISSN 2304-0122 Ufa Mathematical Journal. Vol. 4. No 4 (2012). P. 54-67.

UDC 517.9

FRACTIONAL DIFFERENTIAL EQUATIONS: CHANGE OF VARIABLES AND NONLOCAL SYMMETRIES

R.K. GAZIZOV, A.A. KASATKIN, S.YU. LUKASHCHUK

Abstract. In this paper point transformations of variables in fractional integrals and derivatives of different types are considered. In the general case, fractional integro- differentiation of a function with respect to another function arises after such substitution. The problem of applying a point transformations group to this type of operators is considered, the corresponding prolongation formulae for infinitesimal operator of the group are constructed. Usage of prolongation formulae for finding nonlocal symmetries of equations and checking their admittance is demonstrated on a simple example of an ordinary fractional differential equation. Keywords: fractional derivative, prolongation formulae, nonlocal symmetry.

Introduction Differential equations with fractional derivatives have a wide range of applications as mathematical models of various processes with anomalous kinetics [1, 2]. Investigation of symmetry properties of such equations is an important problem. Meanwhile, unlike the classical integer-order derivatives, there exists a number of differerent definitions of fractional derivatives [3, 4, 5, 6, 7]. These definition differences leads to the fractional differential equations (FDEs) having similar form but significantly different properties. The Riemann-Liouville left-hand side fractional derivative [3] defined as 푛 ∫︁ 푥 훼 1 푑 푦(푡) (푐퐷푥 푦)(푥) = 푛 훼−푛+1 푑푡 (1) Γ(푛 − 훼) 푑푥 푐 (푥 − 푡) and the Caputo left-hand side fractional derivative [4] defined as ∫︁ 푥 (푛) (︀퐶 훼 )︀ 1 푦 (푡) 푐 퐷푥 푦 (푥) = 훼−푛+1 푑푡 (2) Γ(푛 − 훼) 푐 (푥 − 푡) are used most often in practice (here 푛 = [훼] + 1, Γ(푥) is the gamma-function). In general case, a solution of a differential equation with the derivative (1) may contain an integrable singularity (of the order not higher than 1−훼) at the point 푥 = 푐, while the existence of the derivative (2) implies that the solution is bounded at this point. It is known (see, for example, [4]) that if a finite limit lim푥→푐+ 푦(푥) = 푦(푐) exists, then the derivatives (1) and (2) are connected by the relationship 1 푦(푐) ( 퐷훼푦)(푥) = (︀퐶 퐷훼푦)︀ (푥) + . (3) 푐 푥 푐 푥 Γ(1 − 훼) (푥 − 푐)훼 Methods of constructing the point transformation groups admitted by differential equations were developed in the papers [8, 9, 10, 11] for equations with fractional derivatives of the form (1) and (2). Prolongation of the group infinitesimal operator to the fractional integrals and derivatives was constructed. Algorithms of finding the admitted group for equations containing

○c Gazizov R.K., Kasatkin A.A., Lukashchuk S.Yu. 2012. This work was supported by the Ministry of Education and Science of the Russian Federation (contract 11.G34.31.0042 with Ufa State Aviation Technical University and leading scientist prof. N. H. Ibragimov). Submitted on November 9, 2012.

54 FRACTIONAL DIFFERENTIAL EQUATIONS. . . 55 these derivatives were developed, and some problems of group classification of ordinary differential equations and fractional-order partial differential equations were solved. However, it was found that the class of variables substitutions preserving the form of fractional derivatives is very narrow. The general form of such a point substitution for the Riemann-Liouville-type derivatives (1) is defined by the expression

푐푐1 + (푥 − 푐) 푥¯ = , 푦¯ = 휓0(푥) + 푦휓1(푥), 푐1 + 푐2(푥 − 푐) where 푐, 푐1, 푐2 are constants, 휓0(푥), 휓1(푥) are some functions with specific form being determined by the equation under consideration. Nevertheless, the derivatives (1) and (2) are only the particular forms of fractional derivatives, though they are used most frequently. More general definition is a fractional derivative of a function with respect to another function. This type of derivative arise when general point transformation of variables is applied to any fractional derivative of a definite type. FDEs with a fractional derivative of a function with respect to another function are considered, in particular, in the process of constructing invariant solutions of fractional partial differential equations. For example, when constructing scale-invariant solution of anomalous transport equations, one obtains the ordinary differential equations with the Erdélyi-Kober fractional derivatives [8, 12]. The existing methods of sovling such equations are very complicated and work only for a limited class of equations. Using fractional derivatives of a function with respect to another function makes it possible to expand the class of allowed variables substitutions. These variable changes can be considered as a new type of equivalence transformations (brief discussion of this problem can be found in [13]). This approach provides new possibilities for reduction of the variables number, and, in particular, for constructing invariant solutions. In this paper, application of the Lie group analysis methods to the class of FDEs containing fractional derivatives of a function with respect to another function is considered. The first necessary step is the construction of prolongation formulae to extend an infinitesimal operator of the group to the fractional integrals and derivatives of a function with respect to another function. Section 1 of this paper is devoted to this construction. Since fractional derivatives are represented by integro-differential operators (i.e. they are nonlocal), it seems natural that the equations with such derivatives should have nonlocal symmetries. One of the ways to construct such symmetries is to use non-point transformation of variables (containing fractional derivatives and integrals). Then one can determine the form of infinitesimal operators in the space extended to the corresponding nonlocal variables. Using the prolongation formulae for construction and verification of nonlocal symmetries is illustrated by a simple example. Working with fractional derivatives, verification of an operator admittance is often a non-trivial problem as demonstrated in the section 2 of the present paper.

1. A fractional derivative of a function with respect to another function. The prolongation formula In the general case, an arbitrary change of variables푥 ¯ = 휙(푥, 푦), 푦¯ = 휓(푥, 푦) does not preserve the form of the fractional differential operator. In particular, this substitution transforms the Riemann-Liouville fractional derivative (1) of the order 훼 ∈ (0, 1) to the left-hand side derivative of the function 휓(푥, 푦) with respect to the function 휙(푥, 푦): ∫︁ 푥 (︀ 훼 )︀ 1 1 푑 휓[푡]퐷푡휙[푡] 푐퐷휙[푥]휓 [푥] = 훼 푑푡, where푐 ¯ : 휙(푥, 푦(푥))|푥=¯푐 = 푐. Γ(1 − 훼) 퐷푥휙[푥] 푑푥 푐¯ (휙[푥] − 휙[푡]) For the sake of brevity the notation 푓[푥] ≡ 푓(푥, 푦(푥)) is introduced here. The definition and general properties of derivatives of a function with respect to another function are presented, for example, in [3]. Let us give some examples of transforming the Riemann-Liouville operator into other known forms of fractional differentiation operators by the change of variables. 56 R.K. GAZIZOV, A.A. KASATKIN, S.YU. LUKASHCHUK

