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Variational principle of stationary action for fractional nonlocal media and fields
Pacific Journal of Mathematics for Industry volume 7, Article number: 6 (2015)
Abstract
Derivatives and integrals of noninteger orders have a wide application to describe complex properties of physical systems and media including nonlocality of powerlaw type and longterm memory. We suggest an extension of the standard variational principle for fractional nonlocal media with powerlaw type nonlocality that is described by the Riesztype derivatives of noninteger orders. As examples of application of the suggested variational principle, we consider an Ndimensional model of waves in anisotropic fractional nonlocal media, and a onedimensional continuum (string) with powerlaw spatial dispersion. The main advantage of the suggested fractional variational principle is that it is connected with microstructural lattice approach and the lattice fractional calculus, which is recently proposed.
Introduction
Derivatives and integrals of noninteger orders [11, 13, 23, 42, 43] have wide application in physics and mechanics [6, 14, 22, 29, 30]. The tools of fractional derivatives and integrals allow us to investigate the behavior of materials and systems that are characterized by powerlaw nonlocality, powerlaw longterm memory and fractal properties. There are different types of fractional derivatives such as RiemannLiouville, Riesz, GrünwaldLetnikov, Caputo, Marchaud, Hadamard, Weyl and others [11, 23]. Moreover, all these different fractional derivatives are related to each other. For example, the GrünwaldLetnikov derivatives coizncide with Marchaud derivatives for wide class of functions (see Sections 20.2 and 20.3 in [23]), and the Marchaud derivatives are connected with the Liouville derivatives (see Section 5.4 in [23]). All fractional derivatives have a lot of unusual properties that lead to great difficulties in application of fractional calculus. Therefore question about what type of fractional derivatives should be used in applications is not a simple question. Selection of the type of fractional derivative is largely dictated by the properties of objects and materials under consideration.
Nonlocal continuum theory [8, 21] is based on the assumption that the forces between particles of continuum have longrange type thus reflecting the longrange character of interatomic forces. If the nonlocality has a powerlaw type then we can use the fractional calculus for nonlocal continuum mechanics. It is important to apply such types of fractional derivatives that allow us to take into account the longrange particle interactions in microstructural models. Therefore in fractional nonlocal theory, we should use the fractionalorder derivatives that are directly connected with models of lattices with longrange interactions. The microstructural lattice approach, which includes consideration of a continuum limit, allows us to select a type of fractional derivatives that will adequately and correctly describe fractional nonlocal continua.
Fractional nonlocal continuum mechanics is an area of continuum mechanics concerned with the behavior of continua with nonlocalities of powerlaw types that are described by using the theory of derivatives and integrals of noninteger orders. In phenomenological approach, one of the methods to describe the fractional nonlocal continua is based on the variational principle with Lagrangian density that contains fractional derivatives with respect to coordinates. A generalization of traditional calculus of variations for systems that are described by the RiemannLiouville fractional derivatives has been suggested by Agrawal in [1]. Subsequent by extensions of variational calculus for the RiemannLiouville derivatives [5] and other type of fractional derivatives such as the Caputo derivative [15, 18, 19], the Hadamard derivative [3], the Riesz derivatives [2], and fractional integrals [4] have been derived. It should be noted that the extension of variational calculus for the Riesz derivatives, suggested in [2], is really derived for socalled the RieszRiemannLiouville and the RieszCaputo derivatives rather than the usual Riesz fractional derivatives [23]. Unfortunately all suggested extensions of variational calculus are not connected with the microstructural lattice approach. Therefore, an important problem is to formulate a fractional variational principle compatible with the lattice approach.
Let us explain our main motivation to suggest new variational principle with fractionalorder derivatives. Recently, the lattice fractional calculus and the lattice fractional derivatives have been suggested in [37, 39]. In the continuum limit the suggested lattice derivatives are transformed into continuum fractional derivatives of the Riesz type [39]. The Riesztype fractional derivative \(^{RT}\mathbb {D}^{\alpha }_{j}\) is a derivative with respect to one coordinate \(x_{j} \in \mathbb {R}^{1}\) instead of the usual Riesz derivative that is a fractional generalization of Ndimensional Laplacian [11, 20, 23]. The Riesztype operator \(^{RT}\mathbb {D}^{\alpha }_{j}\) can be considered as a partial derivative of noninteger order. Taking into account the direct connection of the Riesztype derivatives with microstructural lattice approach, we suggest to use these fractional derivatives to formulate new fractional variational principle that is compatible with the microstructural lattice approach.
