Partial Fractional Derivatives of Riesz Type and Nonlinear Fractional Differential Equations

Partial fractional derivatives of Riesz type and nonlinear fractional differential equations

Vasily E. Tarasov

Nonlinear Dynamics An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems

ISSN 0924-090X Volume 86 Number 3

Nonlinear Dyn (2016) 86:1745-1759 DOI 10.1007/s11071-016-2991-y

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1 23 Author's personal copy

Nonlinear Dyn (2016) 86:1745–1759 DOI 10.1007/s11071-016-2991-y

ORIGINAL PAPER

Partial fractional derivatives of Riesz type and nonlinear fractional differential equations

Vasily E. Tarasov

Received: 15 July 2015 / Accepted: 27 July 2016 / Published online: 4 August 2016 © Springer Science+Business Media Dordrecht 2016

Abstract Generalizations of fractional derivatives of of noninteger orders. The most famous deﬁnitions were noninteger orders for N-dimensional Euclidean space proposed by Riemann, Liouville, Grünwald, Letnikov, are proposed. These fractional derivatives of the Riesz Marchaud, Riesz, Caputo [6–12]. All fractional-order type can be considered as partial derivatives of non- derivatives have a set of interesting unusual charac- integer orders. In contrast to the usual Riesz deriva- teristic properties [13–19] such as a violation of the tives, the suggested derivatives give the usual partial usual Leibniz rule, a deformations of the usual chain derivatives for integer values of orders. For integer val- rule, a violation of the semi-group property and other. ues of orders, the partial fractional derivatives of the For example, the derivatives and integrals of noninteger Riesz type are equal to the standard partial derivatives orders are noncommutative and nonassociative opera- of integer orders with respect to coordinate. Fractional tors in general. The fractional derivatives violate the generalizations of the nonlinear equations such as sine- usual Leibniz rule, and this violation is a characteris- Gordon, Boussinesq, Burgers, Korteweg–de Vries and tic property for all types of fractional derivatives [20]. Monge–Ampere equations for nonlocal continuum are Fractional-order derivatives can be characterized by a considered. deformations of the usual chain rule [14,15], which becomes more complicated. Keywords Fractional calculus · Fractional derivative · Fractional calculus has a wide application in physics Nonlocal continuum · Fractional dynamics · Nonlinear (for example see [21–27]), since it allows us to describe fractional equations the behavior of systems and media that are characterized by power-law nonlocality and power-law long- Mathematics Subject Classiﬁcation 26A33 term memory (hereditarity). The unusual mathematical properties of fractional-order operators give a possibility of building of new mathematical models for nontriv- 1 Introduction ial and unusual complex continua systems, processes and media. Theory of fractional-order differentiation and integra- In papers [28,29], we propose the fractional deriv- tion has a big history [1–5]. It has been proposed various D± α atives with respect to coordinate x j C j of noninte- types and forms of fractional derivatives and integrals ger order α for N-dimensional Euclidean space. These + B fractional derivatives are given in two forms (even “ ” V. E. Tarasov ( ) and odd “−”). These derivatives cannot give the usual Skobeltsyn Institute of Nuclear Physics, Lomonosov α Moscow State University, Moscow, Russia 119991 local derivatives for some integer values of . The frac- D+ α e-mail: [email protected] tional derivatives of the Riesz type C j give the par- 123 Author's personal copy

1746 V. E. Tarasov tial derivatives of integer orders only for even values Grünwald–Letnikov (see Section 20 of [6]), the Mar- α = > ∈ N D− α 2m 0, where m . The derivatives C j chaud fractional derivative (see Section 5.4 of [6]) and give the partial derivatives of integer orders only for the Caputo derivatives by using equations 2.4.6 and odd values α = 2m + 1 > 0, where m ∈ N.The 2.4.7 of [8]. For example, the composition of left- D− 2m D+ 2m+1 derivatives C j and C j are nonlocal differ- sided and right-sided Grünwald–Letnikov gives the ential operators of integer orders that cannot be repre- fractional Grünwald-Letnikov-Riesz derivatives (see ∂2m/∂ 2m ∂2m+1/∂ 2m+1 sented as x j and x j . As a result, Section 20 of [6]). The partial derivatives of this type the fractional generalization of local continuum mod- are considered and applied in [45](seealso[46,47]). α D± j els by using C j cannot give a local description for It should be noted that the Grünwald-Letnikov frac- some integer values of α j . In addition, we have some tional derivatives coincide with the Marchaud frac- arbitrariness in formulation of fractional nonlocal mod- tional derivatives for the function space L p(R), where α D+ j < ∞ els that is caused by a possibility to use C j and 1 p (see Theorem 20.4 in [6]). Therefore, α D− j the fractional Grünwald–Letnikov–Riesz derivatives C j in the same equation. To minimize this arbitrariness at fractional generalizations of local models, can be considered as a composition of left-sided and we suggested a rule of “fractionalization” in [29]. This right-sided regularized derivatives, one of which is the rule suggests to replace the usual partial derivatives of Marchaud fractional derivative (see Section 5.4 of [6], and about a regularized fractional derivative see also odd orders with respect to x j by the fractional deriv- D− α [48]). atives C j , and the usual partial derivatives of even In this paper, we develop an approach to formula- orders with respect to x j by the fractional derivatives D+ α tion of partial fractional, which is proposed in [28,29]. C j . In this paper, we propose another way to minimize The representation of the suggested partial fractional arbitrariness of fractional generalization of continuum derivative is connected with representation of the Riesz models. We suggest new fractional-order derivatives fractional derivatives in the Lizorkin form, which is α D j that are generalizations of the fractional deriv- suggested in [33] and described in detail in Sections j + α − α 25 and 26 of [6]. A main advantage of the suggested atives D and D . The main advantage of this C j C j approach is based on the direct connection of these generalized derivative is a direct connection with the fractional derivatives with the lattice fractional deriva- usual partial derivatives for all integer orders. For inte- tives that are proposed in [28,29]. Main disadvantage ger values α = n ∈ N, these partial fractional deriva- α of the other approached, which can be based on the tives of the Riesz type D are equal to the standard j Liouville, Caputo, Marchaud fractional derivatives, is partial derivatives of integer orders n with respect to x j an absence of direct relationship with the lattice representation (see Sect. 5 of [28] and also [29]). An n ∂n D = (n ∈ N). existence of a relationship with lattice models is more ∂ n j x j appropriate to get microstructural and nanostructured models of continuum and media with power-law nonlo- This property allows us much easier to build fractional cality, where the long-range intermolecular and inter- nonlocal models of continua and fields. atomic interactions are crucial in determining nonlo- In this paper, we propose the partial fractional deriv- cal properties. Using the suggested operators, we con- atives of the Riesz type. The Riesz fractional deriv- sider fractional generalizations of well-known nonlin- atives and potentials have been suggested in [30,31] ear equations, such as sine-Gordon, Boussinesq, Burg- (see also [6,32]). These operators also are consid- ers, Korteweg–de Vries and Monge–Ampere equa- ered in different works (see [6,8,9,33], [34–41,101] tions. and [28,29,42–44]). There is a relation between the It should be noted that the suggested approach can Riesz fractional derivative (see Sections 25 and 26 of be applied to the Riesz fractional derivatives, which [6]) and the left-sided and right-sided Liouville frac- are defined by the fractional m-dimensional differential tional derivatives, which is shown in Section 12 of operators [31,32]. In this case to define new partial [6](seealso[9]) for one-dimensional case. Another fractional derivatives, we should use these operators way of relating the Riesz derivative to other fractional in the one-dimensional form instead of the derivatives D+ α derivatives is compositions of left-sided and right-sided C j in the corresponding equations. 123 Author's personal copy