1) Translation with respect to 푥 푥¯ = 푥 + 푎, 푦¯ = 푦 conserves the type of the operator but changes the lower limit of integration: 훼 훼 (푐퐷푥 푦)(푥) = (푐¯퐷푥¯ 푦¯) (¯푥), 푐¯ = 푐 + 푎. 2) After substitution of variables 푥¯ = 푥푎, 푦¯ = 푦, the Riemann-Liouville operator is replaced by the Erdélyi-Kober operator [3]: 1 1 푑 ∫︁ 푥¯ 푦¯(푡¯)푡¯푏−1 ( 퐷훼푦)(푥) = 푑푡,¯ 푏 = 1/푎, 푐¯ = 푐푎. 푐 푥 푏−1 푏 ¯푏 훼 Γ(1 − 훼) 푥¯ 푑푥¯ 푐¯ (¯푥 − 푡 ) Such change of variables is often performed to find invariant solutions of the anomalous transport equations with respect to the group of scaling transformations [8]. 3) Change of variables 푥¯ = 푒푥, 푦¯ = 푦 results in the transforming the Riemann-Liouville operator to the fractional derivative operator [3] of the Hadamard type: ∫︁ 푥¯ ¯ ¯ 훼 푥¯ 푑 푦¯(푡) 푑푡 (푐퐷푥 푦)(푥) = (︀ 푥¯ )︀훼 . Γ(1 − 훼) 푑푥¯ 푐¯ 푡¯ 푒 ln 푡¯ Together with the fractional derivative of a function with respect to another function, the definition of the fractional integral of order 훽 > 0 of a function with respect to another function [3] is also used: ∫︁ 푥 ′ (︁ 훽 )︁ 1 푦(푡)푔 (푡) 푐퐼푔(푥)푦 (푥) = 1−훽 푑푡. (4) Γ(훽) 푐 (푔(푥) − 푔(푡)) Here it is assumed [3] that 푔(푥) > 0 within the interval (푐, 푑) and the function 푔(푥) has a continuous derivative 푔′(푥) which is strictly positive or negative (푔′(푥) > 0 or 푔′(푥) < 0 for all 푥). The function 푦(푥) is considered to be Lebesgue integrable within the interval (푐, 푑), i.e. 푦 ∈ 퐿1(푐, 푑). In what follows, for the sake of simplicity, we consider the left-hand side fractional derivative of the order 훼 ∈ (0, 1) of the function 푦(푥) with respect to the function 푔(푥): ∫︁ 푥 ′ (︀ 훼 )︀ 1 푑 (︀ 훼 )︀ 1 1 푑 푦(푡)푔 (푡) 푐퐷푔(푥)푦 (푥) ≡ ′ 푐퐼푔(푥)푦 (푥) = ′ 훼 푑푡. (5) 푔 (푥) 푑푥 Γ(1 − 훼) 푔 (푥) 푑푥 푐 (푔(푥) − 푔(푡)) The fractional derivative (1) for 훼 ∈ (0, 1) is a particular case of (5) when 푔(푥) = 푥. The fractional derivative (5) has two properties which are used below to deduce the prolongation formula. Property 1. The following relationship holds: 훼 훼 1−훼 푐퐷푔(푥) (푔(푥)푦(푥)) = 푔(푥)푐퐷푔(푥)푦(푥) + 훼푐퐼푔(푥) 푦(푥). (6) Proof. ∫︁ 푥 ′ 훼 1 1 푑 푔(푡)푦(푡)푔 (푡) 푐퐷푔(푥) (푔(푥)푦(푥)) ≡ ′ 훼 푑푡 = Γ(1 − 훼) 푔 (푥) 푑푥 푐 [푔(푥) − 푔(푡)] [︂∫︁ 푥 ′ ∫︁ 푥 ]︂ 1 1 푑 푔(푥)푦(푡)푔 (푡) 1−훼 ′ = ′ 훼 푑푡 − [푔(푥) − 푔(푡)] 푦(푡)푔 (푡)푑푡 = Γ(1 − 훼) 푔 (푥) 푑푥 푐 [푔(푥) − 푔(푡)] 푐 1 ∫︁ 푥 푦(푡)푔′(푡) 1 푔(푥) 푑 ∫︁ 푥 푦(푡)푔′(푡) = 훼 푑푡 + ′ 훼 푑푡− Γ(1 − 훼) 푐 [푔(푥) − 푔(푡)] Γ(1 − 훼) 푔 (푥) 푑푥 푐 [푔(푥) − 푔(푡)] ∫︁ 푥 ′ 1 − 훼 푦(푡)푔 (푡) 1−훼 훼 − 훼 푑푡 ≡ 훼푐퐼푔(푥) 푦(푥) + 푔(푥)푐퐷푔(푥)푦(푥). Γ(1 − 훼) 푐 [푔(푥) − 푔(푡)] FRACTIONAL DIFFERENTIAL EQUATIONS. . . 57

Property 2. If lim 푦(푡)(푔(푡) − 푔(푐)) = 0, then the following equality holds: 푡→푐+ ∫︁ 푥 훼 1 퐷푡 [푦(푡)(푔(푡) − 푔(푐))] 푐퐷푔(푥) (푦(푥)(푔(푥) − 푔(푐))) = 훼 푑푡. (7) Γ(1 − 훼) 푐 [푔(푥) − 푔(푡)] Proof. The proof consists of integration by parts and diﬀerentiation of the resulting integral with a variable upper limit: ∫︁ 푥 훼 1 1 푑 푦(푡)(푔(푡) − 푔(푐)) ′ 푐퐷푔(푥)[푦(푥)(푔(푥) − 푔(푐))] = ′ −훼 푔 (푡)푑푡 = Γ(1 − 훼) 푔 (푥) 푑푥 푐 (푔(푥) − 푔(푡)) [︂ 1−훼 ⃒푥 1 1 푑 (푔(푥) − 푔(푡)) ⃒ = ′ − 푦(푡)[푔(푡) − 푔(푐)] ⃒ + Γ(1 − 훼) 푔 (푥) 푑푥 1 − 훼 ⃒푐 ∫︁ 푥 (푔(푥) − 푔(푡))1−훼 ]︂ + 퐷푡[푦(푡)(푔(푡) − 푔(푐))] 푑푡 = 푐 1 − 훼 ∫︁ 푥 1 1 푑 1−훼 = ′ 퐷푡[푦(푡)(푔(푡) − 푔(푐))] (푔(푥) − 푔(푡)) 푑푡 = (1 − 훼)Γ(1 − 훼) 푔 (푥) 푐 푑푥 ∫︁ 푥 1 퐷푡 [푦(푡)(푔(푡) − 푔(푐))] = 훼 푑푡. Γ(1 − 훼) 푐 [푔(푥) − 푔(푡)]

Proposition 1. Let us consider a one-parameter group of point transformations in the infinitesimal form: 푥¯ = 푥 + 푎휉[푥] + 표(푎), 푦¯(¯푥) = 푦(푥) + 푎휂[푥] + 표(푎). ′ Assume that the function 푦(푥) ∈ 퐿1(푐, 푑) has a continuous derivative 푦 (푥) for 푥 ∈ (푐, 푑), the functions 휉[푥] = 휉(푥, 푦(푥)) and 휂[푥] = 휂(푥, 푦(푥)) are sufficiently smooth at every point 푥 ∈ (푐, 푑), 푔(푥) is a monotonous positive twice differentiable function. Then the infinitesimal transformation of the fractional integral (4) for 훽 = 1 − 훼 can be presented in the form