In addition to "lattice" motivation, it is useful to have a variational principle that allows us to derive fractional differential equations of motion that can be solved for a wide class of Lagrangian densities. It is known that the RiemannLiouville, Caputo, Liouville, Marchaud derivatives are defined in the leftsided and the rightsided forms. In general, we should take into account these two forms in the Lagrangian density if we use these derivatives. The corresponding fractional EulerLagrange equations contain the leftsided and the rightsided fractional derivatives as well. In addition, the integration by parts, which is used in derivation of the EulerLagrange equations from the variational principle, transforms the leftsided derivatives into the rightsided and vice versa [23]. As a result, we get a mixture of the left and rightsided derivatives in equations of motion. Unfortunately, these equations can be solved only for a narrow class of examples. In this paper, we suggest a fractional variational principle based on the Riesztype fractional derivatives [37, 39] that are modified Riesz fractional derivatives [11, 20, 23], and have no left and the rightsided forms. In addition, the integration by parts transforms the Riesz fractional derivatives into itself. The corresponding fractional EulerLagrange equations can be solved for a wide class of Lagrangian densities by methods described in [11].
It should be emphasized that the main advantage of the suggested variational approach is that the proposed Riesztype fractional derivatives naturally arise in the fractional continuum mechanics based on the lattice models [31–34, 40], and are directly connected with lattice fractional derivatives suggested in [37, 39].
As examples, we consider Ndimensional model of waves in anisotropic fractional nonlocal media and then we demonstrate that this model is connected with microstructural lattice model. We also consider an elastic string, which is made of a material with a spatial dispersion of powerlaw type, i.e., string with fractional nonlocal material. Using the suggested variational principle, we derive the fractional differential equations and then some solutions of these equations are obtained.
Fractional derivatives and integration by parts
2.1 Marchaud fractional derivatives
In order to derive equations of motion from a fractional variational principle, we should use the integration by parts. Unfortunately the integration by parts for most of fractional derivatives transforms the leftsided derivatives into the rightsided and vice versa. Let us give an example.
Definition 1.
The leftsided (+) and rightsided (−) Marchaud fractional derivatives of order α∈(0;1) (see Section 5.4 of [23]) are defined by the equation
The Marchaud fractional derivatives of orders α>1 (see Section 5.6 of [23]) are defined as
where m is an integer greater than α, and
Remark 1.
Note that the right hand side of (2) does not depend on m if m is an even integer number greater than α (for example, m=2[α/2]+2).
Remark 2.
For α=1,2,…m−1, the expression Γ(1−α)A _{ m }(α) is given (see Eq. 5.81 of [23]) by
Remark 3.
It is important to note that the GrünwaldLetnikov fractional derivatives coincide with the Marchaud fractional derivatives (see Section 20.3 in [23]) for functions from the space \(L_{r}(\mathbb {R})\), where 1≤r<∞ (see Theorem 20.4 in [23]). Moreover both the GrünwaldLetnikov and Marchaud derivatives have the same domain of definition.
Proposition 1.
For the Marchaud fractional derivatives, the integration by parts has the form
Equation (6) holds for functions \(f(x) \in L_{s} (\mathbb {R})\), \(g(x) \in L_{t} (\mathbb {R})\), such that \(\left (\,^{M}D^{\alpha }_{x,} f\right)(x) \in L_{p}(\mathbb {R})\) and \(\left (\,^{M}D^{\alpha }_{x,+} g\right)(x) \in L_{r}(\mathbb {R})\), where p>1, r>1, p ^{−1}+r ^{−1}=1+α, s ^{−1}=p ^{−1}−α, and t ^{−1}=r ^{−1}−α.
Proof.
This proposition is proved in Section 6.3 of [23] (see Equation 6.27 of [23]). □
2.2 Riesztype fractional derivatives
Let us now define the fractional derivatives of the Riesz types for \(\mathbb {R}^{N}\). We will use the Cartesian coordinate system with the basis vectors e _{ j } (j=1,2,…,N), and the radius vector \(\textbf {r} =\sum ^{N}_{j=1} x_{j} \, \textbf {e}_{j}\). The fractional derivatives of the Riesz types have been introduced in [37, 39].
