Conservation laws of (3+α) -dimensional time-fractional diffusion equation

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Conservation laws of (3 + α )-dimensional time-fractional diffusion equation Elham Lashkarian, S. Reza Hejazi, Elham Dastranj * Department of Mathematics, Shahrood University of Technology, Shahrood, Semnan, Iran

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Article history: Received 21 February 2017 Received in revised form 27 September 2017 Accepted 7 October 2017 Available online xxxx Keywords: Fractional symmetry Fractional conservation laws Fractional diffusion equation Laplace operator Noether’s operator

a b s t r a c t The concept of Lie–Backlund symmetry plays a fundamental role in applied mathematics. It is clear that in order to find conservation laws for a given partial differential equations (PDEs) or fractional differential equations (FDEs) by using Lagrangian function, firstly, we need to obtain the symmetries of the considered equation. Fractional derivation is an efficient tool for interpretation of mathematical methods. Many applications of fractional calculus can be found in various fields of sciences as physics (classic, quantum mechanics and thermodynamics), biology, economics, engineering and etc. So in this paper, we present some effective application of fractional derivatives such as fractional symmetries and fractional conservation laws by fractional calculations. In the sequel, we obtain our results in order to find conservation laws of the time-fractional equation in some special cases. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction As we know conservation laws is one of the substantial concepts in physics and mathematics. In this study, conservation laws, for FDEs [1–9], are derived by symmetry group technique. About three centuries ago for finding conservation laws of the considered equation, for the first time, derivatives of noninteger order (α = 21 ) was introduced by Leibniz. Recently this methodology is developed for PDEs for finding exact solutions, symmetries, conservation laws and etc. The concept of conservation laws is given in [10–14]. Furthermore, it has been shown that how we can make conservation laws by using Lie–Backlund symmetry generators. In a number of recent papers, conservation laws by using fractional symmetry for FDEs is developed [15,16]. In this method, the existence of Lagrangian is not necessary. In fact by formal Lagrangian, we yield the components of conservation laws [13]. The classical Lie point symmetry has been investigated for the diffusion equation of the (3 + α ) dimensional of order α ∈ (0, 2) Dαt u = (f (u)ux )x + (g(u)uy )y + (h(u)uz )z ,

(1)

in the sense of Riemann–Liouville derivative. Furthermore, we describe our methodology for calculating of conservation laws for FDEs. In the sequel, as an example and application of the proposed model, we present the calculation of the conservation laws. In the process of computing the symmetry and conservation laws, we use fractional calculating regularly [17–19].

*

Corresponding author. E-mail addresses: [email protected] (E. Lashkarian), [email protected] (S. Reza Hejazi), [email protected] (E. Dastranj).

https://doi.org/10.1016/j.camwa.2017.10.001 0898-1221/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

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The present paper is organized as follows. In section two, we illustrate some fractional results for the considered FDE (1). Third chapter is devoted to find symmetry operators of Eq. (1) in general and special forms. In the fourth chapter, we give a comprehensive analysis of the equation including application of symmetry analysis and conservation laws. 2. Fractional calculus on the Eq. (1) Left sided and right sided time-fractional integral of order n − α are defined as follows. (0 Itn−α )(t , x, y, z) =

(t ITn−α )(t , x, y, z) =

Γ (n − α )

(t − µ)α+1−n

0

Γ (n − α )

u(µ, x, y, z)

T

∫

1

u(µ, x, y, z)

t

∫

1

(µ − t)α+1−n

t

dµ

(2)

dµ.

(3)

Left and right derivatives of Riemann–Liouville and Caputo time-fractional derivatives is described by: α

0 Dt

u=

α

t DT u

=

∂n Γ (n − α ) ∂ t n

∫

∂n Γ (n − α ) ∂ t n

∫

1

(−1)n

( Dt )(t , x, y, z) =

C α t DT u

=

u(µ, x, y, z) (µ − t)α+1−n

0

u(µ, x, y, z)

T t

(µ − t)α+1−n

Γ (n − α )

(−1)n

∫

Γ (n − α )

t

T

t

∫

1

α

C

t

dµ,

(4)

dµ,

(5)

Dn u(µ, x, y, z) (µ − t)α+1−n

0

Dnµ (µ, x, y, z) (µ − t)α+1−n

dµ,

(6)

dµ.

(7)

So, we have α

0 Dt

C α 0 Dt u

u = Dnt (0 Itn−α u),

=0 Itn−α (Dnt u).

In the sequel, four time-fractional generalizations of Eq. (1) are considered. First let us that in the resumption, the classical diffusion equation can be written as ut = C [U ], where C [U ] = (f (u)ux )x + (g(u)uy )y + (h(u)uz )z , four time-fractional generalized to Eq. (1) are ut = Itα C [U ] = Itα (f (u)ux )x + (g(u)uy )y + (h(u)uz )z ,

]

[

(8)

ut = Dt1−α C [U ] = D1t −α (f (u)ux )x + (g(u)uy )y + (h(u)uz )z ,

[

]

(9)

ut = C [Itα U ] = fu Itα u2x + fItα uxx + gu Itα u2y + gItα uyy + hu Itα u2z + hItα uzz ,

(10)

ut = C [D1t −α U ] = fu D1t −α u2x + fD1t −α uxx + gu D1t −α u2y + gD1t −α uyy

+ hu D1t −α u2z + hD1t −α uzz .