Partial fractional derivatives of Riesz type 1747

2 Fractional derivatives of Riesz type Proposition 1 The Fourier transform F of the fractional derivatives of the Riesz type with respect to x j Let us give a definition of the fractional derivatives of has the form the Riesz type that is proposed in [29]. + α α Definition 1 The fractional derivative of the Riesz type F D f (r) (k) =|k | (F f )(k). (3) C j j of the order α with respect to x j is defined by the equation ( ) The property (3) is valid for functions f r from for ∞ 1 + α 1 the Lizorkin space and the space C (R ) of infinitely D f (r) := C ( ,α) differentiable functions on R1 with compact support. j d1 m × 1 (m )( ) ,(<α< ), α+ z f r dz j 0 m (1) Proof The proof of this Proposition is a consequence of R1 |z | 1 j j the properties of the Riesz fractional derivatives [6,8] (m f )(r) m RN = where z j is a finite difference of order of a for with N 1. N function f (r) with the vector step z j = z j e j ∈ R for (− )2m = (− )m the point r ∈ RN . The centered difference with respect Using that i 1 , the fractional derivatives of the Riesz type of even α = 2m, where m ∈ N, to z j are connected with the usual partial derivative of inte- m m! ger orders 2m by the relation (m f )(r) := (−1)n f (r−(m/2−n) z e ). z j ! ( − )! j j = n m n n 0 2m + 2m ∂ f (r) (2) D f (r) = (−1)m . (4) C ∂ 2m j x j The constant d1(m,α)is defined by α = 3/2 For 2, the derivative of the Riesz type is the local π A (α) + d (m,α):= m , operator −∂2/∂x2. The fractional derivatives D 2m 1 2α(1 + α/2)((1 + α)/2) sin(πα/2) j C j for even orders α are local operators. Note that the where D+ 1 derivative C i cannot be considered as a derivative [m/2] ! s−1 m α of first order with respect to x j , i.e., Am(α) := 2 (−1) (m/2 − s) s!(m − s)! s=0 ∂ ( ) D+ 1 ( ) = f r , for the centered difference (2). C f r (5) j ∂x j The constant d1(m,α)is different from zero for all α>0 in the case of an even m and centered difference and it is nonlocal operator. (m f ) (see Theorem 26.1 in [6]). Note that the integral j D+ 1 (1) does not depend on the choice of m >α. Therefore, Remark 2 The derivative C i can be considered as | | we can always choose an even number m so that it is the Hilbert transform of k j [9]. All derivatives of the D+ α α = + greater than parameter α, and we can use the centered Riesz type C j for odd orders 2m 1, where ∈ N difference (2) for all positive real values of α. m , are nonlocal operators that cannot be consid- ∂2m+1/∂ 2m+1 ered as usual partial derivatives x j . Remark 1 Using (1), we can see that the partial frac- D+ α ( ) tional derivative C j f r can be considered as the Riesz fractional derivative in the Lizorkin form [33] 3 Conjugated fractional derivatives of Riesz type (see also Sections 25.4 and 26 of [6], and Section 2.10 ( ) of [8]) of the function f r with respect to one compo- Let us give a definition of the Riesz fractional integra- ∈ R1 ∈ RN nent xj of the vector r , i.e., the operator tion, which is usually called the Riesz potential (see D+ α C j can be considered as a partial fractional deriva- equations 25.31 and 25.32 of [6] and Section 2.10 of tive of the Riesz type. [8]). An important property of the fractional derivatives of the Riesz type is the Fourier transform F of this Definition 2 The Riesz fractional integrals of order operators. α>0 can be defined by the equation 123 Author's personal copy

1748 V. E. Tarasov α ( ) := ( − ) ( ) N ,(α>), The Riesz kernel Rα,1(r) is deﬁned by Ir f r Rα,N r z f z d z 0 RN (( − α)/ ) (6) 1 2 α−1 Rα, (r) = |r| (12) 1 2α π 1/2 (α/2) where the Riesz kernel Rα,N (r) is deﬁned by if α = 2n + 1 where n = 0, 1, 2,,....Forα = 2n + 1, ((N − α)/2) α−N α, ( ) Rα,N (r) = |r| (7) the Riesz kernel R 1 r is 2α π N/2 (α/2) ( ) =− 1 | |α−1 | |, if α = N + 2n and α = N, where n, N ∈ N.For Rα,1 r r ln r (13) γ1(α) α = N + 2n and α = N, the Riesz kernel Rα,N (r) is where 1 α−N Rα,N (r) =− |r| ln |r|, (8) γN (α) n 2n 1/2 γ1(α) = γ1(2n + 1) = (−1) 2 π n! (n + 1/2). where (14)

γN (α) = γN (N + 2n) Note that the integration (11) can be considered as + − / = (−1)n 2N 2n 1π N 2 n! (N/2 + n), (9) integration (6)forN = 1 with respect to one variable 1 2 x j ∈ R . For example, Eq. (11)for(x, y) ∈ R gives = , , ,... and n 0 1 2 . α +∞ I+ ( , ) = ( − ) ( , ) ,(α>), C f x y Rα,1 x z f z y dz 0 The Riesz fractional integrals (6) have the Fourier x −∞ transform F in the form (15) α +∞ I+ ( , ) = ( − ) ( , ) ,(α>). α −α C f x y Rα,1 y z f x z dz 0 F ( ) =| | (F )( ). y −∞ Ir f r k f k (10) (16) Equation (10) holds for the operator (6) if the function ( ) f r belongs to the Lizorkin space [6,8]. Note that the An important property of the Riesz fractional inte- Lizorkin spaces are invariant with respect to application grals of the Riesz type with respect to x j is the Fourier of the Riesz fractional integration. This means that the transform F of this integrals in the form functions obtained by applying the Riesz integration to functions from the Lizorkin space also belong to this + α −α F I f (r) =|k | (F f )(k). (17) space. C j j In papers [28,29], we suggest to define the fractional + α integral I of the Riesz type as the Riesz potential N C j where k = = k j e j . of order α with respect to x . j 1 j Let us note that the distinction between the fractional +α integral of the Riesz type I and the Riesz potential Definition 3 Factional integral of the Riesz type with C j α | |−α | |−α respect to x is defined by the equation Ir is the use of k j instead of k . The continuum j I+ α integral C j of the Riesz type is an integration of α ( ) I+ ( ) = ( − ) f r with respect to one variable x j instead of all N C f r Rα,1 x j z j ,... α j R1 variables x1 xN in Ir . ( ) × f (r + (z j − x j ) e j ) dz j ,(α>0), (11) If f r belongs to the Lizorkin space as a function of x j , then we have [6] the semi-group property where r = N x e is the radius vector, e is the j=1 j j j basis of the Cartesian coordinate system, and + α + β + α + β I I f (r) = I f (r), (18) C j C j C j f (r) = f (x1, x2,...,x j−1, x j , x j+1,...,xN ),

f (r + (z j − x j ) e j ) = f (x1, x2,...,x j−1, z j , x j+1,...,xN ). where α>0 and β>0. 123 Author's personal copy

Partial fractional derivatives of Riesz type 1749 D+ α D− 2 The fractional derivative C j yields an operator Note that the continuum derivative C j cannot +α inverse to the fractional integration I for a special be considered as a local derivative of second order C j D− α space of functions with respect to x j . The derivatives C j for even α = ∈ N orders 2m, where m , are nonlocal operators α α that cannot be considered as usual partial derivatives D+ I+ ( ) = ( ), (α > ). C C f r f r 0 (19) ∂2m/∂ 2m j j x j .