Definition 2.
The Riesztype fractional derivative of order α is defined by the equation
where \((\Delta ^{m}_{z_{j}} u)(\textbf {r})\)is a finite difference of order m of a function f(r) with the vector step \(\textbf {z}_{j}= z_{j} \, \textbf {e}_{j} \in \mathbb {R}^{N}\) for the point \(\textbf {r} \in \mathbb {R}^{N}\). The centered difference
The constant d _{1}(m,α) is defined as
where
for the centered difference (8).
Remark 4.
The constant d _{1}(m,α) depends only on m and α. It is different from zero for all α>0 in the case of an even m (see Theorem 26.1 in [23]). Note that the integral (7) does not depend on the choice of m>α. Therefore, we can always choose an even integer m, so that it is greater than parameter α, and we can use the centered difference (8) for all positive real values of α.
Remark 5.
It should be noted that we can use the noncentered difference instead of the centered difference (8). The noncentered difference is defined by the equation
and the corresponding coefficient \(A^{nc}_{m}(\alpha)\) instead of \({A^{c}_{m}}(\alpha)\), where
In the case of noncentered differences the constant d _{1}(m,α) vanishes if and only if α=1,3,5,…,2[m/2]−1. Therefore the noncentered differences (9) can be used only for noninteger positive orders α and for odd integer values of α.
Remark 6.
Using (7), we can see that the Riesztype fractional derivative \(^{RT}\mathbb {D}^{\alpha _{j}}_{j} \, f(\textbf {r})\) is the Riesz derivative [23] of the function f(r) with respect to one coordinate \(x_{j} \in \mathbb {R}^{1}\) instead of the usual Riesz operator defined for the vector \(\textbf {r} \in \mathbb {R}^{N}\) as a fractional generalization of Ndimensional Laplacian. The Riesztype fractional derivative \(^{RT}\mathbb {D}^{\alpha _{j}}_{j}\) is the Riesz fractional derivative with respect to x _{ j } for \(\mathbb {R}^{1}\). The Riesztype operator \(^{RT}\mathbb {D}^{\alpha }_{j}\) can be considered as a partial derivative of noninteger order.
Remark 7.
In [37, 39] the Riesztype fractional derivative \(^{RT}\mathbb {D}^{\alpha }_{j}\) is denoted as \(\mathbb {D}_{C}^{+} \left [ \alpha \atop j \right ] \) and it is a continuum analog of lattice fractional derivatives that are suggested in these papers.
Remark 8.
It should be noted that an extension of the variational calculus for the Riesz derivatives, which is suggested in [2], is really considered for the socalled RieszRiemannLiouville derivatives and RieszCaputo derivatives. In paper [2], the usual Riesz fractional derivatives are not considered. Moreover extensions of variational calculus suggested in [2], are not connected with microstructural lattice approach. The main advantage of our variational approach is that the Riesztype fractional derivatives are related to the lattice fractional derivatives [37, 39].
Remark 9.
Using that (−i)^{2m}=(−1)^{m}, the Riesztype fractional derivatives for even α=2m, where \(m \in \mathbb {N}\), are connected with the usual partial derivative of integer orders 2m by the relation
where we use the notation
For α=2, the Riesztype derivative is the local operator \( \partial ^{2}/\partial {x^{2}_{j}}\), i.e.,
and so on. The fractional derivatives \(^{RT}\mathbb {D}^{2m}_{j}\) for even orders α are local operators. Note that the Riesz derivative \({~}^{RT}\mathbb {D}^{1}_{j}\) cannot be considered as a derivative of the first order with respect to x _{ j }, i.e., \(\,^{RT}\mathbb {D}^{1}_{j} \, f(\textbf {r}) \, \ne \, {D^{1}_{j}} f(\textbf {r})\). Note that the Riesztype derivatives of odd orders α=2m+1, where \(m \in \mathbb {N}\), are nonlocal operators that cannot be considered as usual derivatives \(D^{2m+1}_{j}={\partial ^{2m+1}}/{\partial x^{2m+1}_{j}}\). For α=1 the derivative \(^{RT}\mathbb {D}^{1}_{j}\) is a nonlocal operator that can be viewed as a "square root of the 1D Laplacian".