(︁ 1−훼 )︁ (︁ 1−훼 )︁ 푐퐼푔(¯푥) 푦¯ (푥) = 푐퐼푔(푥) 푦 (푥) + 푎휁훼−1[푥] + 표(푎), where 휁훼−1 is determined by the prolongation formula 1−훼 ′ ′ (︀ 훼 )︀ 휁훼−1[푥] = 푐퐼푔(푥) (휂 − 휉푦 )(푥) + 휉[푥]푔 (푥) 푐퐷푔(푥)푦 (푥). (8) Proof. Let us write the fractional integration operator in the new variables푥, ¯ 푦¯: ∫︁ 푥¯ ′ ∫︁ 푥+푎휉[푥] ′ (︁ 1−훼 )︁ 1 푦¯(¯휏)푔 (¯휏)푑휏¯ 1 푦¯(¯휏)푔 (¯휏)푑휏¯ 푐퐼푔(¯푥) 푦¯ (¯푥) ≡ 훼 = 훼 + 표(푎). Γ(1 − 훼) 푐 [푔(¯푥) − 푔(¯휏)] Γ(1 − 훼) 푐 [푔(푥 + 푎휉[푥]) − 푔(¯휏)] (9) To make the substitution of the function푦 ¯(¯휏), a certain substitution of the integration variable휏 ¯ is necessary. The most natural type of the substitution휏 ¯ = 휏 + 푎휉[휏] allows one to turn easily from푦 ¯(¯휏) to 푦(휏) (¯푦(¯휏) = 푦(휏) + 푎휂[휏] + 표(푎)). However, after this substitution the lower limit of integration becomes a function of the parameter 푎, which complicates further transformations signiﬁcantly and requires imposing additional restrictions on the type of the function 휉[푥] [9]. The substitution of variables 휏¯ = 푡 + 푎ℎ(푥, 푡), where 푔′(푥) 푔(푡) − 푔(푐) ℎ(푥, 푡) = 휉[푥] · (10) 푔(푥) − 푔(푐) 푔′(푡) 58 R.K. GAZIZOV, A.A. KASATKIN, S.YU. LUKASHCHUK is more optimal. Here 푡 is a new variable of integration. Such a substitution preserves the integration limits because 푡 = 푐 transforms to휏 ¯ = 푐, and 푡 = 푥 transforms to휏 ¯ = 푥 + 푎휉[푥]. Carrying out this substitution in (9) we obtain ∫︁ 푥 (︂ )︂⃒ (︁ 1−훼 )︁ 1 −훼 ′ 푑휏¯ ⃒ 푐퐼푔(¯푥) 푦¯ (푥) = 푦¯(¯휏) · [푔(푥 + 푎휉[푥]) − 푔(¯휏)] · 푔 (¯휏) ⃒ 푑푡 + 표(푎). Γ(1 − 훼) 푐 푑푡 ⃒휏¯=푡+푎ℎ(푥,푡) (11) Let us consider transformation of every subintegral factor in detail. To express푦 ¯(¯휏) via 푡 it is necessary to substitute in 푦(휏)+푎휂[휏]+표(푎) the argument 휏, which is transformed exactly into휏 ¯ during the substitution. It is known that the inverse inﬁnitesimal substitution has the form 휏 =휏 ¯− 푎휉[¯휏] + 표(푎). Expressing휏 ¯ via 푡 determined earlier, we obtain 휏 = 푡 + 푎ℎ(푥, 푡) − 푎휉[푡] + 표(푎). As a result we have

푦¯(¯휏)|휏¯=푡+푎ℎ(푥,푡) = (푦(휏) + 푎휂[휏] + 표(푎))|휏=푡+푎ℎ(푥,푡)−푎휉[푡]+표(푎) = (10) = 푦(푡 + 푎ℎ(푥, 푡) − 푎휉[푡]) + 푎휂[푡] + 표(푎) = 푦(푡) + 푎푦′(푡)(ℎ(푥, 푡) − 휉[푡]) + 푎휂[푡] + 표(푎) = ′ (10) 휉[푥]푔 (푥) 푔(푡) − 푔(푐) = 푦(푡) + 푎휂[푡] − 푎휉[푡]푦′(푡) + 푎 · 푦′(푡) + 표(푎). (12) 푔(푥) − 푔(푐) 푔′(푡) Then,

′ ′ (푔(푥 + 푎휉[푥]) − 푔(¯휏))|휏¯=푡+푎ℎ(푥,푡) = 푔(푥) + 푎휉[푥]푔 (푥) − 푔(푡) − 푎ℎ(푥, 푡)푔 (푡) + 표(푎) = (︂ 푔(푡) − 푔(푐) 푔′(푥) )︂ = 푔(푥) − 푔(푡) + 푎 휉[푥]푔′(푥) − 휉[푥] 푔′(푡) + 표(푎) = 푔(푥) − 푔(푐) 푔′(푡) (︂ 휉[푥]푔′(푥) )︂ = (푔(푥) − 푔(푡)) 1 + 푎 + 표(푎), 푔(푥) − 푔(푐) whence (︂ ′ )︂ −훼⃒ −훼 휉[푥]푔 (푥) [푔(푥 + 푎휉[푥]) − 푔(¯휏)] ⃒ = (푔(푥) − 푔(푡)) 1 − 훼푎 + 표(푎). (13) 휏¯=푡+푎ℎ(푥,푡) 푔(푥) − 푔(푐) Finally, (︂ )︂⃒ ′ 푑휏¯ ⃒ ′ ′′ (10) 푔 (¯휏) ⃒ = (푔 (푡) + 푎푔 (푡)ℎ(푥, 푡))(1 + 푎ℎ푡(푥, 푡)) + 표(푎) = 푑푡 ⃒휏¯=푡+푎ℎ(푥,푡) ′ (10) 휉[푥]푔 (푥) (︂ 푔(푡) − 푔(푐) 푑 푔(푡) − 푔(푐))︂ = 푔′(푡) + 푎 푔′′(푡) + 푔′(푡) + 표(푎) = 푔(푥) − 푔(푐) 푔′(푡) 푑푡 푔′(푡) (︂ 휉[푥]푔′(푥) )︂ = 푔′(푡) 1 + 푎 + 표(푎). (14) 푔(푥) − 푔(푐) Substituting (12),(13),(14) into (11), we obtain ∫︁ 푥 (︂ )︂⃒ (︁ 1−훼 )︁ 1 −훼 ′ 푑휏¯ ⃒ 푐퐼푔(¯푥) 푦¯ (¯푥) = 푦¯(¯휏)[푔(푥 + 푎휉[푥]) − 푔(¯휏)] 푔 (¯휏) ⃒ 푑푡 + 표(푎) = Γ(1 − 훼) 푐 푑푡 ⃒휏¯=푡+푎ℎ(푥,푡) ∫︁ 푥 (︂ ′ )︂ 1 ′ 휉[푥]푔 (푥) 푔(푡) − 푔(푐) ′ = 푦(푡) + 푎휂[푡] − 푎휉[푡]푦 (푡) + 푎 ′ 푦 (푡) × Γ(1 − 훼) 푐 푔(푥) − 푔(푐) 푔 (푡) (︂ 휉[푥]푔′(푥) )︂ (︂ 휉[푥]푔′(푥) )︂ × (푔(푥) − 푔(푡))−훼 1 − 훼푎 푔′(푡) 1 + 푎 푑푡 + 표(푎) = 푔(푥) − 푔(푐) 푔(푥) − 푔(푐) FRACTIONAL DIFFERENTIAL EQUATIONS. . . 59