(11)

α

where It is the left-sided time-fractional integral of order α and 1 − α of the Riemann–Liouville type. Note that since Dαt Itα u = u, Eq. (8) , can be rewritten as

Dt1−α

is the left-sided time-fractional derivative of order

1 Dα+ u = Dαt ut = (f (u)ux )x + (g(u)uy )y + (h(u)uz )z . t

(12)

Substituting Dt1−α = Itα−1 and Itα−1 ut = C Dαt u into Eq. (9) yields C

Dαt ut = (f (u)ux )x + (g(u)uy )y + (h(u)uz )z . α

α

α 2

Taking v = It u, It uxx = vxx , It ux = v , ut =

(

2 x

D1t +α

(13)

v from Eq. (10) we have

)

D1t +α vt = (f (v )vx )x + (g(v )vy )y + (h(v )vz )z , Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

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consequently, if we denote v = u in Eq. (10), then we obtain Dt1+α ut = (f (u)ux )x + (g(u)uy )y + (h(u)uz )z ,

(14)

and, by taking D1t −α u = v , Eq. (11) can be written as Dαt ut = (f (u)ux )x + (g(u)uy )y + (h(u)uz )z ,

(15)

when v = u. Eqs. (12)–(15) are four different time-fractional extensions of the classic diffusion equation. Thus a more general form of time-fractional diffusion equation could be written in the form of µ(α )

F (t , x, y, z , Dt µ(α )

Dt

u, ux , uxx , uy , uyy , uz , uzz ) =

u − (f (u)ux )x + (g(u)uy )y + (h(u)uz )z .

(16)

3. Symmetry analysis for FDEs (1) Suppose X = ξ0

∂ ∂ ∂ ∂ ∂ + ξ1 + ξ2 + ξ3 +η , ∂t ∂x ∂y ∂z ∂u

(17)

be a symmetry operator for Eq. (1) (which is known in the literature as an infinitesimal operator or generator of the symmetry group), the second order prolonged vector field , (Pr (α,2) X ), annihilates (1) on its solution, namely, Pr (α,2) X (∆) |∆=0 = 0,

∆ = Dαt u − (f (u)ux )x − (g(u)uy )y − (h(u)uz )z .

(18)

So we have Pr (α,2) X = X + ζ α,t ∂∂tα u + ζ11 ∂ux + ζ21 ∂uxx + ζ12 ∂uy + ζ22 ∂uyy + ζ13 ∂uz + ζ23 ∂uzz , where ζ11 , ζ12 , ζ13 , ζ21 , ζ22 and ζ23 satisfy

ζ11 = Dx η − ut Dx ξ 0 − ux Dx ξ 1 − uy Dx ξ 2 − uz Dx ξ 3 , ζ21 = Dx ζ11 − uxt Dx ξ 0 − uxx Dx ξ 1 − uxy Dx ξ 2 − uxz Dx ξ 3 , ζ12 = Dy η − ut Dy ξ 0 − ux Dy ξ 1 − uy Dy ξ 2 − uz Dy ξ 3 , ζ22 = Dy ζ12 − uyt Dy ξ 0 − uyx Dy ξ 1 − uyy Dy ξ 2 − uyz Dy ξ 3 , ζ13 = Dz η − ut Dz ξ 0 − ux Dz ξ 1 − uy Dz ξ 2 − uz Dz ξ 3 , ζ23 = Dz ζ13 − uzt Dz ξ 0 − uxz Dz ξ 1 − uyz Dz ξ 2 − uzz Dz ξ 3 ,

(19)

where Dx , Dy and Dz are total derivative operators. Solving (18) along with (19), the following characteristic system is driven by:

ξt1 = ξt2 = ξt3 = ξu1 = ξu2 = ξu3 = ξt0 = ξx0 = ξy0 = ξz1 = 0, ∞ ∑ 3 1 2f ξxu + 2f ′ ξx3 + 2hξzu + 2h′ ξz1 = 0, ∂tα ηu − Dnt +1 ξ 0 = 0, n=1

2f ξ

+ 2f ξx + 2g ξ + 2g ξy = 0,

2g ξ

+ 2g ξy + 2hξ + 2h ξz = 0,

2 xu 3 yu

′ 2

1 yu 2 zu

′ 3

u∂tα ηu − ∂tα η + f ηxx + g ηyy + hηzz = 0,

′ 1

0 0 f ξxx + g ξyy + hξzz0 = 0,

′ 2

3 3 2hηzu + 2h′ ηz − f ξxx − g ξyy − hξzz3 = 0,

f ′ η − 2f ξx1 + α f ξt0 = 0,

2g ηyu + 2g ηy − f ξ − ξ

− hξ = 0,

g η − 2g ξ + α ξ = 0,

2f ηxu + 2f ηx − f ξ − ξ

− hξ = 0 ,

h′ η − 2hξ + α ξ = 0,

2 xx 1 xx

′

′

f η + f ηu − 2f ξx − ′′

′ 1

′

2 g yy 1 g yy 1 2f xu 2 2g yu 3 2h zu

2 zz 1 zz

2 y 3 z 2 x

′

ξ + f ηuu + α f ξt = 0,

f ξ + ξ = 0,

′ 0

g η + g ηu − 2g ξy −

ξ + g ηuu + α g ξt = 0,

f ξx3 + ξ = 0,

h′′ η + h′ ηu − 2h′ ξz3 −

ξ + hηuu + α h′ ξt0 = 0,

g ξy3 + ξ = 0.