2.3 Integration by parts for Riesztype fractional derivatives
Let us now describe a connection of the Riesztype fractional derivative and the Marchaud fractional derivatives.
Proposition 2.
The Riesztype fractional derivative \(\,{~}^{RT}\mathbb {D}^{\alpha }_{x}\) defined in (7), can be expressed in terms the Marchaud fractional derivatives \(\,{~}^{M}D^{\alpha }_{x,\pm }\) defined in (2), as follows
where
Proof.
The rightsided Marchaud fractional derivative \(\left ({~}^{M}D^{\alpha }_{x,} f\right) (x)\) can be written as
Then the sum of the leftsided and rightsided Marchaud fractional derivatives is given by
Using (7) and expression (16), we obtain the relation of the Riesztype and Marchaud fractional derivatives in the form (13). □
Representation (13) and Eq. (6) allow us to prove that the integration by parts transforms the Riesztype fractional derivatives into themselves. We have the following statement.
Proposition 3.
The integration by parts for the Riesztype fractional derivatives (7) has the form
Proof.
Using (13) and then (6), we get
□
As a result, integration by parts (17) does not change the form of the Riesztype fractional derivative. Note also that this integration by parts does not change the sign in front of integral.
Variational principle for fractional nonlocal continuum
Let us now consider a variational approach to describe the Ndimensional nonlocal continuum. Equations for the fractional nonlocal continuum can be derived as the EulerLagrange equation for the action functional
where \(\mathcal {L} \left (\varphi, {D^{1}_{t}} \varphi, \,{~}^{RT}\mathbb {D}^{\alpha _{1}}_{j} \varphi, \,{~}^{RT}\mathbb {D}^{\alpha _{2}}_{j} \varphi \right)\) is the Lagrangian density that define the Ndimensional continuum or the field model, φ=φ(r,t) is a scalar field. \(\,^{RT}\mathbb {D}^{\alpha _{1}}_{j}\) and \(\,{~}^{RT}\mathbb {D}^{\alpha _{2}}_{j}\) are the Riesztype fractional derivatives with respect to x _{ j } with j=1,2,…,N. Note that x _{ j } are dimensionless values in the fractional dynamical models [29]. In general, the action functional (18) can be considered for a bounded region \(R \subset \mathbb {R}^{N+1}\) by using the Lagrangian density
Let us formulate the principle of stationary action for the functional (18).
Proposition 4.
Let the Lagrangian density \(\mathcal {L}\left (\varphi, {D^{1}_{t}} \varphi,\right. \left.{~}^{RT}\mathbb {D}^{\alpha _{1}}_{x} \varphi, \,{~}^{RT}\mathbb {D}^{\alpha _{2}}_{x} \varphi \right)\) be a function with continuous first and second (partial) derivatives with respect to all its arguments, and the function φ=φ(r,t) has the continuous Riesztype fractional derivatives of orders α _{1}>0 and α _{2}>0 with respect to x _{ j } with j=1,2,…,N; in particular, the function φ belongs to the Lizorkin space. Then the holonomic variational equation
which describes the principle of stationary action for the functional (18), gives the EulerLagrange equation
if we assume that the variations δ φ and δ x are equal to zero on the boundary ∂ R of the region \(R \subset \mathbb {R}^{N+1}\).
Proof.
The first variation of the action functional (18) with respect to φ(r,t) and its derivatives has the form
If the considered continuum and fields do not contain some nonholonomic constraints, then the fractional derivatives and the variation commute:
Using integration by parts (17), the variation (21) can be represented in the form
The principle of stationary action is defined by the holonomic variational equation
As a result, the stationary action principle for fractional nonlocal continuum gives eq. (20) that is the fractional EulerLagrange equation for the model described by the Lagrangian density \(\mathcal {L} = \mathcal {L}\left (\varphi, {D^{1}_{t}} \varphi, \,{~}^{RT}\mathbb {D}^{\alpha _{1}}_{j} \varphi, \,{~}^{RT}\mathbb {D}^{\alpha _{2}}_{j} \varphi \right)\). □
In order to derive equations of motion from variational principles, the integration by parts should be used. It is wellknown that for wide class of of fractionalorder derivatives the integration by parts transforms the leftsided derivatives into the rightsided and vice versa. To avoid this problem, we proposed the variational principle with the Riesztype fractional derivatives. As we proved, the integration by parts (17) does not change the form of this fractionalorder derivative.