1 ∫︁ 푥 푦(푡)푔′(푡)푑푡 푎 ∫︁ 푥 푔′(푡)푑푡 = 훼 + 훼 × Γ(1 − 훼) 푐 (푔(푥) − 푔(푡)) Γ(1 − 훼) 푐 (푔(푥) − 푔(푡)) (︂ 휉[푥]푔′(푥) [︂푔(푡) − 푔(푐) ]︂)︂ × 휂[푡] − 휉[푡]푦′(푡) + 푦′(푡) + (1 − 훼)푦(푡) + 표(푎) = 푔(푥) − 푔(푐) 푔′(푡) (︁ 1−훼 )︁ 1−훼 ′ = 푐퐼푔(푥) 푦 (푥) + 푎푐퐼푔(푥) (휂 − 휉푦 )(푥)+ 푎휉[푥]푔′(푥) ∫︁ 푥 ((푔(푡) − 푔(푐))푦′(푡) + (1 − 훼)푦(푡)푔′(푡) + 훼 푑푡 + 표(푎). Γ(1 − 훼)(푔(푥) − 푔(푐)) 푐 (푔(푥) − 푔(푡))

Let us use properties 1 and 2 to transform the last integral:

∫︁ 푥 ∫︁ 푥 ′ 1 1 퐷푡[푦(푡)(푔(푡) − 푔(푐))] − 훼푦(푡)푔 (푡) (7) ... = 훼 푑푡 = Γ(1 − 훼) 푐 Γ(1 − 훼) 푐 (푔(푥) − 푔(푡)) (7) 훼 1−훼 훼 훼 1−훼 (6) = 푐퐷푔(푥)[푦(푥)(푔(푥) − 푔(푐))] − 훼푐퐼푔(푥) 푦(푥) = 푐퐷푔(푥)(푔(푥)푦(푥)) − 푔(푐)푐퐷푔(푥)푦(푥) − 훼푐퐼푔(푥) 푦(푥) =

(6) 훼 1−훼 훼 1−훼 = 푔(푥)푐퐷푔(푥)푦(푥) + 훼푐퐼푔(푥) 푦(푥) − 푔(푐)푐퐷푔(푥)푦(푥) − 훼푐퐼푔(푥) 푦(푥) = (︀ 훼 )︀ = (푔(푥) − 푔(푐)) 푐퐷푔(푥)푦 (푥).

This ﬁnally results in

(︁ 1−훼 )︁ (︁ 1−훼 )︁ (︁ 1−훼 ′ ′ (︀ 훼 )︀ )︁ 푐퐼푔(¯푥) 푦¯ (¯푥) = 푐퐼푔(푥) 푦 (푥) + 푎 푐퐼푔(푥) (휂 − 휉푦 )(푥) + 휉[푥]푔 (푥) 푐퐷푔(푥)푦 (푥) + 표(푎), which proves the proposition.

Proposition 2. In the conditions of proposition 1 the infinitesimal transformation of the fractional derivative (5) of the order 훼 ∈ (0, 1) has the form

(︀ 훼 )︀ (︀ 훼 )︀ 푐퐷푔(¯푥)푦¯ (¯푥) = 푐퐷푔(푥)푦 (푥) + 푎휁훼[푥] + 표(푎), where

훼 ′ ′ (︁ 훼+1 )︁ 휁훼[푥] = 푐퐷푔(푥)(휂 − 휉푦 )(푥) + 휉[푥]푔 (푥) 푐퐷푔(푥) 푦 (푥). (15)

Proof. By the deﬁnition, we have

1 푑 (︁ )︁ (︀ 퐷훼 푦¯)︀ (¯푥) ≡ 퐼1−훼푦¯ (¯푥). 푐 푔(¯푥) 푔′(¯푥) 푑푥¯ 푐 푔(¯푥)

Using the inﬁnitesimal expansions

푑 (︂푑푥¯)︂−1 푑 푑 = = (1 − 푎퐷 휉[푥] + 표(푎)) , 푑푥¯ 푑푥 푑푥 푥 푑푥

1 1 (︂ 푔′′(푥) )︂ = 1 − 푎휉[푥] + 표(푎) , 푔′(¯푥) 푔′(푥) 푔′(푥)

(︁ 1−훼 )︁ (︁ 1−훼 )︁ 푐퐼푔(¯푥) 푦¯ (¯푥) = 푐퐼푔(푥) 푦 (푥) + 푎휁훼−1[푥] + 표(푎), 60 R.K. GAZIZOV, A.A. KASATKIN, S.YU. LUKASHCHUK we obtain 1 (︂ 푔′′ )︂ 푑 (︀ 퐷훼 푦¯)︀ (¯푥) = 1 − 푎휉 + 표(푎) (1 − 푎퐷 휉 + 표(푎)) (︀ 퐼1−훼푦 + 푎휁 + 표(푎))︀ = 푐 푔(¯푥) 푔′ 푔′ 푥 푑푥 푐 푔 훼−1 1 (︂ 퐷 (휉푔′))︂ = 1 − 푎 푥 (︀퐷 ( 퐼1−훼푦) + 푎퐷 휁 )︀ + 표(푎) = 푔′ 푔′ 푥 푐 푔 푥 훼−1 ′ 1 푎 (︂ 퐷 (휉푔 ) )︂ (8) = 퐷 ( 퐼1−훼푦) + 퐷 휁 − 푥 퐷 ( 퐼1−훼푦) + 표(푎) = 푔′ 푥 푐 푔 푔′ 푥 훼−1 푔′ 푥 푐 푔 (8) 푎 = 퐷훼푦 + (︀퐷 ( 퐼1−훼 (휂 − 휉푦′)) + 퐷 (휉푔′ 퐷훼푦) − 퐷 (휉푔′) 퐷훼푦)︀ + 표(푎) = 푐 푔 푔′ 푥 푐 푔 푥 푐 푔 푥 푐 푔 (︂ 1 )︂ = 퐷훼푦 + 푎 퐷 ( 퐼1−훼 (휂 − 휉푦′)) + 휉퐷 ( 퐷훼푦) + 표(푎) = 푐 푔 푔′ 푥 푐 푔 푥 푐 푔 훼 (︀ 훼 ′ ′ 훼+1 )︀ = 푐퐷푔 푦 + 푎 푐퐷푔 (휂 − 휉푦 ) + 휉푔 푐퐷푔 푦 + 표(푎). Here we omit the argument 푥 for all functions for the sake of brevity. Remark 1. When 푔(푥) = 푥 (15) transforms into the prolongation formula for the Riemann- Liouville type derivative obtained earlier [9], and in case of integer 훼, it coincides with the known classical prolongation formulae for integer-order derivatives [14]. Remark 2. Unlike integer-order derivatives, it is impossible to expand brackets in the right- hand side of (15) in the general case, because the fractional derivative of separate summands 휂 ′ and 휉푦 may not exist. An example of an operator with such coeﬃcients is 푋1 from section 3. Remark 3. One can show that the formulae (8) and (15) are valid for fractional integrals and derivatives of an arbitrary order, respectively.