′′

′ 2

′

(20)

g t0 h t0 g y1 h z1 h z2

′ 0

The solution of system (20), for arbitrary f (u), g(u), h(u) and α ∈ (0, 2), is

ξ 0 = c1 t ,

ξ1 =

c1 α 2

x + c3 ,

ξ2 =

c1 α 2

y + c4 ,

ξ3 =

c1 α 2

z + c5

, η = 0,

(21)

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

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where c1 , c2 , c3 and c5 are arbitrary constants. Thus, we obtain the corresponding four-dimensional Lie algebra X1 =

∂ , ∂x

X2 =

∂ , ∂y

X3 =

∂ , ∂z

X4 =

2 ∂ ∂ ∂ ∂ t +x +y +z . α ∂t ∂x ∂y ∂z

(22)

Case 1: If consider f (u) = g(u) = h(u) = 1, there are nine infinitesimal operators for Eq. (1) with arbitrary α ∈ (0, 2).

∂ ∂ ∂ ∂ ∂ ∂ ∂ , X2 = , X3 = , X4 = x + y + z +2 ∂x ∂y ∂z ∂x ∂y ∂z ∂u ∂ ∂ ∂ ∂ X5 = y − x , X6 = z −x , ∂x ∂y ∂x ∂z ( ) ∂ ∂ α ∂ ∂ ∂ ∂ X7 = , X8 = t + x +y +z , X9 = . ∂t ∂t 2 ∂x ∂y ∂z ∂u X1 =

(23)

Case 2: For f (u) = g(u) = h(u) = eu , there are eight infinitesimal operators for Eq. (1) with arbitrary α ∈ (0, 2).

∂ ∂ ∂ ∂ ∂ ∂ ∂ , X2 = , X3 = , X4 = x + y + z +2 ∂x ∂y ∂z ∂x ∂y ∂z ∂u ∂ ∂ ∂ ∂ X5 = y − x , X6 = z −x , ∂x ∂y ∂x ∂z ∂ ∂ ∂ ∂ X7 = z − y , X8 = t − α . ∂y ∂z ∂t ∂u X1 =

(24)

Case 3: And finally if take f (u) = g(u) = h(u) = uA , seven infinitesimal operators for Eq. (1) with arbitrary α ∈ (0, 2) are obtained as follows:

∂ ∂ ∂ ∂ ∂ ∂ ∂ , X2 = , X3 = , X4 = x + y + z +2 ∂x ∂y ∂z ∂x ∂y ∂z ∂u ∂ ∂ ∂ ∂ ∂ ∂ X5 = y − x , X6 = z − x , X7 = z −y . ∂x ∂y ∂x ∂z ∂y ∂z

X1 =

(25)

4. Conservation laws for FDEs The theory of finding of conservation laws for FDEs is an extension of Ibragimov’s method [13]. So many of definitions and tricks such as formal Lagrangian, Euler–Lagrange operator and components of conservation laws except time component, have similar intentions [13], Suppose C t = C t (t , x, y, z , u, . . .),

C x = C x (t , x, y, z , u, . . .),

C = C (t , x, y, z , u, . . .),

C z = C z (t , x, y, z , u, . . .),

y

y

are components of conservation laws, then they should satisfy the identity Dt (C t ) + Dx (C x ) + Dy (C y ) + Dz (C z ) = 0,

(26)

and a formal Lagrangian for Eq. (1) can be written as

[

µ(α )

L = v (t , x, y, z) Dt

]

(u) − (f (u)ux )x − (g(u)uy )y − (h(u)uz )z ,

(27)

where v (t , x, y, z) is a new dependent variable. Euler–Lagrange operator is defined by

δ ∂ ∂ ∂ ∂ ∂ µ(α ) = + (Dt )∗ − Dx − Dy − Dz µ (α ) δu ∂u ∂ ux ∂ uy ∂ uz ∂ (Dt ) ∂ ∂ ∂ + D2x + D2y + D2z , ∂ uxx ∂ uyy ∂ uzz µ(α )

µ(α )

where (Dt )∗ is the adjoint operator of Dt derivatives are defined differently, (0 Dαt )∗ = (−1)nt ITn−α (Dnt ) ≡ Ct DαT ,

(28) µ(α ) ∗

. Adjoint operators (Dt

) for Riemann–Liouville and Caputo

(C0 Dαt )∗ = (−1)n Dnt (t ITn−α ) ≡ t DαT .

fractional

(29)

µ(α )