In the next section we consider this fractional variational approach to a model of waves in anisotropic fractional nonlocal media and for a fractional generalization of vibrating string model.
Application of the variational principle to waves in anisotropic fractional nonlocal media
4.1 Continuum model of waves in fractional nonlocal media
Let us consider waves in anisotropic medium with spatial dispersion of powerlaw type, i.e., nonlocality of fractional order. We will describe the medium by a scalar field φ=φ(r,t) in the Euclidean space \((\textbf {r},t) \in \mathbb {R}^{N} \times \mathbb {R}\). The order of nonlocality along different directions (along the x _{ j }axis) will be denoted by α _{ j }/2>0. The Lagrangian density for this field is
If α _{ j }=2 for all j=1,2,…,N, this Lagrangian density defines usual (local) classical field theory.
For the Lagrangian density (23), we get
Substitution of (24) and (25) into (20) gives the EulerLagrange equation of the form
which is the fractional differential equation for the fractional nonlocal anisotropic medium.
For homogeneous media, A _{ j } are independent of r, and the fractional differential Eq. (26) can be rewritten in the form
For wide class of functions φ(r,t) from the Lizorkin space [11] for example, Eq. (27) can be represented in the form
In the case A _{ j }=E and α _{ j }=α for all j=1,2,…,N, Eq. (28) can be written as
where
is the fractional Laplacian of the Riesz type [37]. Equation (29) describes waves in isotropic nonlocal medium. For α=2 and V(φ)=0, Eq. (29) is the classical wave equation since ^{RT} Δ ^{α}=−Δ.
4.2 Lattice model of waves in fractional nonlocal media
Let us consider an Ndimensional unbounded lattice with the noncoplanar vectors a _{ j }, (j=1,2,…,N) that define the distance a _{ j }=a _{ j } between particles with mass M. For simplification, we will consider mutually orthogonal vectors a _{ j }. In the general case, the Cartesian coordinate system does not depend on the choice of lattice vectors a _{ j }. However, conveniently choose the basis vectors e _{ j } of the Cartesian coordinate system such that a _{ j }=a _{ j } e _{ j }. Sites of this lattice will be characterized by the number vector n=(n _{1},n _{2},…,n _{ N }), where n _{ j } (j=1,2,…,N) are integers. We assume that the positions of particles in the lattice coincide with the lattice sites, so that the vector n is a number vector of the corresponding particle.
Let us consider a model of lattice with longrange interactions that is described by the Lagrangian
with the kernel
where _{1} F _{2} is the Gauss hypergeometric function [7]. The first term of (31) defines the kinetic energy. The second and third terms give the potential energy. If we consider V(φ _{ n }(t))=J _{ n }(t)φ _{ n }(t), where J _{ n }(t) is the external force, then we get the linear theory. For \(V(\varphi _{\textbf {n}}(t))= (\lambda /4) \, \varphi ^{4}_{\textbf {n}}(t)\), we have the wellknown nonlinear φ ^{4}theory.
The equation of motion can be obtained, as usual, from the stationary condition for action
The stationary condition for the functional (33) with the Lagrangian (31) gives the EulerLagrange equation
in the form
where F is the potential force
In papers [37, 39], definition of lattice fractional derivatives of the positive real orders α _{ j } in the directions e _{ j }=a _{ j }/a _{ j } are presented.
Definition 3.
Lattice fractional partial derivatives of orders α _{ j }>0 are operators defined by the equations
where \(\alpha _{j} \in \mathbb {R}\), α _{ j }>0, \(n_{j}, m_{j} \in \mathbb {Z}\), \(\textbf {m} \in \mathbb {Z}^{3}\), and the interaction kernel \(K^{+}_{\alpha _{j}}(n_{j}m_{j})\) is defined by Eq. (32).
Using (37), the EulerLagrange Eq. (35) can be rewritten as
There is a connection between the lattice fractional derivatives and the Riesztype fractional derivatives that is described by the following proposition established in [37, 39].
Proposition 5.
In the continuum limit the lattice fractional derivatives (37) are transformed into the continuum fractional derivatives of orders α _{ j } that are the Riesztype fractional derivatives with respect to coordinates x _{ j } by
where \({\mathcal F}_{\Delta }\) is the Fourier series transform, Lim is the passage to the limit a _{ j }→0, and \({\mathcal F}^{1}\) is the inverse Fourier integral transform [39].