Equation (19) holds for f (r) belonging to the Lizorkin 4 Partial fractional derivatives of the Riesz type space of functions with respect to xi ∈ R. Moreover, this property is also valid for the fractional integration 1 Let us give a definition of the partial fractional deriva- in the framework of a set of L p-spaces L p(R ) where 1 p < 1/α (see Theorem 26.3 in [6]). tives. Using the property (10), we can see that the frac- Definition 5 The partial fractional derivatives of the D+ α α = tional derivative C j with 1 cannot be consid- Riesz type of the order α is the linear operator ered as the standard derivative of first order with respect to x j . Therefore, we define new fractional derivative α πα α πα α −α + − D of the Riesz type. D := cos D + sin D . C j j 2 C j 2 C j Definition 4 The conjugate Riesz fractional derivative (23) of the order α with respect to x is defined by the equa- j D± α tion where C j are defined by Eqs. (1) and (20). ⎧ ∂ + α− ⎪ D 1 α>1 The partial fractional derivatives (23)havethefol- ⎨ ∂x j C j − α ∂ lowing Fourier transform. D := ∂ α = 1 (20) C ⎪ x j j ⎩ ∂ + −α I 1 0 <α<1. Proposition 2 The Fourier transform F of the partial ∂x j C j fractional derivatives of the Riesz type has the form <α< D− α For 0 1, the operator C j is analogous to the conjugate Riesz derivative [10]. Therefore, the α πα ( )/ α F D ( ) ( ) = i sgn k j 2| | (F )( ), D− α α> f r k e k j f k operator C j for all values 0 is called conjugate j derivative of the Riesz type [29]. (24) Using (3) and (17), we can see that the Fourier transform F of the fractional derivative (20) is given by where α>0. Equation (24) is valid for functions f (r) ∞(R1) − α from for the Lizorkin space and the space C of F D f (r) (k) R1 C j infinitely differentiable functions on with compact α−1 α support. = ikj |k j | (F f )(k) = i sgn(k j ) |k j | (F f )(k). (21) Proof The proof is based on relations (3), (21) and the Euler’s formula. Using (23), we get α For the odd values of α,Eqs.(4) and (20)givethe F D f (r) (k) relation j πα α = F D+ ( ) ( ) cos C f r k 2m + 1 ∂2m+1 f (r) 2 j D− ( ) = (− )m ,(∈ N). f r 1 + m πα α C j ∂ 2m 1 − x j + sin F D f (r) (k) 2 C j (22) πα α πα α = cos |k j | + i sgn(k j ) sin |k j | 2 2 Equation (22) means that the fractional derivatives πα ( )/ α ×(F )( ) = i sgn k j 2| | (F )( ). D− α α f k e k j f k C i of the odd orders are local operators represented by the usual derivatives of integer orders. 123 Author's personal copy

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The partial fractional derivatives of the Riesz type Remark 3 Let us compare the suggested fractional (23) of integer order α>0 are directly related with the derivatives with the regularised fractional derivative, partial derivatives of integer orders. which is proposed in [48], the unified fractional derivatives, which is considered in [49], the Marchaud deriva- Proposition 3 The partial fractional derivatives of the α tive [6] and other. The derivatives D have the follow- Riesz type of orders α for integer positive values α = j ing advantages compared with the above mentioned: m ∈ N are usual partial derivative of integer orders m, (1) The Marchaud and regularised fractional deriva- m ∂m f (r) tives are defined as ordinary differential operators D f (r) = , (25) j ∂xm in the standard definitions. The proposed deriva- j D α tives j are the partial derivatives with respect to where m ∈ N. variable x j of order α j in general. D α (2) The derivatives j have analogs for physical lat- Proof The proof of this proposition follows from rela- tices as opposed to the fractional centered deriv- tions (4) and (22). Let us first consider the even values atives, that is based on the central differences of α = ∈ N of orders. For even 2m, where m , the partial noninteger orders (for details see Sect. 5 of [28]). fractional derivatives of the Riesz type are given by The fractional derivatives, which are based on the D 2m ( ) = (π ) D+ 2m ( ) fractional central differences of type 2, correspond f r cos m C f r j j to interaction of lattice particles with virtual parti- − 2m cles with half-integer numbers that do not exist in + sin(π m) D f (r) C j the physical lattices. 2m + 2m ∂ f (r) = (−1)m D f (r) = (−1)2m (3) The regularised fractional derivatives [48]are C j ∂ 2m x j based on the Grünwald–Letnikov fractional deriv- ∂2m ( ) atives. These derivatives have the lattice and finite- = f r . ∂ 2m difference analogs that can be considered as an x j asymptotic discretization only (for details see Sec- For odd α = 2m+1, where m ∈ N, the partial fractional tion 2.2 of [29] and [50]). The Fourier transform derivatives of the Riesz type are F of the Grünwald–Letnikov fractional differences α 2m + 1 + 2m + 1 ∇ D f (r) = cos(π m + π/2) D f (r) h,± is given by j C j α α F{∇ ,± f (x)}(k) = (1 − exp{±ikh}) F{ f (x)}(k) + h + (π + π/ ) D− 2m 1 ( ) ( ) ∈ (R) sin m 2 C f r for any function f x L1 (see Property 2.30 j of [8]). Therefore, these fractional differences (and + = (π ) D− 2m 1 ( ) lattice derivatives) cannot be considered as an exact cos m C f r j discretization of the fractional derivatives [50]. − 2m + 1 Note that the standard Leibniz rule does not hold = (−1)m D f (r) C j for the Grünwald–Letnikov differences of integer α ∂2m+1 ( ) ∂2m+1 ( ) order. The suggested fractional derivatives D are = (− )2m f r = f r . j 1 + + based on the lattice derivatives (T -differences) that ∂x2m 1 ∂x2m 1 j j can be considered as an exact discrete analogs of As a result, we have the fractional derivatives [50]. The Fourier transform F of the T -ifference T α of order α is given α ∂α f (r) h D f (r) = (α ∈ N). (26) by j ∂xα j T α πα ( )/ α F{ ( )}( ) = i sgn k j 2 | | F{ ( )}( ), h f x k e k j f x k which is analog of (24). The main characteristic This property greatly simplifies the construction of property of these differences is that the standard fractional nonlocal generalizations of models of con- Leibniz rule holds for the T -differences of integer tinua and fields. order (for details see [50,51]). 123 Author's personal copy

Partial fractional derivatives of Riesz type 1751

5 Properties of partial fractional derivatives of the The partial fractional derivatives of the Riesz type for Riesz type different variables xi and x j , where i = j, commute α α α α Let us describe some properties of the partial fractional D 1 D 2 f (r) = D 2 D 1 f (r)(i = j) i j j i derivatives of the Riesz type. All these properties are similar to properties of the Riesz derivatives of non- (29) integer orders [8,31,32]. As the properties of the par- for sufficiently good functions f (r). The commutation tial fractional derivatives, we will consider the linear- relation (29) with α1 = α2 = 1is ity, (non)commutativity, semigroup property and the 1 1 1 1 violation of the chain and Leibniz rules. It should be D D f (r) = D D f (r), (i = j) noted that the violation of the standard Leibniz and i j j i chain rules can be considered as characteristic proper- (30) ties of the derivatives of noninteger orders [15,18–20]. that can be rewritten in the form The performance of commutative and semigroup prop- ∂2 ( ) ∂2 ( ) erties cannot be considered as fundamental properties f r = f r . ∂ ∂ ∂ ∂ (31) that any partial derivative should satisfy. To prove this xi x j x j xi fact, we demonstrate that these properties cannot be It is well known that the commutation relation (31) regarded as the fundamental properties even for stan- may be broken for discontinuous functions f (r) and dard derivatives of integer orders. For standard partial if the partial derivatives of f (r) are not continuous. derivatives of integer orders and the suggested partial For example, the commutativity may be broken if the derivatives of noninteger orders, the implementation function does not have differentiable partial derivatives. or violation of these two properties is dependent on the It is possible when the Clairaut’s theorem on equality considered function spaces. Therefore, the suggested of mixed partial derivatives is not satisfied, i.e., when partial fractional derivatives of the Riesz type are rea- the second partial derivatives are not continuous. sonable to consider as operators that satisfy these main Let us introduce the notation x = xi and y = x j , properties of partial derivatives for fractional and inte- where i = j, and a function f (x, y) = g(r).Inthe ger orders. The fact that the proposed fractional deriva- general case, the partial derivatives are not commute, tives coincide with the usual partial derivative for inte- i.e., ger orders and satisfy the same algebraic properties is a necessary and sufficient condition for operators to be ∂ ∂ f ∂ ∂ f (x, y) = (x, y). (32) considered an promising partial fractional derivatives. ∂x ∂y ∂y ∂x 1) Linearity. The partial fractional derivatives of the Riesz type are As an example, we can consider the function the linear operators xy(x2 − y2) f (x, y) := ((x, y) = (0, 0)), x2 + y2 α α α f (x, y) := 0 ((x, y) = 0). (33) D af(r)+bg(r) = a D f (r)+b D g(r), j j j The partial derivatives of the function (33)exist (27) and everywhere continuous. The limit of difference quotients shows that , ∈ R where a b . ∂ f ∂ f 2) Commutative property. (0, 0) = 0, (0, 0) = 0. (34) ∂x ∂y In general, the partial fractional derivatives with respect to the same variable x j are not commute However, the second partial derivatives are not continuous at the point (x, y) = (0, 0). Using that α1 α2 α2 α1 D D f (r) = D D f (r), (α1 = α2). j j j j ∂ f ∂ f (x, 0) = x, (0, y) =−y, (35) (28) ∂y ∂x 123 Author's personal copy

