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Partial Fractional Derivatives of Riesz Type and Nonlinear Fractional Differential Equations

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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 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 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 character- ized 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 possibil- ity 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 arbi- trariness 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 rep- resentation (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 frac- tional 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 equa- tion ( ) 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.
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