Proof.
This proposition is proved in Section 5 of [39]. □
Using this proposition, we can get that the continuum limit transforms of lattice Eqs. (38) into (28), which is the fractional partial differential equation of the fractional nonlocal continuum.
As a result, proposed fractional variational principle allows us to get the EulerLagrange equations that are directly connected with microstructural lattice models of fractional nonlocal media [31–34, 40], and the lattice field theories [35, 38].
Application of the variational principle to fractional nonlocal onedimensional continuum
5.1 Continuum and lattice models of string with fractional nonlocality
Let us now consider a string, which is made of a material with spatial dispersion of powerlaw type. The Lagrangian density of this string with nonlocality of two orders α _{1}>0 and α _{2}>0 can be presented in the form
For an incompressible elastic solid, the displacement field φ(x)=u _{ y }(x) is transversal, or orthogonal to the longitudinal axis (in our case, the xaxis) of wave propagation. Note that x and \({l^{2}_{s}} (\alpha _{2})\) for fractional nonlocal models are dimensionless values [29]. The first term represents the kinetic energy, where ρ is the linear density (i.e., mass per unit length). The second and third terms represent the potential energy due to internal forces, and T is the string tension. The fourth term represents the potential energy due to the external load J(x).
For local materials, we have α _{1}=1, α _{2}=2, and we should use the Lagrangian density in the form
Note that the Lagrangian density (40) with α _{1}=1 and α _{2}=2 similar to (41) but contains minus sign in front of the third term. Therefore expression (40) cannot give (41) for α _{1}=1, α _{2}=2.
For the Lagrangian density (40) of the string model, we get the derivatives
Substitution of (42) and (43) into (20) gives the EulerLagrange equation
which is the fractional differential equation for the fractional nonlocal string.
If the continuum is homogeneous, then T is independent of x, and the fractional string equation has simple form
Using the properties of the fractional Riesz derivatives, for functions φ(x), for which \(\left ({~}^{RT}\mathbb {D}^{\alpha _{2}}_{x}\right)^{2} \varphi = \,^{RT}\mathbb {D}^{2\alpha _{1}}_{x} \varphi \), Eq. (45) can be represented in the form
In the general case, we should consider an effective source term J _{ eff }(x,t) instead of J(x,t), where J _{ eff }(x,t) contains the function J(x,t) and deviations from the semigroup property for the Riesztype derivatives.
It should be noted that fractional differential Eq. (46) is directly connected with the lattice model described by the equation
that can be derived from stationary action principle with the Lagrangian
where the kernel \(K^{+}_{\alpha } (nm)\) is defined by Eq. (32).
If we have materials without nonlocality and memory, then α _{1}=1, α _{2}=2, and Eq. (46) has the form
where we use \(\,^{RT}\mathbb {D}^{2}_{x} =  \, {D^{2}_{x}}\), and \(\,^{RT}\mathbb {D}^{4}_{x} = + \, {D^{4}_{x}}\).
Note that despite the fact that the Lagrangian density (40) cannot give Lagrangian density (41) for α _{1}=1, α _{2}=2, the eqs. (46) gives (49) for α _{1}=1, α _{2}=2. This is due to the fact that
Equation (49) is the differential equation for string which was made of a gradient elasticity material [16, 17, 36, 41]. If we use α _{1}=1, α _{2}=0, we get the equation
It is the usual wellknown equation for string without nonlocality and memory.
5.2 Solution of fractional differential equation
For the static case (\({D^{2}_{t}} \varphi =0\) and J(x,t)=J(x)), Eq. (46) has the form
The particular solution of Eq. (52) can be presented (see Corollary 5.14 of [11]) in the form
where \(G_{\alpha _{1},\alpha _{2}}(x)\) is the Greentype function
If the external force J(x) is applied at the point x=0, then
and the field φ(x) has a simple form \(\phantom {\dot {i}\!}\varphi (x) = (J_{0}/T)\, G_{2\alpha _{1},2\alpha _{2}}(x)\), where \(G_{2\alpha _{1},2\alpha _{2}}(x)\phantom {\dot {i}\!}\) is defined by (54). As a result, the field φ(x) is given by
The plane wave
can be traveling in fractional nonlocal continua with its frequency ω, where the fractional differential equation for the field φ _{ p }(x) is
Here we use J(x,t)=e ^{−iωt} J _{ p }(x). For a wide class of functions φ _{ p }(x) Eq. (57) can be rewritten in the form
The particular solution of Eq. (58) can be given (Theorem 5.24 of [11]) in the form
where \( G^{(\omega)}_{2\alpha _{1},2\alpha _{2}}(x)\) is the Greentype function
Here \({l^{2}_{s}} (\alpha _{2}) \ne 0\) and ρ ω ^{2}≠0. For the case (55), the solution (59) is
This is solution of the fractional string equation for external load J(x) applied at a point (55).