2. Nonlocal symmetries Nonlocal symmetries of differential equations with integer-order derivatives have been known for a long time [15] and provide possibilities to construct additional invariant solutions and con- servation laws in a number of cases. Meanwhile, it should be mentioned that there exists no constructive algorithm of finding such symmetries. Several heuristic approaches are known that allow one to construct particular types of nonlocal symmetries. One of them is the introduction of nonlocal variables and extension of the transformation action to these variables. This approach can be successfully applied for equations with fractional-order derivatives. In this case the prolongation formulae (8), (15) constructed in the previous section can be used both for construction of nonlocal symmetries and for verification of their admittance by the equation. Let us demonstrate it by a simple example. We consider the equation 훼+1 0퐷푥 푦 = 0, 훼 ∈ (0, 1), (16) 훼−1 which has the well-known general solution 푦 = 푥 (푐1푥 + 푐2)(푐1, 푐2 are arbitrary constants). According to the definition of the fractional derivative, the equation (16) can be written in the form 2 1−훼 퐷푥(0퐼푥 푦) = 0, where ∫︁ 푥 (︀ 1−훼 )︀ 1 푦(푡) 0퐼푥 푦 (푥) = 훼 푑푡 Γ(1 − 훼) 0 (푥 − 푡) is the left sided integral of the fractional order 1 − 훼. 1−훼 Upon the nonlocal substitution 푧 = 0퐼푥 푦, the equation (16) is written in the form 푧′′ = 0, (17) which admits the known eight-parameter group [9] determined by the infinitesimal operators 휕 휕 휕 휕 푋 = , 푋 = , 푋 = 푥 , 푋 = 푧 , 1 휕푥 2 휕푧 3 휕푥 4 휕푥 FRACTIONAL DIFFERENTIAL EQUATIONS. . . 61

휕 휕 휕 휕 휕 휕 푋 = 푥 , 푋 = 푧 , 푋 = 푥2 + 푥푧 , 푋 = 푥푧 + 푧2 . 5 휕푧 6 휕푧 7 휕푥 휕푥 8 휕푥 휕푧 1−훼 1−훼 By virtue of the identity 0퐷푥 0퐼푥 푦 = 푦 it is possible to reverse the nonlocal substitution: 1−훼 푦 = 0퐷푥 푧. Applying the prolongation formula (15) with 푔(푥) = 푥, we can construct the prolongation of 1−훼 each operator to the fractional derivative 0퐷푥 푧: 1−훼 ′ 2−훼 휁1−훼 = 0퐷푥 (휂 − 휉푧 ) + 휉 0퐷푥 푧. (18) We omit the lower indices 0 and 푥 in the operators of fractional differentiation and integration for the sake of simplicity . The known relationship between the Riemann-Liouville and the Caputo derivatives (3) can be written in this case as 푓(0)푥−훽 퐷훽푓 ≡ 퐷퐼1−훽푓 = 퐼1−훽푓 ′ + , 훽 ∈ (0, 1). (19) Γ(1 − 훽) Differentiating (19), we have the relation 푓(0)푥−훽−1 퐷훽+1푓 = 퐷훽푓 ′ + . (20) Γ(−훽) The relations (19) and (20) allow one to write 푧(0)푥훼−1 푧(0)푥훼−2 퐼훼푧′ = 퐷1−훼푧 − , 퐷1−훼푧′ = 퐷2−훼푧 − (21) Γ(훼) Γ(훼 − 1) 1−훼 ′ when 훽 = 1 − 훼. Since the value 푧(0) exists, then the fractional derivative 0퐷푥 푧 also exists. The Leibniz rule for the fractional differentiation of the product of two functions (see [3]) also appears useful for construction of the prolongations: ∞ ∑︁ (︂훽)︂ 퐷훽(푓푔) = 퐷훽−푘푓 퐷푘푔. (22) 푘 푘=0 (︀훽)︀ 훽−푘 푘−훽 Here 푘 are binomial coefficients, 퐷 푓 = 퐼 푓 when 푘 > 훽. In particular, 퐷훽(푥푓) = 푥퐷훽(푓) + 훽퐷훽−1(푓), (23) 퐷훽(푥2푓) = 푥2퐷훽(푓) + 2훽푥퐷훽−1(푓) + 훽(훽 − 1)퐷훽−2(푓). (24) The fractional derivative of the power function has the form [3] Γ(훾 + 1) 퐷훼푥훾 = 푥훾−훼, 훾 > −1, 훼 ∈ . (25) Γ(훾 − 훼 + 1) R

Prolongation of the operator 푋1: Here 휉 = 1, 휂 = 0 and 훼−2 (21) 푧(0)푥 휁 = −퐷1−훼(푧′) + 퐷2−훼푧 = . 1−훼 Γ(훼 − 1)

Prolongation of the operator 푋2: 훼−1 (25) 푥 휁 = 퐷1−훼(1) = . 1−훼 Γ(훼)

Prolongation of the operator 푋5: 훼 (25) 푥 휁 = 퐷1−훼(푥) = . 1−훼 Γ(훼 + 1)

Prolongation of the operator 푋6: 1−훼 휁1−훼 = 퐷 (푧). 62 R.K. GAZIZOV, A.A. KASATKIN, S.YU. LUKASHCHUK

Prolongation of the operator 푋3: 1−훼 ′ 2−훼 휁1−훼 = −퐷 (푥푧 ) + 푥퐷 (푧).

The assumption that the ﬁnite value 푧(0) exists entails that (푥푧)|푥=0 = 0. Then, due to (19) we have 퐷1−훼(푥푧)′ = 퐷2−훼(푥푧) and substituting 푥푧′ as (푥푧)′ − 푧, we obtain

2−훼 1−훼 2−훼 (23) 1−훼 휁1−훼 = −퐷 (푥푧) + 퐷 푧 + 푥퐷 (푧) = (훼 − 1)퐷 푧.

Prolongation of the operator 푋4: 1−훼 ′ 2−훼 휁1−훼 = −퐷 (푧푧 ) + 푧퐷 (푧). Applying the equation (17), the Leibniz rule (22) and the representations (21) it is possible to eliminate nonlinearity under the operator of fractional differentiation: ∞ (︂ )︂ (22) ∑︁ 1 − 훼 (17) (21) 휁 = − 퐷푛(푧) 퐷1−훼−푛푧′+푧퐷2−훼(푧) = −푧퐷1−훼푧′−(1−훼)푧′퐼훼푧′+푧퐷2−훼(푧) = 1−훼 푛 푛=0 훼−1 훼−2 ′ 훼−1 훼−2 (21) (︂ 푧(0)푥 )︂ 푧푧(0)푥 푧 푧(0)푥 푧푧(0)푥 = −(1−훼)푧′ 퐷1−훼푧 − + = (훼−1)푧′퐷1−훼푧− + . Γ(훼) Γ(훼 − 1) Γ(훼 − 1) Γ(훼 − 1) The presented form of the coefficient of the prolonged operator is not the only one possible. In particular, we can eliminate the variable 푧′ by applying the representation of the fractional derivative 퐷1−훼푧 to the equation (17): ∞ (︂ )︂ 훼−1 ′ 훼 (22) ∑︁ 1 − 훼 (17),(25) 푧푥 (1 − 훼)푧 푥 퐷1−훼푧 = 퐷푛(푧) 퐷1−훼−푛1 = + , 푛 Γ(훼) Γ(훼 + 1) 푛=0 whence, due to relation Γ(훼 + 1) = 훼Γ(훼) we have 푧 (1 − 훼)푧′ = Γ(훼 + 1)푥−훼퐷1−훼푧 − 훼 . 푥 As a result we find (︁ 푧 )︁ (︂ 푧(0)푥훼−1 )︂ 푧푧(0)푥훼−2 휁 = − Γ(훼 + 1)푥−훼퐷1−훼푧 − 훼 퐷1−훼푧 − + = 1−훼 푥 Γ(훼) Γ(훼 − 1) (퐷1−훼푧)2 훼푧퐷1−훼푧 푧(0)Γ(훼 + 1)퐷1−훼푧 (︂ 1 훼 )︂ = −Γ(훼 + 1) + + + − 푧푧(0)푥훼−2 = 푥훼 푥 푥Γ(훼) Γ(훼 − 1) Γ(훼) (퐷1−훼푧)2 훼(푧 + 푧(0))퐷1−훼푧 푧푧(0)푥훼−2 = −Γ(훼 + 1) + − . 푥훼 푥 Γ(훼)

Prolongation of the operator 푋7: 1−훼 2 ′ 2 2−훼 휁1−훼 = 퐷 (푥푧 − 푥 푧 ) + 푥 퐷 푧.