Also adjoint operators (Dt )∗ , for each of Eqs. (12)–(15) are defined by µ(α ) 1 ∗ For Eq. (12) (Dt )∗ ≡ (Dα+ ) = t DαT Dt , t µ(α ) ∗ C α ∗ for Eq. (13) (Dt ) ≡ ( Dt ) = t DαT , Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

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Table 1 W

−ux

C x,y,z

Component of conservation laws for Eq. (12) (f = g = h = 1)

Cx

y(az + b)(c1 (T − t)α + c2 ) [−ux + xuxx ]

C

x(az + b)(c1 (T − t)α + c2 ) −ux + yuyx

[

y

xy(c1 (T − t)α + c2 ) [−aux + (az + b)uzx ]

Cz C

−uy

y(az + b)(c1 (T − t)α + c2 ) −uy + xuxy

[ ] [ ] x(az + b)(c1 (T − t)α + c2 ) −uy + yuyy [ ] xy(c1 (T − t)α + c2 ) −auy + (az + b)uzy

x

Cy Cz

−uz

Cx

y(az + b)(c1 (T − t)α + c2 ) [−uz + xuxz ]

Cy

x(az + b)(c1 (T − t)α + c2 ) −uz + yuyz

C

[

C

xuz − zux

xy(c1 (T − t) + c2 ) [−auz + (az + b)uzz ] y(az + b)(c1 (T − t)α + c2 ) xuy − yux − x(xuxy − yuxx + uy ) α

y

xy(c1 (T − t)α + c2 ) a(xuy − yux ) − (az + b)(xuyz − yuxz )

Cx

y(az + b)(c1 (T − t)α + c2 ) [xuz − zux − x(xuxz − zuxx + uz )]

C

[

x(az + b)(c1 (T − t)α + c2 ) xuz − zux − y(xuyz − zuxy )

[

y

y(az + b)(c1 (T − t)α + c2 ) yuz − zuy − x(yuxz − zuxy )

[ [

]

x(az + b)(c1 (T − t)α + c2 ) yuz − zuy − y(yuyz − zuyy + uz )

Cz

xy(c1 (T − t)α + c2 ) a(yuz − zuy ) − (az + b)(yuzz − zuzy − uy )

[

Cx

y(az + b)(c1 (T − t)α + c2 ) [−ut + xuxt ]

Cy

x(az + b)(c1 (T − t)α + c2 ) −ut + yuyt

[

] ]

]

α

z

xy(c1 (T − t) + c2 ) [−aut + (az + b)uzt ] y(az + b)(c1 (T − t)α + c2 ) W + x(tuxt +

[ [

α

Cz

] ] x(az + b)(c1 (T − t)α + c2 ) W + y(tuyt + α2 (uy + xuxy + yuyy + zuyz )) [ ] xy(c1 (T − t)α + c2 ) W + (az + b)(tuzt + α2 (uz + xuzx + yuzy + zuzz ))

Cx

y(az + b)(c1 (T − t)α + c2 ) [u − xux ]

Cy

u

]

xy(c1 (T − t)α + c2 ) [a(xuz − zux ) − (az + b)(xuzz − zuxz − ux )]

x

Cx

−tut − 2 (xux + yuy + zuz )

]

Cy

C α

] ]

x(az + b)(c1 (T − t) + c2 ) xuy − yux − y(xuyy − yuxy − ux )

Cz

−ut

[ [

Cz

C

yuz − zuy

]

α

z

Cx xuy − yux

]

C

x(az + b)(c1 (T − t)α + c2 ) u − yuy

[

y

2

(ux + xuxx + yuxy + zuxz ))

]

xy(c1 (T − t)α + c2 ) [u − (az + b)uz ]

Cz

µ(α )

for Eq. (14) (Dt )∗ ≡ (D1t +α )∗ = Ct D1T +α , µ(α ) for Eq. (15) (Dt )∗ ≡ (Dαt )∗ = Ct DαT . (n−α ) Here t IT is the right-sided fractional integral (3), t DαT and Ct DαT are the right-sided Riemann–Liouville and Caputo fractional derivative of order α (4), (5), respectively. In [13] the adjoint equation is defined for PDEs. In a similar way, the adjoint equation for non-linear FDEs can be defined µ(α )

F ∗ = F ∗ (t , x, y, z , u, v, Dt

µ(α ) ∗

u, (Dt

) v, ux , vx , uxx , vxx , . . .) =

δL , δu

(30)

where L is the formal Lagrangian (27). By calculations of Eq. (30), simplified form of adjoint equation is obtained as follows: µ(α ) ∗

F ∗ = (Dt

) v − f vxx − g vyy − hvzz = 0,

n − 1 < α < n.