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we get ∂2 f 1 (x ) := lim f (x + 2 h) ∂ 2 0 → 2 0 x h 0 h 2 ∂ f ∂ f − ( + ) − ( ) . (0, 0) = 1, (0, 0) =−1. (36) 2 f x0 h f x0 (41) ∂x ∂y ∂y ∂x Using (40), the repeated action of the first derivatives As a result, we have inequality (32) at the point (x, y) = is defined by the equation (0, 0) since (36)gives1=−1. Therefore, the commutative property for is not satisfied even for the partial ∂ ∂ f (x0) derivatives of integer orders in the general case. The ∂x ∂x commutativity of partial derivatives is satisfied only on 1 ∂ f ∂ f := lim (x0 + h1) − (x0) function spaces, where the Clairaut’s theorem is satis- h →0 h ∂x ∂x 1 1 fied. For example, the commutativity of partial deriva- 1 = lim lim f (x + h + h ) tives holds for the space of polynomial functions. → → 0 1 2 h1 0 h2 0 h1 h2 3) Semigroup property. − ( + ) − ( + ) + ( ) . In the general case, the semigroup property is not sat- f x0 h1 f x0 h2 f x0 (42) isfied Therefore, the repeated action of the first derivatives is represented by the double limit instead of the single passage to the limit in the second derivative (41). α α α + α 1 2 1 2 In the general case, the repeated action of the first D D f (r) = D f (r), (α1,α2 > 0). j j j derivatives is not equal to the action of the first deriva- (37) tive (for example, see Section 122 of [52]), i.e., The property (37) leads to the fact that action of two 2 repeated partial fractional derivatives of order α does ∂ ∂ f ∂ f (x0) = (x0). (43) not equivalent to the action of fractional derivative of ∂x ∂x ∂x2 double order 2α, For example, we can consider the function α α 2 α D D f (r) = D f (r), (α1 > 0). (38) j j j − f (x) := x3 sin(x 1)(x = 0), f (x) := 0 (x = 0). It should be noted that the semigroup property is not sat- (44) isfied for standard partial derivatives of integer order. Therefore, the semigroup property cannot be consid- For this function, the first derivative exists ered as fundamental property that should be satisfied for any partial derivative. To demonstrate a violation of ∂ f − − (x) = 3 x2 sin(x 1) − x cos(x 1)(x = 0), the semigroup property, we consider standard partial ∂x derivatives of a function f (x) = g(r) with respect to ∂ f (0) := 0, (45) the variable x = x j . Let us define the derivatives of ∂x integer order by the equation but the second derivative at x = x0 = 0 does not exist since the following limit ∂n f f n f (x) (x) := lim , (39) ∂xn h→0 hn 1 ∂ f ∂ f lim (x + h) − (x ) (46) → ∂ 0 ∂ 0 where f n f (x) is the standard forward finite differ- h 0 h x x ence of order n ∈ N. This formula defines the derivative of order n ∈ N by a single passage to the limit. Equation does not exist at x0 = 0, where

(39)gives 1 ∂ f ∂ f 1 − − (0 + h) − (0) = 3 h2 sin(h 1) − h cos(h 1) ∂ f 1 h ∂x ∂x h ( ) := ( + ) − ( ) , − − x lim f x0 h f x0 (40) = 3 h sin(h 1) − cos(h 1). ∂x h→0 h 123 Author's personal copy

Partial fractional derivatives of Riesz type 1753

At the same time, the following limit at x = x0 = 0 direct connection of the suggested fractional differen-

1 tial equations and lattice (discrete) models with long- lim f (x0 + 2 h) − 2 f (x0 + h) + f (x0) range interactions. We can state that the suggested non- h→0 h2 linear fractional differential equations with the Riesz 1 − − = lim 8 h3 sin(2h 1) − 2 h3 sin(2h 1) fractional derivatives can be considered as an contin- h→0 h2 uum limit of the lattice models with long-range inter- −1 −1 = lim 8 h sin(2h ) − 2 h sin(h ) = 0 (47) actions of power-law type. h→0 From a mathematical point of view, the standard lat- = exists at x0 0. tice models are based on mathematical approach that 4) The Leibniz rule. is used the forward, backward and central finite differ- It should be noted that the Leibniz rule for the suggested ences. From the physical point of view, this approach α = partial fractional derivative of order 1 does not assumes a short-ranged and nearest-neighbor approxi- satisfied mation. However, there exist different physical cases, when continuum cannot be described in the framework of this approximation. For example, the excitation α α α D ( fg) = g D f + f D g (α > 0,α= 1). transfer [59] is due to the transition dipole-dipole inter- j j j action; the vibron energy transport [60] is due to the (48) transition dipole-dipole interaction that corresponds to the interaction of the type 1/|n −m|3, where n, m ∈ Z. This property is a characteristic property of all types of In systems, where the dispersion curves of two ele- fractional-order derivatives [19,20]. mentary excitations are intersect or close, there is an 5) Chain rule. effective long-range transfer. Such situations, which It should be noted that the standard chain rule for the are called the polariton effects [59], arise for excitons, partial fractional derivative of order α = 1 does not phonons and photons in semiconductors and molec- satisfied ular crystals. Polyatomic molecules contain charged α α groups with a long-range Coulomb interaction that cor- D f (g(r)) = (D f(g)) = ( )D g(r) j g g g r j responds to 1/|n − m|1 between them. One of the most (α > 0,α= 1), (49) widely used type of long-range interactions [61–66], which are considered for nonlinear differential equa- D where g is a derivative of integer or noninteger order tions of corresponding nonlocal continuum, is given with respect to the variable g. This property is can be by the interaction kernel considered as a characteristic property of fractional- order derivatives [15]. 1 Kα(n − m) = , (50) |n − m|α