Acting similarly to the procedure of the operator 푋3 prolongation, we ﬁnd 1−훼 1−훼 2 1−훼 2 2−훼 휁1−훼 = 퐷 (푥푧) − 퐷 퐷(푥 푧) + 퐷 (2푥푧) + 푥 퐷 푧 = (23),(24) = 3퐷1−훼(푥푧) − 퐷2−훼(푥2푧) + 푥2퐷2−훼푧 = 3푥퐷1−훼푧 + (3 − 3훼)퐼훼푧 − (4 − 2훼)푥퐷1−훼푧− − (2 − 훼)(1 − 훼)퐼훼푧 = (2훼 − 1)푥퐷1−훼푧 + (1 − 훼2)퐷훼푧.

Prolongation of the operator 푋8: 1−훼 2 ′ 2−훼 휁1−훼 = 퐷 (푧 − 푥푧푧 ) + 푥푧퐷 푧. (26) Applying the Leibniz rule (22), due to the equation (17) we ﬁnd ∞ (︂ )︂ (22) ∑︁ 1 − 훼 (17) 퐷1−훼(푧2) = 퐷푛푧 퐷1−훼−푛푧 = 푧퐷1−훼푧 + (1 − 훼)푧′퐼훼푧. (27) 푛 푛=0 FRACTIONAL DIFFERENTIAL EQUATIONS. . . 63

Similarly, applying the Leibniz rule for 푥푧 · 푧′ and taking into account that due to the equation (17) 퐷3(푥푧) = 0, we have

(22),(17) (21) − 퐷1−훼(푥푧푧′) = −푥푧퐷1−훼(푧′) − (1 − 훼)(푧 + 푥푧′)퐼훼푧′ − (1 − 훼)(−훼)푧′퐼훼+1푧′ = 훼−2 훼−1 훼 (21) 푧(0)푥 [︂ 푧(0)푥 ]︂ [︂ 푧(0)푥 ]︂ = −푥푧퐷2−훼푧+푥푧 −(1−훼)(푧+푥푧′) 퐷1−훼푧 − +훼(1−훼)푧′ 퐼훼푧 − = Γ(훼 − 1) Γ(훼) Γ(훼 + 1) 푧푧(0)푥훼−1 [︂ 푧푧(0)푥훼−1 푧′푧(0)푥훼 ]︂ = −푥푧퐷2−훼푧+ −(1−훼) 푧퐷1−훼푧 − + 푥푧′퐷1−훼푧 − + Γ(훼 − 1) (훼 − 1)Γ(훼 − 1) (훼 − 1)Γ(훼 − 1) [︂ 푧′푧(0)푥훼 ]︂ + 훼(1 − 훼) 푧′퐼훼푧 − = 훼(훼 − 1)Γ(훼 − 1) = −푥푧퐷2−훼푧 − (1 − 훼)푧퐷1−훼푧 − (1 − 훼)푥푧′퐷1−훼푧 + 훼(1 − 훼)푧′퐼훼푧. (28) Substituting (27) and (28) into (26) we obtain 1−훼 2 ′ 훼 ′ 1−훼 휁1−훼 = 훼푧퐷 푧 + (1 − 훼 )푧 퐼 푧 − (1 − 훼)푥푧 퐷 푧. As a result, the prolonged operators take the form 휕 푧(0)푥훼−2 휕 푋˜ = + , 1 휕푥 Γ(훼 − 1) 휕푧(1−훼) 휕 푥훼−1 휕 푋˜ = + , 2 휕푧 Γ(훼) 휕푧(1−훼) 휕 휕 푋˜ = 푥 + (훼 − 1)푧(1−훼) , 3 휕푥 휕푧(1−훼) 휕 (︂ 푧′푧(0)푥훼−1 푧푧(0)푥훼−2 )︂ 휕 푋˜ = 푧 + (훼 − 1)푧′푧(1−훼) − + , 4 휕푥 Γ(훼 − 1) Γ(훼 − 1) 휕푧(1−훼) 휕 푥훼 휕 푋˜ = 푥 + , 5 휕푧 Γ(훼 + 1) 휕푧(1−훼) 휕 휕 푋˜ = 푧 + 푧(1−훼) , 6 휕푧 휕푧(1−훼) 휕 휕 휕 푋˜ = 푥2 + 푥푧 + [︀(2훼 − 1)푥푧(1−훼) + (1 − 훼2)푧(−훼)]︀ , 7 휕푥 휕푧 휕푧(1−훼) 휕 휕 휕 푋˜ = 푥푧 + 푧2 + [︀훼푧푧(1−훼) − (1 − 훼)푥푧′푧(1−훼) + (1 − 훼2)푧′푧(−훼)]︀ , 8 휕푥 휕푧 휕푧(1−훼) (1−훼) 1−훼 1−훼 where 푧 ≡ 0퐷푥 푧. Whence, after the reverse substitution of the variables 푧 = 0퐼푥 푦, we ﬁnd symmetries of the equation (16): 휕 푦(훼−1)(0)푥훼−2 휕 푋 = + , 1 휕푥 Γ(훼 − 1) 휕푦 휕 휕 휕 푋 = 푥훼−1 , 푋 = 푥 + (훼 − 1)푦 , 2 휕푦 3 휕푥 휕푦 휕 (︂ 푦(훼)푦(훼−1)(0)푥훼−1 푦(훼−1)푦(훼−1)(0)푥훼−2 )︂ 휕 푋 = 푦(훼−1) + (훼 − 1)푦푦(훼) − + , 4 휕푥 Γ(훼 − 1) Γ(훼 − 1) 휕푦 휕 휕 푋 = 푥훼 , 푋 = 푦 , 5 휕푦 6 휕푦 휕 휕 푋 = 푥2 + [(2훼 − 1)푥푦 + (1 − 훼2)퐼푦] , 7 휕푥 휕푦 휕 휕 푋 = 푥푦(훼−1) + [훼푦푦(훼−1) − (1 − 훼)푥푦푦(훼) + (1 − 훼2)푦(훼)퐼푦] . 8 휕푥 휕푦 64 R.K. GAZIZOV, A.A. KASATKIN, S.YU. LUKASHCHUK

(훼−1) 1−훼 Here 푦 ≡ 0퐼푥 푦, 퐼푦 ≡ 0퐼푥푦. The symmetries 푋2, 푋3, 푋5, 푋6 are local, other symmetries are nonlocal. Let us note that (훼−1) the initial value 푦 (0) contained in the operators 푋1 and 푋4 is a natural initial condition for the formulation of the Cauchy problem for fractional differential equations. Let us show that the coefficients of the operators 푋1 . . . 푋8 satisfy the definite equation

휁훼+1|퐷훼+1푦=0 = 0, which takes the form 훼+1 ′ ⃒ 퐷 (휂 − 휉푦 )⃒퐷훼+1푦=0 = 0 for the equation (16). Operators 푋2,푋5,푋6. The veriﬁcation is trivial. For 푋6, we have 훼+1 ⃒ 퐷 (푦)⃒퐷훼+1푦=0 = 0. ′ 훼 For 푋2, we have 휂 − 휉푦 = 푥 . By virtue of (25) we obtain 퐷훼+1푥훼 = 0, because the gamma-function has poles of the ﬁrst order at the points 푥 = 0, 푥 = −푛, 푛 ∈ R. 훼+1 훼−1 Likewise for 푋5: 퐷 푥 = 0.