(31)

With the extension of the definition of non-linear self-adjoint for FDEs, we realize that Eq. (1) is a non-linear self-adjoint. Because the adjoint equation (31) satisfies all solution of Eq. (1). Substituting v = ϕ (t , x, y, z) = φ (t)ψ (x)ψ (y)ψ (z) ̸ = 0 into adjoint equation (31), yields µ(α )

ψ (x)ψ (y)ψ (z)(Dt )∗ (φ (t)) [ ] −φ (t) f ψ (x)′′ ψ (y)ψ (z) + g ψ (x)ψ (y)′′ ψ (z) + hψ (x)ψ (y)ψ (z)′′ = 0. Therefore

{

f ψ (x)′′ ψ (y)ψ (z) + g ψ (x)ψ (y)′′ ψ (z) + hψ (x)ψ (y)ψ (z)′′ = 0 µ(α ) (Dt )∗ (φ (t)) = 0

(32)

The first equation in (32) is a second order PDE, that by taking f = g = h = 1, eu , uA many exact solutions for this equation are obtained. Three of them are as follows:

ψ1 (x, y, z) = xy(az + b), ψ2 (x, y, z) = a1 a2 (az + b), √ √ ψ3 (x, y, z) = ex+y (d1 sin 2z + d2 cos 2z).

(33)

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

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Table 2 W

− ux

− uy

C x,y,z

Component of conservation laws for Eq. (12) (f = g = h = 1)

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )uxx

C

y

a1 a2 (az + b)(c1 (T − t)α + c2 )uyx

C

z

a1 a2 (az + b)(c1 (T − t)α + c2 )uxy

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 )uyy

Cz

a1 a2 (c1 (T − t)α + c2 ) −auy + (az + b)uzy a1 a2 (az + b)(c1 (T − t) + c2 )uxz a1 a2 (az + b)(c1 (T − t)α + c2 )uyz

Cz

a1 a2 (c1 (T − t)α + c2 ) [−auz + (az + b)uzz ]

[ ] −a1 a2 (az + b)(c1 (T − t)α + c2 ) xuxy − yuxx + uy [ ] −a1 a2 (az + b)(c1 (T − t)α + c2 ) xuyy − yuxy − ux [ ] a1 a2 (c1 (T − t)α + c2 ) a(xuy − yux ) − (az + b)(xuyz − yuxz )

x

Cy Cz

Cz

−a1 a2 (az + b)(c1 (T − t)α + c2 ) [xuxz − zuxx + uz ] [ ] −a1 a2 (az + b)(c1 (T − t)α + c2 ) xuyz − zuxy α a1 a2 (c1 (T − t) + c2 ) [a(xuz − zux ) − (az + b)(xuzz − zuxz − ux )] [ ] −a1 a2 (az + b)(c1 (T − t)α + c2 ) yuxz − zuxy [ ] α −a1 a2 (az + b)(c1 (T − t) + c2 ) yuyz − zuyy + uz [ ] a1 a2 (c1 (T − t)α + c2 ) a(yuz − zuy ) − (az + b)(yuzz − zuzy − uy )

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )uxt

y

a1 a2 (az + b)(c1 (T − t)α + c2 )uyt

Cx xuz − zux

Cy Cz Cx

yuz − zuy

Cy

− ut

C

α

−tut − 2 (xux + yuy + zuz )

Cz

a1 a2 (c1 (T − t)α + c2 ) [−aut + (az + b)uzt ]

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 ) tuxt +

α

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 ) tuyt +

α

Cz

a1 a2 (c1 (T −t)α +c2 ) aW + (az + b)(tuzt +

C u

]

α

Cy

C xuy − yux

[

x

C

− uz

a1 a2 (c1 (T − t)α + c2 ) [−aux + (az + b)uzx ]

Cx

[ [

] ]

2

(ux + xuxx + yuxy + zuxz )

2

(uy + xuxy + yuyy + zuyz )

[

α 2

(uz + xuzx + yuzy + zuzz ))

]

α

x

−a1 a2 (az + b)(c1 (T − t) + c2 )ux −a1 a2 (az + b)(c1 (T − t)α + c2 )uy a1 a2 (c1 (T − t)α + c2 ) [au − (az + b)uz ]

Cy Cz

The second equation in (32) must be solved separately for each of Eqs. (12)–(15). For Eq. (12), (Dtα+1 )∗ φ (t) = t DαT (φ ′ (t)) = 0, so φ (t) = c1 (T − t)α + c2 , For Eq. (13), (C Dαt )∗ φ (t) = t DαT (φ (t)) = 0, so φ (t) = c1 (T − t)α−1 , For Eq. (14), (D1t +α )∗ φ (t) = Ct D1T +α (φ (t)) = 0. { } Using Laplace property for Caputo time-fractional derivative L Ct DαT φ (t) = sα φ (s) − 1, we find the exact solution φ (t) = c1 t n + c2 t n−1 . By consideration 0 < α < 1 and 1 < α < 2, we have φ (t) = c1 t + c2 , and φ (t) = c1 t 2 + c2 t, respectively. For Eq. (15), we have (Dαt )∗ φ (t) = Ct DαT (φ (t)) = 0, For 0 < α < 1, φ (t) = c1 and for 1 < α < 2, φ (t) = c1 t Similar to PDEs, Noether’s identity is a fundamental tool to derive components of conservation laws. Consider Dt (ξ 0 )I + Dx (ξ 1 )I + Dy (ξ 2 )I + Dz (ξ 3 )I = W