6 Nonlinear fractional-order differential equations where α is a positive real number. In this case, we have nonlocal coupling given by the power-law function (50) The partial fractional derivatives of the Riesz type can with a physical relevant parameter α, which charac- be used in the partial fractional-order differential equa- terizes the weakening of the interaction. Some integer tions of continua and ﬁelds with power-law nonlocality values of α correspond to the well-known physical sit- [53–58]. In this section, we consider fractional gener- uations that correspond to the Coulomb potential for alizations of nonlinear equations, such as sine-Gordon, α = 1 and the dipole-dipole interaction for α = 3. Boussinesq, Burgers, Korteweg–de Vries and Monge– Quantum and classical lattice systems with long- α D j Ampere equations. We can use the derivatives j range interactions have been actively investigated with different values α j (αi = α j for i = j) to describe beginning with the Dyson’s papers [67,68], where the anisotropy in nonlocal media. the interaction kernels of the form (50) are applied. Let us give some comments about motivation of frac- The long-range interactions of different types have tional nonlocal generalization of the nonlinear differ- been studied in lattice systems as well as in their ential equations. This physical motivation is based on a continuous analogs. Different lattice and chain mod- 123 Author's personal copy

1754 V. E. Tarasov els with long-range interactions have been considered As an example, we can consider N( f ) = sin( f ). in [67–75]. An infinite one-dimensional Ising model In this case, equation (51) has the form of the frac- with long-range interactions is presented in [67–69]. tional three-dimensional sine-Gordon equation d The -dimensional classical Heisenberg model with 2 3 ∂ f (r, t) α j long-range interaction is described in [74], and their − A j D f (r, t) + sin( f (r, t)) = 0, ∂t2 j quantum generalization has been proposed in [70–73]. j=1 Kinks in lattice models with long-range interactions (53) are described in [76]. Time periodic spatially localized solutions, which are called breathers, on lattices The corresponding integer-order differential equa- α = and chains with long-range interactions are studied in tion is defined by j 2 such that [77–79]. Decay properties of breathers for discrete sys- ∂2 f (r, t) 3 ∂2 f (r, t) tems with long-range interactions, which are described − A j + sin( f (r, t)) = 0. ∂t2 ∂x2 by the discrete nonlinear Schrodinger equations, are j=1 i described in [61–66,80]. Properties of dynamical chaos (54) and synchronization in lattice models with long-range (2) In general, the nonlinear term can contain fractional- interaction of the type (50) are considered in works order derivatives [107–111]. The fractional three- [81–83]. We can note that a characteristic property of dimensional Boussinesq equation is the lattice models with power-like long-range interactions is that the solutions of differential equations have ∂2 f (r, t) 3 α − A D j f (r, t) power-like tails [77,79,81–84]. Similar characteristic ∂t2 j j j=1 property has the corresponding continuum models that are described by fractional differential equations with 3 β + B D j f (r, t) Riesz derivatives of noninteger orders. In the works j j j=1 [28,29,85,86], we prove that the equations with the Riesz fractional derivatives are directly related to lat- 3 γ + C D j f 2(r, t) = . tice models with long-range interactions of power-law j 0 (55) = j type. j 1 The three-dimensional fractional-order generaliza- The third term of equation (55) describes the gradi- tions of the well-known nonlinear differential equations ent contribution to the dynamics of continuum. For for nonlocal anisotropic continuum can be written in α j = 2, βx = 4, γx = 2, we get the integer-order the following forms. differential equation in the form 3 (1) The fractional-order differential equation of nonlin- ∂2 f (r, t) ∂2 f (r, t) − A j ear waves in nonlocal continuum with nonlocality ∂t2 ∂x2 j=1 j can be presented in the form 3 ∂4 f (r, t) + B j ∂ 4 2 3 = x ∂ f (r, t) α j j 1 j − A j D f (r, t) + N( f (r, t)) = 0, ∂t2 j 3 2 2 j=1 ∂ f (r, t) + C j = 0. (56) (51) ∂x2 j=1 j In the one-dimensional case, Eq. (56) gives nonlin- where N( f ) is the nonlinear term. The integer- ear partial differential equation order differential equation is defined by α j = 2 for (51), and we obtain ∂2 f (x, t) ∂2 f (x, t) ∂2 f 2(x, t) − A + B ∂t2 x ∂x2 x ∂x2 ∂4 f (x, t) ∂2 f (r, t) 3 ∂2 f (r, t) + C = . x 4 0 (57) − A j + N( f (r, t)) = 0. ∂x ∂t2 ∂x2 j=1 j Equation (57) has been derived by analysis of (52) long waves in shallow water. This equation is also 123 Author's personal copy

Partial fractional derivatives of Riesz type 1755

applied to problems in the percolation of water in For one-dimensional case, Eq. (62) has the form porous subsurface strata. ∂ f (x, t) ∂3 f (x, t) ∂ f 2(x, t) + A − B = 0. (3) Complex nonlinear media can be described by ∂t x ∂x3 x ∂x differential equations of the ﬁrst order in time (63) [102–106]. The fractional three-dimensional Burg- This is the well-known one-dimensional Korteweg– ers equation can be considered in the form de Vries equation that is derived in the analysis ∂ f (r, t) 3 α of shallow waves in canals. Now the Korteweg– − A D j f (r, t) ∂t j j de Vries equations are used in a different phys- j=1 ical phenomena as models for solitons, travel- 3 β ing waves, shock wave formation, boundary layer + B D j f 2(r, t) = 0. (58) j j behavior, mass transport and turbulence. The three- j=1 dimensional generalization of the KdV equation For α j = 2 and βi = 1, Eq. (58) is the second- has been considered in [91]. The three-dimensional order differential equation is deﬁned by in the form KdV equation with power functions is considered in [92]. 3 3 ∂ f (r, t) ∂2 f (r, t) ∂ f 2(r, t) (5) The fractional generalization of the Monge–Ampere − A j + B j = 0. ∂t ∂x2 ∂x equation can be considered in the form j=1 j j=1 j β β (59) A D 1 f (x, y)D 2 f (x, y) 1 2 The special form of one-dimensional fractional α α 2 − D 1 D 2 f (x, y) Burgers equation has been discussed in [87]. In one- 1 2 dimensional case, Eq. (59)gives 2 α α ∂ ( , ) ∂2 ( , ) ∂ 2( , ) i j f x t f x t f x t + Aij D D f (x, y) + C = 0, − A1 + B1 = 0. i j ∂t ∂x2 ∂x i, j=1 (60) (64)

This is the well-known Burgers Eq. [88], that is where A, Bij = B ji, C are functions depending on α D j = applied in fluid dynamics as simplified mathemat- x, y, f , and j f . We also use notation x1 x ical model for turbulence, mass transport, shock and x2 = y.Forα j = 1 and β j = 2, equation (64) wave and boundary layer behavior. gives the well-known Monge–Ampere equation (4) The fractional generalization of the three-dimensional ∂2 f (x, y) ∂2 f (x, y) ∂2 f (x, y) 2 Korteweg–de Vries equation can be considered in A − ∂x2 ∂y2 ∂x∂y the from 2 ∂ ( , ) 3 α ∂2 f (x, y) f r t j + B + C = 0. (65) + A j D f (r, t) ij ∂ ∂ ∂t j , = xi x j j=1 i j 1 The Monge–Ampere Eq. (65) often appears in 3 β − B D j f 2(r, t) = 0. (61) differential geometry. For example, this equation j j j=1 appear in the Minkowski problem and in the Weyl problem in differential geometry of surfaces. The one-dimensional fractional KdV equation has Note that there are a lot of problems to formu- been suggested in papers [89,90]. For α = 3 and j late a fractional-order generalization of differen- β = 1, equation (61) gives the third-order differ- j tial geometry [17,18], since the usual chain rule ential equation in the form [14,15], which defines the coordinate transforma- tions, becomes much more complicated. 3 3 ∂ f (r, t) ∂3 f (r, t) ∂ f 2(r, t) + A j − B j = 0. For computer simulations of the fractional-order dif- ∂t ∂x3 ∂x j=1 j j=1 j ferential equations with the partial fractional deriva- (62) tives of the Riesz type we can used methods suggested 123 Author's personal copy