Operator 푋1: (︂푦(훼−1)(0)푥훼−2 )︂ 휁 = 퐷훼+1 − 푦′ . 훼+1 Γ(훼 − 1) Note that derivatives 퐷훼+1푦′ and 퐷훼+1푥훼−2 do not exist, therefore it is impossible to apply the operator 퐷훼+1 to the separate summands in this case. The relationship (19) with 푓 = 퐼1−훼푦 allows one to write the following representation of 푦: (퐼1−훼푦)(0) · 푥훼−1 푦(훼−1)(0)푥훼−1 푦 = 퐷1−훼퐼1−훼푦 = 퐼훼퐷퐼1−훼푦 + = 퐼훼퐷훼푦 + , (29) Γ(훼) Γ(훼) whence 푦(훼−1)(0)푥훼−2 푦′ = 퐷1−훼퐷훼푦 + (30) Γ(훼 − 1) due to (훼 − 1)Γ(훼 − 1) = Γ(훼). Then 훼+1 1−훼 훼 휁훼+1 = −퐷푥 (퐷푥 퐷푥 푦). Due to the relationship (19) and the equation (16) we have 훼 훼−1 훼 훼−1 (19) (퐷 푦)(0)푥 (16) (퐷 푦)(0)푥 퐷1−훼퐷훼푦 = 퐷퐼훼퐷훼푦 = 퐼훼퐷훼+1푦 + = (31) Γ(1 − 훽) Γ(1 − 훽) (the existence of (퐷훼푦)(0) follows from the formulation of the Cauchy problem for the original equation or from the existence of 푧′(0)). Due to (25) the 훼 + 1 fractional derivative of the expression (31) is equal to zero, whence

휁훼+1|퐷훼+1푦=0 = 0.

Operator of the dilation group 푋3: 훼+1 ′ 휁훼+1 = 퐷 ((훼 − 1)푦 − 푥푦 ). Applying the representation 푥푦′ = (푥푦)′ −푦 and the relationship 퐷훼+1(푥푦)′ = 퐷훼+2(푥푦) (which holds due to (푥푦)|푥=0 = 0), we obtain

훼+1 ′ 훼+1 훼+2 (23) 훼+2 훼+1 휁훼+1 = 퐷 (훼푦 − (푥푦) ) = 훼퐷 푦 − 퐷 (푥푦) = −푥퐷 (푦) − 2퐷 (푦), and

휁훼+1|퐷훼+1푦=0 = 0. FRACTIONAL DIFFERENTIAL EQUATIONS. . . 65

Operator 푋4: (︂ 푦(훼)푦(훼−1)(0)푥훼−1 푦(훼−1)푦(훼−1)(0)푥훼−2 )︂ 휁 = 퐷훼+1 (훼 − 1)푦푦(훼) − + − 푦(훼−1)푦′ . 훼+1 푥 Γ(훼 − 1) Γ(훼 − 1) Applying the Leibniz rule (22) it is easy to see that due to the equation (16) the fractional derivatives of the ﬁrst and the second summands vanish: ∞ ⃒ (22) (︂훼 + 1)︂ ⃒ 훼+1 (훼) ⃒ ∑︁ 훼+1−푛 훼+푛 ⃒ 퐷 (푦푦 )⃒ 훼+1 = 퐷 푦 · 퐷 푦 = 0, 푥 퐷 푦=0 푛 ⃒ 푛=0 ⃒퐷훼+1푦=0 ∞ ⃒ (22) (︂훼 + 1)︂ ⃒ (25) 훼+1 훼−1 (훼) ⃒ ∑︁ 훼+1−푛 훼−1 훼+푛 ⃒ 퐷 (푥 푦 )⃒ 훼+1 = 퐷 푥 · 퐷 푦 = 0. 푥 퐷 푦=0 푛 ⃒ 푛=0 ⃒퐷훼+1푦=0 To simplify the remaining parts of the expression we use the representation 푦′ (30):

(훼−1) (훼−1) 훼−2 (︂푦 (0)푦 푥 )︂ (30) (22) 퐷훼+1 − 푦(훼−1)푦′ = −퐷훼+1 (︀퐼1−훼푦 · 퐷1−훼퐷훼푦)︀ = 푥 Γ(훼 − 1) 푥 ∞ (︂ )︂ (22) ∑︁ 훼 + 1 = − 퐷훼+1−푛(퐷1−훼퐷훼푦) · 퐷푛(퐼1−훼푦). 푛 푛=0 By virtue of the equation 퐷훼+푛 = 0 and all summands with 푛 > 1 vanish. The ﬁrst two summands are also equal to zero due to the relationship (31), holding true for the equation: (31) −퐷훼+1(퐷1−훼퐷훼푦) · 퐼1−훼푦 − (훼 + 1)퐷훼(퐷1−훼퐷훼푦) · 퐷훼푦 = 0.

Remark. It is clear from the proof that an operator of a simpler form than 푋4 is admitted: 휕 푦(훼−1)푦(훼−1)(0)푥훼−2 휕 푋ˆ = 푦(훼−1) + . 4 휕푥 Γ(훼 − 1) 휕푦

Operator 푋7: 훼+1 2 2 ′ 휁훼+1 = 퐷 ((2훼 − 1)푥푦 + (1 − 훼 )퐼푦 − 푥 푦 ). The following equalities hold: 퐷훼+1퐼푦 = 퐷2퐼1−훼퐼푦 = 퐷2퐼(퐼1−훼푦) = 퐷퐼1−훼푦 = 퐷훼푦, 퐷훼(푥푦′) = 퐷훼(푥푦)′ − 퐷훼푦 = 퐷훼+1(푥푦) − 퐷훼푦 = 푥퐷훼+1푦 + 훼퐷훼푦, 퐷훼+1(푥푦′) = 퐷퐷훼(푥푦′) = (훼 + 1)퐷훼+1푦 + 푥퐷훼+2푦. Thus, 훼+1 훼 훼 ′ 훼 훼+1 ′ ⃒ 퐷 퐼푦 = 퐷 푦, 퐷 (푥푦 )|퐷훼+1푦=0 = 훼퐷 푦, 퐷 (푥푦 )⃒퐷훼+1푦=0 = 0. (32) Then, 훼+1 2 훼+1 훼+1 ′ 휁훼+1 = (2훼 − 1)퐷 (푥푦) + (1 − 훼 )퐷 퐼푦 − 퐷 (푥 · 푥푦 ) = = (2훼 − 1)푥퐷훼+1(푦) + (2훼 − 1)(훼 + 1)퐷훼푦 + (1 − 훼2)퐷훼푦 − 푥퐷훼+1(푥푦′) − (훼 + 1)퐷훼(푥푦′). After substituting the equation (16) and using the relationships (32), we obtain 2 2 훼 훼 휁훼+1|퐷훼+1푦=0 = (2훼 + 훼 − 1 + 1 − 훼 )퐷 푦 − 0 − 훼(훼 + 1)퐷 푦 = 0.