Conclusion
To obtain equations of fractional nonlocal theory for media and fields with powerlaw nonlocality, we propose a new fractional variational principle for Lagrangians with fractionalorder derivatives. This fractional variational principle is based on the Riesztype fractional derivatives. The characteristic properties of the proposed approach are the following:

(1)
The Riesztype of derivatives do not have the leftsided and the rightsided forms, in contrast to the RiemannLiouville, Caputo, Liouville and Marchaud fractional derivatives.

(2)
The integration by parts transforms the Riesztype fractional derivatives into themselve. For fractional derivatives of RiemannLiouville, Caputo, Liouville, and Marchaud, the integration by parts, which should be used in derivation of the EulerLagrange equations from variational principle, transforms the leftsided derivatives into the rightsided and vice versa.

(3)
The corresponding fractional EulerLagrange equations can be solved for a wide class of Lagrangian densities by methods described in [11].

(4)
The Riesztype fractional derivative \(^{RT}\mathbb {D}^{\alpha _{j}}_{j}\) is a derivative with respect to one coordinate \(x_{j} \in \mathbb {R}^{1}\) contrary to the usual Riesz derivative [11, 23], which is a fractional generalization of Ndimensional Laplacian. The Riesztype derivative \(^{RT}\mathbb {D}^{\alpha _{j}}_{j}\) can be viewed as a partial derivative of noninteger order.

(5)
The Riesztype fractional derivatives naturally arise in the fractional continuum mechanics based on the lattice models [31–34, 40], since they directly connected with lattice fractional derivatives that are recently proposed [37, 39].
The main advantage of the suggested fractional variational principle is the fact that it is connected with microstructural lattice approach and the lattice fractional calculus that is recently proposed. In the papers [37, 39] it was proves that the Riesztype fractional derivative is a continuum limit of the lattice fractional derivative.
Nonlocal continuum theory is based on the assumption that the forces between particles of continuum have longrange type, thus reflecting the longrange character of interatomic forces. Nonlocality of the powerlaw type allows us to use fractional derivatives and integrals in nonlocal continuum mechanics. In fractional nonlocal theory it is important to apply fractional derivatives that take into account the longrange particle interactions in microstructural models. The microstructural lattice approach, which includes consideration of continuum limit, allows us to select a type of fractional derivatives that will be more adequately and correctly describe fractional nonlocal continua. Therefore we propose to use the Riesz type fractionalorder derivatives that are directly connected with models of lattices with longrange interactions. It allow us to get more correct models of fractional nonlocal media by fractional variational principle.
In this paper we consider a fractional principle of stationary action with Riesztype fractional derivatives. This principle is represented by the holonomic variational equation δ S[φ]=0. In the general case, we should use the variational principles that are represented by nonholonomic variational equations. To consider nonlocal media with dissipation and nonpotential forces, nonLagrangian systems, we should apply a fractional generalization of the Sedov nonholonomic variational equation [24–26] instead of the stationary action principle.
The proposed fractional variational principle also allows us to obtain exact analytical solutions of the fractional differential equations for models of a wide class of media with fractional nonlocality. A characteristic feature of the behavior of a fractional nonlocal continuum is the spatial powertails of noninteger orders [12, 33]. The fractional nonlocal models, which are used to describe complex media, can be characterized by a common or universal spatial behavior media by analogy with the universal temporal behavior of lowloss dielectrics [9, 10, 27, 28]. The proposed fractional variational principle can be important in the fractional field theory [38] and in the fractional quantum theory [35].
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DOI
PACS
 45.10.Hj
 04.20.Fy
MSC
 26A33
 034A08
 49S05