1756 V. E. Tarasov in [93–97] and [28,29,50]. Let us note the main differ- assume that the suggested nonlinear fractional differences between the methods of lattice fractional calcu- ential equations with the derivatives of the Riesz type lus, which is recently proposed in [28,29,50], and the can demonstrate new effects such discrete solitons, numerical approach for fractional differential equations kinks, breathers, dynamical chaos and synchronization that are considered in [93–97]. by computer simulations. The discrete models, which are proposed in papers [28,29], correspond to the continuum models exactly [50], and these models can be considered as microstruc- 7 Conclusion tural basis in the form of physical lattices. They are not asymptotically equivalent, i.e., they are not an approxi- In this paper, we propose new partial fractional-order mation. Equations of lattice models exactly correspond derivatives that are generalizations of the Riesz frac- to fractional differential equations without any approx- tional derivatives, which are suggested in [28,29]. The imation. (For details about exact and asymptotic con- main advantage of these partial fractional derivatives nections of lattice and continuum models, see [28]). of the Riesz type is that these derivatives for integer The numerical methods for partial fractional differ- orders are equal to the usual partial derivatives of inte- ential equations with the Riesz space fractional deriv- ger orders n with respect to x . This property allows atives, which are considered in [93,94], replace the j us much easier to build fractional models of nonlo- Riesz fractional derivatives by the finite differences cal continua. The other advantage of the suggested with power-law weights (the finite-difference approxi- approach is a direct connection [28,29,50,85,86]ofthe mation). The same type of replacements is used in the suggested fractional-order derivatives with equations finite difference methods [95–97]. The lattice fractional of lattice models with long-range interactions and lat- calculus approach is based on special type of infinite tice fractional calculus that is proposed in recent papers fractional differences that describe long-range inter- [28,29]. The lattice approach can be more appropriate actions in physical lattices. In general, the finite dif- to get microstructural models of media with power-law ferences correspond to models with nearest-neighbor nonlocality, where the intermolecular and interatomic and next-nearest-neighbor interactions [98,99]. Non- interactions are crucial in determining continuum prop- local continuum theory is based on the assumption that erties. We assume that the suggested Riesz fractional the forces between particles are a long-range type, thus derivatives can be used in the fractional vector calculus reflecting the long-range character of interatomic and [28,100]. For computer simulations of the fractional intermolecular forces. In papers [28,29,50], we sug- partial differential equations with generalized Riesz gest physical lattice models with long-range interac- space derivatives, the finite-difference methods [93,94] tions of power-law type. Discrete (lattice) models with and the lattice calculus [29,50] can be used. long-range type of interactions are important in fractional nonlocal models. The long-range interactions, which are represented by infinite differences, describe nonlocality at micro- and nanoscales. Therefore, the References suggested lattice models with long-range interactions more correctly describe the continuum media with non- 1. Letnikov, A.V.:Historical development of the theory of dif- locality of power-law type. ferentiation of fractional order. Mat. Sb. 3, 85–119 (1868). (in Russian) In the works [28,29,50] have been given exact dis- 2. Debnath, L.: A brief historical introduction to fractional crete (lattice) analogs of the Riesz fractional derivatives calculus. Int. J. Math. Educ. Sci. Technol. 35(4), 487–501 of integer and noninteger orders that have the form (2004) (50). The computer simulations are actively used for 3. Tenreiro Machado, J., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. the linear and nonlinear systems with long-range inter- Numer. Simul. 16(3), 1140–1153 (2011) actions of the form (50) with integer and noninteger 4. Tenreiro Machado, J.A., Galhano, A.M., Trujillo, J.J.: Sci- α (for example, see [61,63–66,79,83,84]). Therefore, ence metrics on fractional calculus development since we can assume that computer simulations of nonlin- 1966. Fract. Calc. Appl. Anal. 16(2), 479–500 (2013) 5. Tenreiro Machado, J.A., Galhano, A.M.S.F., Trujillo, J.J.: ear differential equations with the suggested fractional- On development of fractional calculus during the last fifty order derivatives can be successfully realized. We years. Scientometrics 98(1), 577–582 (2014) 123 Author's personal copy

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6. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Inte- 26. Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, grals and Derivatives Theory and Applications. Gordon and D.: Fractional Calculus with Applications in Mechanics. Breach, New York (1993) Wiley-ISTE, London (2014) 7. Podlubny, I.: Fractional Differential Equations. Academic 27. Tarasov, V.E.: Review of some promising fractional phys- Press, San Diego (1998) ical models. Int. J. Mod. Phys. B 27(9), 1330005 (2013) 8. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and 28. Tarasov, V.E.: Toward lattice fractional vector calculus. J. Applications of Fractional Differential Equations. Elsevier, Phys. A 47(35), 355204 (2014). (51 pages) Amsterdam (2006) 29. Tarasov, V.E.: Lattice fractional calculus. Appl. Math. 9. Ortigueira, M.D.: Fractional Calculus for Scientists and Comput. 257, 12–33 (2015) Engineers. Springer, Netherlands (2011) 30. Riesz, M.: L’integrale de Riemann–Liouville et le prob- 10. Uchaikin, V.V.: Fractional Derivatives for Physicists and leme de Cauchy pour l’equation des ondes. Bull. de la Soc. Engineers, Vol. I. Background and Theory. Springer, Math. de France. Supplement. 67, 153–170 (1939). https:// Higher Education Press (2012) eudml.org/doc/86724 11. Zhou, Y.: Basic Theory of Fractional Differential Equa- 31. Riesz, M.: L’intégrale de Riemann-Liouville et le probléme tions. World Scientiﬁc, Singapore (2014) de Cauchy. Acta Math. 81(1), 1–222 (1949). doi:10.1007/ 12. Valerio, D., Trujillo, J.J., Rivero, M., Tenreiro Machado, BF02395016. (in French) J.A., Baleanu, D.: Fractional calculus: a survey of use- 32. Prado, H., Rivero, M., Trujillo, J.J., Velasco, M.P.: New ful formulas. Eur. Phys. J. Spec. Top. 222(8), 1827–1846 results from old investigation: a note on fractional m- (2013) dimensional differential operators. The fractional Lapla- 13. Ortigueira, M.D., Tenreiro Machado, J.A.: What is a frac- cian. Fract. Calc. Appl. Anal. 18(2), 290–306 (2015) r (Rn) tional derivative? J. Comput. Phys. 293, 4–13 (2015) 33. Lizorkin, P.I.: Characterization of the spaces L p in 14. Liu, Cheng-shi: Counterexamples on Jumarie’s two basic terms of difference singular integrals. Mat. Sb. 81(1), 79– fractional calculus formulae. Commun. Nonlinear Sci. 91 (1970). (in Russian) Numer. Simul. 22(1–3), 92–94 (2015) 34. Samko, S.: Convolution and potential type operators in 15. Tarasov, V.E.: On chain rule for fractional derivatives. L p(x). Integral Transforms Spec. Funct. 7(3–4), 261–284 Commun. Nonlinear Sci. Numer. Simul. 30(1–3), 1–4 (1998) (2016) 35. Samko, S.: Convolution type operators in L p(x).Integral 16. Tarasov, V.E.: Local fractional derivatives of differentiable Transforms Spec. Funct. 7(1–2), 123–144 (1998) functions are integer-order derivatives or zero. Int. J. Appl. 36. Samko, S.: On a progress in the theory of Lebesgue spaces Comput. Math. 2(2), 195–201 (2016) with variable exponent: maximal and singular operators. 17. Tarasov, V.E.: Comments on Riemann–Christoffel tensor Integr. Transf. Spec. Funct. 16(5–6), 461–482 (2005) in differential geometry of fractional order application to 37. Samko, S.: A new approach to the inversion of the Riesz fractal space-time, [Fractals 21 (2013) 1350004]. Fractals potential operator. Fract. Calc. Appl. Anal. 1(3), 225–245 23(2), 1575001 (2015) (1998) 18. Tarasov, V.E.: Comments on the Minkowski’s space-time 38. Rafeiro, H., Samko, S.: Approximative method for the is consistent with differential geometry of fractional order, inversion of the Riesz potential operator in variable [Physics Letters A 363 (2007) 5–11]. Phys. Lett. A 379(14– Lebesgue spaces. Fract. Calc. Appl. Anal. 11(3), 269–280 15), 1071–1072 (2015) (2008) 19. Tarasov, V.E.: Leibniz rule and fractional derivatives of 39. Rafeiro, H., Samko, S.: On multidimensional analogue of power functions. J. Comput. Nonlinear Dyn. 11(3), 031014 Marchaud formula for fractional Riesz-type derivatives in (2016) domains in Rn. Fract. Calc. Appl. Anal. 8(4), 393–401 20. Tarasov, V.E.: No violation of the Leibniz rule. No frac- (2005) tional derivative. Commun. Nonlinear Sci. Numer. Simul. 40. Almeida, A., Samko, S.: Characterization of Riesz and 18(11), 2945–2948 (2013) Bessel potentials on variable Lebesgue spaces. J. Funct. 21. Carpinteri, A., Mainardi, F. (eds.): Fractals and Fractional Spaces Appl. 4(2), 113–144 (2006) Calculus in Continuum Mechanics. Springer, New York 41. Samko, S.G.: On spaces of Riesz potentials. Math. USSR- (1997) Izv. 10(5), 1089–1117 (1976) 22. Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): 42. Ortigueira, M.D., Laleg-Kirati, T.-M., Tenreiro Machado, Advances in Fractional Calculus. Theoretical Develop- J.A.: Riesz potential versus fractional Laplacian. J. Stat. ments and Applications in Physics and Engineering. Mech. Theory Exp. 2014(9), P09032 (2014) Springer, Dordrecht (2007) 43. Cerutti, R.A., Trione, S.E.: The inversion of Marcel Riesz 23. Mainardi, F.: Fractional Calculus and Waves in Linear ultrahyperbolic causal operators. Appl. Math. Lett. 12(6), Viscoelasticity: An Introduction to Mathematical Models. 25–30 (1999) World Scientiﬁc, Singapore (2010) 44. Cerutti, R.A., Trione, S.E.: Some properties of the gener- 24. Tarasov, V.E.: Fractional Dynamics: Applications of Frac- alized causal and anticausal Riesz potentials. Appl. Math. tional Calculus to Dynamics of Particles, Fields and Media. Lett. 13(4), 129–136 (2000) Springer, New York (2011) 45. Tarasov, V.E.: Lattice model of fractional gradient and 25. Uchaikin, V.V.: Fractional Derivatives for Physicists and integral elasticity: Long-range interaction of Grunwald– Engineers. Vol. II. Applications. Springer, Higher Educa- Letnikov–Riesz type. Mech. Mater. 70(1), 106–114 (2014). tion Press (2012) arXiv:1502.06268