δ + Dt (N t ) + Dx (N x ) + Dy (N y ) + Dz (N z ), δu

(34)

where I is the identity operator, X˜ is prolongation for the Lie point group generator (17), W is characteristic for generator X defined by W = η − ξ 0 ut − ξ 1 ux − ξ 2 uy − ξ 3 uz , and finally N x , N y , N z , N t are the Noether’s operator that are given by N x = ξ 1I + W

N y = ξ 2I + W

(

(

∂ ∂ − Dx ∂ ux ∂ uxx

)

∂ ∂ − Dy ∂ uy ∂ uyy

)

+ Dx (W )

+ Dy (W )

∂

,

(35)

∂ , ∂ uyy

(36)

∂ uxx

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

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7

Table 3 W

−ux

C x,y,z

Component of conservation laws for Eq. (12) (f = g = h = 1)

Cx

ex+y (d1 sin

Cy

[ ] 2z)(c1 (T − t)α + c2 ) −ux + uyx [ √ ] √ √ √ √ ex+y (c1 (T − t)α + c2 ) − 2((d1 cos 2z − d2 sin 2z))ux + (d1 sin 2z + d2 cos 2z)uzx √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) −uy + uxy √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) −uy + uyy [ √ ] √ √ √ √ ex+y (c1 (T − t)α + c2 ) − 2((d1 cos 2z − d2 sin 2z))uy + (d1 sin 2z + d2 cos 2z)uzy √ √ ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) [−uz + uxz ] √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) −uz + uyz [ √ ] √ √ √ √ ex+y (c1 (T − t)α + c2 ) − 2((d1 cos 2z − d2 sin 2z))uz + (d1 sin 2z + d2 cos 2z)uzz √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) xuy − yux − (xuxy − yuxx + uy ) √ √ [ ] x+y α e (d1 sin 2z + d2 cos 2z)(c1 (T − t) + c2 ) xuy − yux − (xuyy − yuxy − ux ) [√ ] √ √ √ √ ex+y (c1 (T − t)α + c2 ) 2((d1 cos 2z − d2 sin 2z))(xuy − yux ) − (d1 sin 2z + d2 cos 2z)(xuyz − yuxz ) √ √ ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) [xuz − zux − (xuxz − zuxx + uz )] √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) xuz − zux − (xuyz − zuxy ) x+y α e [(c1 (T − t) + ] √ √ √ √ √ c2 ) 2((d1 cos 2z − d2 sin 2z))(xuz − zux ) − (d1 sin 2z + d2 cos 2z)(xuzz − zuxz − ux ) √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) yuz − zuy − (yuxz − zuxy ) √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) yuz − zuy − (yuyz − zuyy + uz ) [√ ] √ √ √ √ xy(c1 (T − t)α + c2 ) 2((d1 cos 2z − d2 sin 2z))(yuz − zuy ) − (d1 sin 2z + d2 cos 2z)(yuzz − zuzy − uy ) √ √ ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) [−ut + uxt ] √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) −ut + uyt [ √ ] √ √ √ √ e(x+y) (c1 (T − t)α + c2 ) − 2((d1 cos 2z − d2 sin 2z))ut + (d1 sin 2z + d2 cos 2z)uzt √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ) W + tuxt + α2 (ux + xuxx + yuxy + zuxz ) √ √ ] [ α x+y α e (d1 sin 2z + d2 cos 2z)(c1 (T − t) + c2 ) W + tuyt + 2 (uy + xuxy + yuyy + zuyz ) ex+y[(c1 (T − t)α + ] √ √ √ √ √ c2 ) 2((d1 cos 2z − d2 sin 2z))W + (d1 sin 2z + d2 cos 2z)(uzt + α2 (uz + xuzx + yuzy + zuzz )) √ √ ex+y (d1 sin √2z + d2 cos √2z)(c1 (T − t)α + c2 ) [u − ux ] [ ] α x+y e (d1 sin 2z + d2 cos [√ 2z)(c1 (T√− t) + c2 ) √u − uy ] √ √ ex+y (c1 (T − t)α + c2 ) 2((d1 cos 2z − d2 sin 2z))u − (d1 sin 2z + d2 cos 2z)uz ex+y (d1 sin

Cz Cx

−uy

Cy Cz Cx

−uz

Cy Cz Cx

xuy − yux

Cy Cz Cx

xuz − zux

Cy Cz

Cx yuz − zuy

Cy Cz Cx

−ut

Cy Cz

−tut −

α

Cx Cy

(xux 2 +yuy + zuz )

Cz

Cx Cy

u

Cz

N z = ξ 3I + W

(

∂ ∂ − Dz ∂ uz ∂ uzz

√

√ 2z + d2 cos

√

2z)(c1 (T − t)α + c2 ) [−ux + uxx ]

√

2z + d2 cos

) + Dz (W )

∂ . ∂ uzz

(37)

For the Riemann–Liouville time-fractional derivative, the Noether operator N t is as follows: N t = ξ 0I +

n−1 ∑

∂

k=0

∂ (Dαt u)

1−k (−1)k Dα− (W )Dkt t

( − (−1)n J W , Dnt

∂ ∂ (Dαt u)

)

,

(38)

and for the Caputo time-fractional derivative, operator N t is given by N =ξ I+ t

0

n−1 ∑

1−k Dkt (W )Dα− T

k=0

∂ ∂ (Dαt u)

( −J

Dnt (W )

,

∂ ∂ (Dαt u)

)

,

(39)

where J as following integral. J(f , g) =

1

Γ (n − α )

∫ t∫ 0

T t

f (τ , x, y, z)g(µ, x, y, z) (µ − τ )n−α

dµdτ .