Operator 푋8: 훼+1 (훼−1) (훼) 2 (훼) (훼−1) ′ 휁훼+1 = 퐷 [훼푦푦 − (1 − 훼)푥푦푦 + (1 − 훼 )푦 퐼푦 − 푥푦 푦 ]. Let us use the Leibniz rule (22) for representation of every summand taking into account that 퐷훼+푛푦 = 0 due to the equation (16) when 푛 > 0. ∞ (︂ )︂ (22) ∑︁ 훼 + 1 (16) 퐷훼+1[훼푦퐼1−훼푦] = 훼 퐷훼+1−푛푦 퐷푛+훼−1푦 = 훼(훼 + 1)(퐷훼푦)2, 푛 푛=0 66 R.K. GAZIZOV, A.A. KASATKIN, S.YU. LUKASHCHUK

∞ (︂ )︂ (22) ∑︁ 훼 + 1 (16) (23) 퐷훼+1[(훼 − 1)푥푦퐷훼푦] = (훼 − 1) 퐷훼+1−푛(푥푦) 퐷푛+훼푦 = (훼 − 1)퐷훼+1(푥푦) = 푛 푛=0 (23) (16) = (훼 − 1)푥퐷훼+1(푦) + (훼 − 1)(훼 + 1)(퐷훼푦)2 = (훼2 − 1)(퐷훼푦)2,

∞ (︂ )︂ (22) ∑︁ 훼 + 1 (16) 퐷훼+1[(1 − 훼2)퐷훼푦 퐼푦] = (1 − 훼2) 퐷훼+1−푛(퐼푦) 퐷푛+훼푦 = 푛 푛=0 (16) (32) = (1 − 훼2)퐷훼+1(퐼푦) 퐷훼푦 = (1 − 훼2)(퐷훼푦)2,

∞ (︂ )︂ (22) ∑︁ 훼 + 1 (16) 퐷훼+1[−푥푦′퐼1−훼푦] = − 퐷훼+1−푛(푥푦′) 퐷푛+훼−1푦 = 푛 푛=0 (16) (32) = −퐷훼+1(푥푦′)퐼1−훼푦 − (훼 + 1)퐷훼(푥푦′)퐷훼푦 = −훼(훼 + 1)(퐷훼푦)2. It is easy to see that the sum of right-hand sides turns to zero, which was to be proved. Note that simpler operators are also admitted: 휕 휕 휕 푋ˆ = 푥푦(훼−1) + 훼푦푦(훼−1) , 푋¯ = (︀(훼 − 1)푥푦푦(훼) + (1 − 훼2)푦(훼)퐼푦)︀ . 8 휕푥 휕푦 8 휕푦 Remark. In our previous paper [9] ﬁve local symmetries, including the projective operator 휕 휕 푋 = 푥2 + 훼푥푦 9 휕푥 휕푦 were obtained from the principle of invariance of the equation (16). The present operator cannot be combined from 푋1, . . . , 푋8, but the closest to it is 푋7 (obtained from the projective operator for the equation 푧′′ = 0). It is easy to verify that the nonlocal operator 휕 푋 ≡ 푋 − 푋 = [(훼 − 1)푥푦 + (1 − 훼2)퐼푦] 10 7 9 휕푦 is admitted by the equation (16). Meanwhile, in the boundary case 훼 = 1 the operator 푋10 turns to zero, i.e. 푋7 matches 푋9.

Conclusion The prolongation formulae obtained in the paper give an opportunity to investigate symmetry properties of a new class of differential equations, containing fractional derivatives of a function with respect to another function. Meanwhile, the following important problem to be solved is the development of a method that allows to solve the resulting determining equations. The main difficulty in this case is caused by splitting rules for the determining equation. Another direction of the further research is systematization of results considering nonlocal symmetries of fractional differential equations and developing new algorithms of their construction. The problem of determining classification rules for nonlocal symmetries of such equations also seems to be important.

BIBLIOGRAPHY 1. R. Metzler, J. Klafter The random walk’s guide to anomalous diffusion: A fractional dynamic approach // Phys. Rep., 2000, V. 339, P. 1–77. 2. Uchaikin V.V. Method of fractional derivatives (in Russian). — Artishok publ., Ulyanovsk, 2008. 512 p 3. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and derivatives: Theory and Appli- cations. — Gordon and Breach Science Publishers, Yverdon, 1993. FRACTIONAL DIFFERENTIAL EQUATIONS. . . 67

4. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo Theory and applications of fractional differential equations. — Elsevier, Amsterdam, 2006. 5. Nahushev A.M. Fractional calculation and its application. (In Russian). — M.: Phismatilit. 2003. 272 p. 6. G. Jumarie Modified Riemann-Liouville derivative and fractional Taylor series of nondifferen- tiable functions further results // Computers and Mathematics with Applications. Vol. 51, 2006. P. 1367–1376. 7. K.M. Kolwankar, A.D. Gangal H¨olderexponents of irregular signals and local fractional derivatives // Pramana J. Phys. V. 48, No. 1 (1997). P. 49–68. 8. E Buckwar, Yu. Luchko Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations // J. Math. Anal. Appl., 1998, V. 227, P. 81–97. 9. Gazizov R.K., Kastkin A.A., Lukashchuk S.Yu. Continuous groups of transformations of fractional differential equations // Vestnik USATU. 2007. V. 9, 3 (21). P. 125–135. 10. R.K. Gazizov, A.A. Kasatkin, S.Yu. Lukashchuk Symmetry properties of fractional diffusion equations // Physica Scripta. 2009. T 136, 014016. 11. R.K. Gazizov, A.A. Kasatkin, S.Yu. Lukashchuk Group-Invariant Solutions of Fractional Differ- ential Equations. ”— Nonlinear Science and Complexity, Springer. 2011. P. 51–59. 12. R. Sahadevan, T. Bakkyaraj Invariantanalysis of timefractional generalized Burgers and Ko- rtewegde Vries equations // Journal of Mathematical Analysis and Applications, V. 393, Issue 2. P. 341–347. 13. Gazizov R.K., Kasatkin A.A., Lukashchuk S.Yu. Symmetry properties of differential equations of the fractional order translation // Works of the Institute of mechanics by the name of R.R. Mav- lyutov USC RAS. Issue. 9. / Works of the 5th Russian conference with the international par- ticipation ”Multi-phase systems: theory and application” (Ufa, July 2-5, 2012.). Part 1. — Ufa: Neftegazovoye delo, 2012. P. 59–64 14. N.H. Ibragimov (ed.) CRC Handbook of Lie Group Analysis of Differential Equations. ”— CRC Press, Boca Raton. V. 1. 1994. 430 p. 15. Akhatov I.Sh., Gazizov R.K., Ibragimov N.H. Nonlocal symmetries. Heuristic approach. ”— Modern problems of mathematics. The latest achievements. Results of science and technology. M.: VINITI, 1989. V. 34. P. 3–83.

Rafail Kavyevuch Gazizov, Ufa State Aviation Technical University, Laboratory ”Group analysis of mathematical models of natural sciences, techniques and technologies”, 12, Karl Marx str., Ufa, Russia, 450000 E-mail: [email protected] Alexey Alexandrovich Kasatkin, Ufa State Aviation Technical University, Laboratory ”Group analysis of mathematical models of natural sciences, techniques and technologies”, 12, Karl Marx str., Ufa, Russia, 450000 E-mail: alexei [email protected] Stanislav Yurjevich Lukashchuk, Ufa State Aviation Technical University, Laboratory ”Group analysis of mathematical models of natural sciences, techniques and technologies”, 12, Karl Marx str., Ufa, Russia, 450000 E-mail: [email protected]