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46. Tarasov, V.E.: Fractional-order difference equations for 65. Mingaleev, S.F., Gaididei, Y.B., Mertens, F.G.: Solitons physical lattices and some applications. J. Math. Phys. in anharmonic chains with ultra-long-range interatomic 56(10), 103506 (2015) interactions. Phys. Rev. E 61, R1044–R1047 (2000). 47. Tarasov, V.E.: Three-dimensional lattice models with long- arxiv:patt-sol/9910005 range interactions of Grunwald–Letnikov type for frac- 66. Korabel, N., Zaslavsky, G.M.: Transition to chaos in tional generalization of gradient elasticity. Meccanica discrete nonlinear Schrödinger equation with long-range 51(1), 125–138 (2016). doi:10.1007/s11012-015-0190-4 interaction. Phys. A 378(2), 223–237 (2007) 48. Ortigueira, M.D., Magin, R.L., Trujillo, J.J., Velasco, M.P.: 67. Dyson, F.J.: Existence of a phase-transition in a one- A real regularised fractional derivative. Signal Image Video dimensional Ising ferromagnet. Commun. Math. Phys. 12, Process. 6(3), 351–358 (2012) 91–107 (1969) 49. Ortigueira, M.D., Trujillo, J.J.: A unified approach to frac- 68. Dyson, F.J.: Non-existence of spontaneous magnetization tional derivatives. Commun. Nonlinear Sci. Numer. Simul. in a one-dimensional Ising ferromagnet. Commun. Math. 17(12), 5151–5157 (2012) Phys. 12, 212–215 (1969) 50. Tarasov, V.E.: Exact discretization by Fourier transforms. 69. Dyson, F.J.: An Ising ferromagnet with discontinuous long- Commun. Nonlinear Sci. Numer. Simul. 37, 3161 (2016) range order. Commun. Math. Phys. 21, 269–283 (1971) 51. Tarasov, V.E.: United lattice fractional integro- 70. Nakano, H., Takahashi, M.: Quantum Heisenberg chain differentiation. Fract. Calc. Appl. Anal. 19(3) (2016) with long-range ferromagnetic interactions at low tem- accepted for publication peratures. J. Phys. Soc. Jpn. 63, 926–933 (1994). 52. Fichtenholz, G.M.: Differential and Integral Calculus, vol. arxiv:cond-mat/9311034 1, 7th Ed. Nauka, Moscow, 1969. (in Russian) 71. Nakano, H., Takahashi, M.: Quantum Heisenberg model 53. Tarasov, V.E.: Non-linear fractional field equations: weak with long-range ferromagnetic interactions. Phys. Rev. B non-linearity at power-law non-locality. Nonlinear Dyn. 50, 10331–10334 (1994) 80(4), 1665–1672 (2015) 72. Nakano, H., Takahashi, M.: Quantum Heisenberg ferro- 54. Tarasov, V.E.: Large lattice fractional Fokker–Planck equa- magnets with long-range interactions. J. Phys. Soc. Jpn. tion. J. Stat. Mech. Theory Exp. 2014(9), P09036 (2014). 63, 4256–4257 (1994) arXiv:1503.03636 73. Nakano, H., Takahashi, M.: Magnetic properties of quan- 55. Tarasov, V.E.: Fractional Liouville equation on lattice tum Heisenberg ferromagnets with long-range interactions. phase-space. Phys. A Stat. Mech. Appl. 421, 330–342 Phys.Rev.B52, 6606–6610 (1995) (2015). arXiv:1503.04351 74. Joyce, G.S.: Absence of ferromagnetism or antiferromag- 56. Tarasov, V.E.: Fractional quantum field theory: from lat- netism in the isotropic Heisenberg model with long-range tice to continuum. Adv. High Energy Phys. 2014, 957863 interactions. J. Phys. C 2, 1531–1533 (1969) (2014). 14 pages 75. Sousa, J.R.: Phase diagram in the quantum XY model with 57. Tarasov, V.E., Aifantis, E.C.: Non-standard extensions of long-range interactions. Eur. Phys. J. B 43, 93–96 (2005) gradient elasticity: fractional non-locality, memory and 76. Braun, O.M., Kivshar, Y.S., Zelenskaya, I.I.: Kinks in the fractality. Commun. Nonlinear Sci. Numer. Simul. 22(13), Frenkel–Kontorova model with long-range interparticle 197–227 (2015). (arXiv:1404.5241) interactions. Phys. Rev. B 41, 7118–7138 (1990) 58. Tarasov, V.E.: Discretely and continuously distributed 77. Flach, S.: Breathers on lattices with long-range interaction. dynamical systems with fractional nonlocality. In: Cattani, Phys.Rev.E58, R4116–R4119 (1998) C., Srivastava, H.M., Yang, X.-J. (eds.) Fractional Dynam- 78. Flach, S., Willis, C.R.: Discrete breathers. Phys. Rep. 295, ics (De Gruyter Open, Warsaw, Berlin, 2015) Chapter 3. 181–264 (1998) pp. 31–49. doi:10.1515/9783110472097-003 79. Gorbach, A.V., Flach, S.: Compactlike discrete breathers 59. Davydov, A.S.: Theory of Molecular Excitons. Plenum, in systems with nonlinear and nonlocal dispersive terms. New York (1971) Phys.Rev.E72, 056607 (2005) 60. Scott, A.C.: Davydov’s soliton. Phys. Rep. 217, 1–67 80. Laskin, N., Zaslavsky, G.M.: Nonlinear fractional dynam- (1992) ics on a lattice with long-range interactions. Phys. A 368, 61. Gaididei, YuB, Mingaleev, S.F., Christiansen, P.L., Ras- 38–54 (2006). arxiv:nlin.SI/0512010 mussen, K.O.: Effects of nonlocal dispersive interactions 81. Tarasov, V.E., Zaslavsky, G.M.: Fractional dynamics of on self-trapping excitations. Phys. Rev. E 55, 6141–6150 coupled oscillators with long-range interaction. Chaos (1997) 16(2), 023110 (2006). arxiv:nlin.PS/0512013 62. Rasmussen, K.O., Christiansen, P.L., Johansson, M., Gai- 82. Tarasov, V.E., Zaslavsky, G.M.: Fractional dynamics of didei, YuB, Mingaleev, S.F.: Localized excitations in dis- systems with long-range interaction. Commun. Nonlinear crete nonlinear Schröedinger systems: effects of nonlocal Sci. Numer. Simul. 11(8), 885–898 (2006) dispersive interactions and noise. Phys. D 113, 134–151 83. Zaslavsky, G.M., Edelman, M., Tarasov, V.E.: Dynamics of (1998) the chain of oscillators with long-range interaction: from 63. Gaididei, Yu., Flytzanis, N., Neuper, A., Mertens, F.G.: synchronization to chaos. Chaos 17(4), 043124 (2007) Effect of nonlocal interactions on soliton dynamics in 84. Korabel, N., Zaslavsky, G.M., Tarasov, V.E.: Coupled anharmonic lattices. Phys. Rev. Lett. 75, 2240–2243 (1995) oscillators with power-law interaction and their fractional 64. Mingaleev, S.F., Gaididei, Y.B., Mertens, F.G.: Solitons dynamics analogues. Commun. Nonlinear Sci. Numer. in anharmonic chains with power-law long-range interac- Simul. 12(8), 1405–1417 (2007). arxiv:math-ph/0603074 tions. Phys. Rev. E 58, 3833–3842 (1998)