(40)

If we effect two sides of Eq. (34) on the Lagrangian, the left-hand side of this equality is equal to zero X˜ L + Dt (ξ 0 )L + Dx (ξ 1 )L + Dy (ξ 2 )L + Dz (ξ 3 )L|(1) = 0.

(41)

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

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E. Lashkarian et al. / Computers and Mathematics with Applications (

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Table 4 C x,y,z

W

− ux

− uy

C

x

C

y

− ux + yuyx

] ]

Cx

y(az + b)(c1 (T − t)α + c2 )eu [(1 − 2xux )(xuz − zux ) − x(xuxz − zuxx + uz )]

Cy

x(az + b)(c1 (T − t)α + c2 )eu (1 − 2yuy )(xuz − zux ) − y(xuyz − zuxy )

Cx Cy

C

z

Cx C

y

[

]

[

α

xy(c1 (T − t) + c2 )e [(a − 2uz (az + b))(xuz − zux ) − (az + b)(xuzz − zuxz − ux )] y(az + b)(c1 (T − t)α + c2 ) (1 − 2xux )(yuz − zuy ) − x(yuxz − zuxy )eu

[ [

α

Cx

y(az + b)(c1 (T − t)α + c2 )eu [(2xux − 1)(α + tut ) + xtuxt ]

Cz C

x

]

x(az + b)(c1 (T − t) + c2 ) (1 − 2yuy )(yuz − zuy ) − y(yuyz − zuyy + uz )eu xy(c1 (T − t)α + c2 )e

y

]

u

Cz

C

2 − xux − yuy − zuz

2yu2y

Cz

Cz

−α − tut

x(az + b)(c1 (T − t) + c2 )e

[ [

Cx

Cy

yuz − zuy

u

[ ] y(az + b)(c1 (T − t)α + c2 )eu 2xux uy − uy + xuxy [ ] x(az + b)(c1 (T − t)α + c2 )eu 2yu2y − uy + yuyy [ ] xy(c1 (T − t)α + c2 )eu 2(az + b)uy uz − auy + (az + b)uzy [ ] y(az + b)(c1 (T − t)α + c2 )eu 2xu − xuy − uz + xuxz [ ] x(az + b)(c1 (T − t)α + c2 )eu 2yuy uz − uz + yuyz [ ] xy(c1 (T − t)α + c2 )eu 2(az + b)u2z − auz + (az + b)uzz [ ] y(az + b)(c1 (T − t)α + c2 )eu (1 − 2xux )(xuy − yux ) − x(xuxy − yuxx + uy ) [ ] x(az + b)(c1 (T − t)α + c2 )eu (1 − 2yuy )(xuy − yux ) − y(xuyy − yuxy − ux ) [ ] xy(c1 (T − t)α + c2 )eu (a − 2uz (az + b))(xuy − yux ) − (az + b)(xuyz − yuxz )

Cx

xuz − zux

α

xy(c1 (T − t)α + c2 )eu 2ux uy (az + b) − aux + (az + b)uzx

Cz

xuy − yux

y(az + b)(c1 (T − t)α + c2 )eu 2xu2x − ux + xuxx

Cz Cy

− uz

Component of conservation laws for Eq. (12) (f = g = h = eu )

[ u

]

(a − 2uz (az + b))(yuz − zuy ) − (az + b)(yuzz − zuzy − uy )

x(az + b)(c1 (T − t)α + c2 )eu (2yuy − 1)(α + tut ) + ytuyt

[

]

]

xy(c1 (T − t)α + c2 )eu [(2(az + b)uz − a)(α + tut ) + t(az + b)uzt ] y(az + b)(c1 (T − t)α + c2 )eu W (1 − 2xux ) + x(xuxx + yuxy + zuxz + ux )

[ [ u

] ]

Cy

x(az + b)(c1 (T − t)α + c2 )e

Cz

xy(c1 (T − t)α + c2 )eu W (a − 2(az + b)uz ) + (az + b)(xuxz + yuzy + zuzz + uz )

W (1 − 2yuy ) + y(xuxy + yuyy + zuyz + uy )

[

]

Table 5 W

C x,y,z C

− ux

− uy

yuz − zuy

−α − tut

]

a1 a2 (c1 (T − t)α + c2 )eu [2ux uz (az + b) − aux + (az + b)uzx ]

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )eu 2ux uy + uxy

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 )e

Cz

a1 a2 (c1 (T − t)α + c2 )eu 2(az + b)uy uz − auy + (az + b)uzy

[ [ u

2uy ux + uyx

2u2y + uyy

] ]

]

[

α

x

a1 a2 (az + b)(c1 (T − t) + c2 )e [2ux uz + uxz ]