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85. Tarasov, V.E.: Continuous limit of discrete systems with 100. Tarasov, V.E.: Fractional vector calculus and fractional long-range interaction. J. Phys. A 39(48), 14895–14910 Maxwell’s equations. Ann. Phys. 323(11), 2756–2778 (2006). arXiv:0711.0826 (2008) 86. Tarasov, V.E.: Map of discrete system into continuous. J. 101. Samko, S.: On local summability of Riesz potentials in the Math. Phys. 47(9), 092901 (2006). arXiv:0711.2612 case Reα>0. Anal. Math. 25, 205–210 (1999) 87. Biler, P., Funaki, T., Woyczynski, W.A.: Fractal Burger 102. Webb, G.M., Zank, G.P.: Painleve analysis of the three- equation. J. Differ. Equ. 14, 9–46 (1998) dimensional Burgers equation. Phys. Lett. A 150(1), 14–22 88. Burgers, J.: The Nonlinear Diffusion Equation. Reidel, (1990) Dordrecht (2008) 103. Shandarin, S.F.: Three-dimensional Burgers equation 89. Momani, S.: An explicit and numerical solutions of the as a model for the large-scale structure formation fractional KdV equation. Math. Comput. Simul. 70, 110– in the Universe, Chapter In: The IMA Volumes in 1118 (2005) Mathematics and its Applications (1996) pp. 401–413. 90. Miskinis, P.: Weakly nonlocal supersymmetric KdV hier- arXiv:astro-ph/9507082 archy. Nonlinear Anal. Model. Control 10, 343–348 (2005) 104. Dai, Ch-Q, Yu, F.-B.: Special solitonic localized structures 91. de Bouard, A., Saut, J.-C.: Solitary waves of generalized for the (3+1)-dimensional Burgers equation in water waves. Kadomtsev–Petviashvili equations, Annales de l’Institut Wave Motion 51(1), 52–59 (2014) Henri Poincare. Anal. Non Lineaire 14(2), 211–236 (1997) 105. Korpusov, M.O.: Blowup of solutions of the three- 92. Jones, K.L.: Three-dimensional Korteweg–de Vries equa- dimensional Rosenau–Burgers equation. Theor. Math. tion and traveling wave solutions. Int. J. Math. Math. Sci. Phys. 170(3), 280–286 (2012) 24(6), 379–384 (2000) 106. Wazwaz, A.-M.: A variety of (3+1)-dimensional Burgers 93. Yang, Q., Liu, F., Turner, I.: Numerical methods for frac- equations derived by using the Burgers recursion operator. tional partial differential equations with Riesz space frac- Math. Methods Appl. Sci. (2014). doi:10.1002/mma.3255 tional derivatives. Appl. Math. Model. 34(1), 200–218 published online: 18 AUG (2010) 107. Johnson, R.S.: A two-dimensional Boussinesq equation 94. Shen, S., Liu, F., Anh, V., Turner, I.: The fundamental for water waves and some of its solutions. J. Fluid Mech. solution and numerical solution of the Riesz fractional 323(1), 65–78 (1996) advection-dispersion equation. IMA J. Appl. Math. 73(6), 108. El-Sabbagh, M.F., Ali, A.T.: New exact solutions for (3+1)- 850–872 (2008) dimensional Kadomtsev–Petviashvili equation and gener- 95. Li, ChP, Zeng, F.H.: Finite difference methods for frac- alized (2+1)-dimensional Boussinesq equation. Int. J. Non- tional differential equations. Int. J. Bifurc. Chaos 22(4), linear Sci. Numer. Simul. 6(2), 151–162 (2005) 1230014 (2012) 109. Yong-Qi, Wu: Periodic wave solution to the (3+1)- 96. Li, ChP, Zeng, F.H.: The ﬁnite difference methods for frac- dimensional Boussinesq equation. Chin. Phys. Lett. 25(8), tional ordinary differential equations. Numer. Funct. Anal. 2739–2742 (2008) Optim. 34(2), 149–179 (2013) 110. Huan, Zhang, Bo, Tian, Hai-Qiang, Zhang, Tao, Geng, 97. Huang, Y.H., Oberman, A.: Numerical methods for Xiang-Hua, Meng, Wen-Jun, Liu, Ke-Jie, Cai: Periodic the fractional Laplacian: a ﬁnite difference-quadrature wave solutions for (2+1)-dimensional Boussinesq equation approach. SIAM J. Numer. Anal. 52(6), 3056–3084 (2014). and (3+1)-dimensional Kadomtsev–Petviashvili equation. arXiv:1311.7691 Commun. Theor. Phys. 50(5), 1169–1176 (2008) 98. Tarasov, V.E.: General lattice model of gradient elasticity. 111. Moleleki, L.D., Khalique, C.M.: Symmetries, travel- Mod. Phys. Lett. B. 28(7), 1450054 (2014) (17 pages). ing wave solutions, and conservation laws of a (3+1)- arXiv:1501.01435 dimensional Boussinesq equation. Adv. Math. Phys. 2014. 99. Tarasov, V.E.: Lattice model with nearest-neighbor and (2014) Article ID 672679 next-nearest-neighbor interactions for gradient elasticity. Discontinuity Nonlinearity Complex. 4(1), 11–23 (2015). arXiv:1503.03633

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