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 )eu 2uy uz + yuyz

[

α

]

u

[

]

Cz

] − auz + (az + b)uzz [ ] a1 a2 (az + b)(c1 (T − t)α + c2 )eu −2ux (xuy − yux ) − (xuxy − yuxx + uy ) [ ] a1 a2 (az + b)(c1 (T − t)α + c2 )eu −2uy (xuy − yux ) − (xuyy − yuxy − ux ) [ ] a1 a2 (c1 (T − t)α + c2 )eu (a − 2uz (az + b))(xuy − yux ) − (az + b)(xuyz − yuxz )

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )eu [−2ux (xuz − zux ) − (xuxz − zuxx + uz )]

z

Cy

C

y

a1 a2 (c1 (T − t) + c2 )e

u

2(az +

b)u2z

a1 a2 (az + b)(c1 (T − t)α + c2 )eu −2uy (xuz − zux ) − (xuyz − zuxy )

[

]

Cz

a1 a2 (c1 (T − t)α + c2 )eu [(a − 2uz (az + b))(xuz − zux ) − (az + b)(xuzz − zuxz − ux )]

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )eu −2ux (yuz − zuy ) − (yuxz − zuxy )

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 )e

C

z

C

x

Cy

α

a1 a2 (c1 (T − t) + c2 )e

u

[

[ [ u

]

−2uy (yuz − zuy ) − (yuyz − zuyy + uz )

a1 a2 (az + b)(c1 (T − t) + c2 )e [2ux (α + tut ) + tuxt ] a1 a2 (az + b)(c1 (T − t)α + c2 )eu 2uy (α + tut ) + tuyt

[

α

]

a1 a2 (c1 (T − t) + c2 )e [(2(az + b)uz − a)(α + tut ) + t(az + b)uzt ] a1 a2 (az + b)(c1 (T − t)α + c2 )eu −2ux W + xuxx + yuxy + zuxz + ux

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 )e

C

]

u

Cx z

]

(a − 2uz (az + b))(yuz − zuy ) − (az + b)(yuzz − zuzy − uy )

α

z

C 2 − xux − yuy − zuz

[ [ u

Cz

Cx

xuz − zux

a1 a2 (az + b)(c1 (T − t)α + c2 )eu 2u2x + uxx a1 a2 (az + b)(c1 (T − t)α + c2 )e

C xuy − yux

Component of conservation laws for Eq. (12) (f = g = h = eu )

Cy

C

− uz

x

u

[ [ u

] ]

−2uy W + xuxy + yuyy + zuyz + uy

a1 a2 (c1 (T − t)α + c2 )eu W (a − 2(az + b)uz ) + (az + b)(xuxz + yuzy + zuzz + uz )

[

]

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

E. Lashkarian et al. / Computers and Mathematics with Applications (

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9

Table 6 C x,y,z Component of conservation laws for Eq. (12) (f = g = h = eu )

W

√

√

2z)(c1 (T − t)α + c2 )eu 2u2x − ux + uxx

Cx

ex+y (d1 sin

Cy

ex+y (d1 sin

Cz

ex+y (c1 (T − t)α + c2 )eu (d1 sin

Cx

ex+y (d1 sin

Cy

ex+y (d1 sin

Cz

ex+y (c1 (T − t)α + c2 )eu (d1 sin

Cx

ex+y (d1 sin

−ux

2z + d2 cos

√

√

√

√

√

2z + d2 cos

√

√

2z)(c1 (T − t)α + c2 )eu 2u2y − uy + uyy

[

√

√

2z)(2uy uz + uyz ) −

√ 2(d1 cos

√

2z − d2 sin

2z)uy

]

2z)(c1 (T − t)α + c2 )eu [2ux uz + uxz ]

√

√

2z)(c1 (T − t)α + c2 )eu 2uy uz + yuyz

[

]

Cz

ex+y (c1 (T − t)α + c2 )eu (d1 sin

Cx

ex+y (d1 sin

Cy

ex+y (d1 sin

Cz

ex+y (c1 (T − t)α + c2 )eu

Cx

ex+y (d1 sin

Cy

ex+y (d1 sin

Cz

ex+y (c1 (T − t)α + c2 )eu

Cx

ex+y (az + b)(c1 (T − t)α + c2 )eu (1 − 2ux )(yuz − zuy ) − (yuxz − zuxy )

Cy

ex+y (az + b)(c1 (T − t)α + c2 )eu (1 − 2uy )(yuz − zuy ) − (yuyz − zuyy + uz )

Cz

ex+y (c1 (T − t)α + c2 )eu (d1 sin

yuz − zuy

]

√

2z + d2 cos

ex+y (d1 sin

2z + d2 cos

[

xuz − zux

2z)ux

]

Cy

xuy − yux

√ 2z − d2 sin

]

√

2z + d2 cos

√

2(d1 cos

[

√

2z + d2 cos

]

2z)(2ux uz + uxz ) −

2z)(c1 (T − t)α + c2 )eu 2ux uy − uy + uxy

[

−uz

[

√

2z + d2 cos

√

]

2z)(c1 (T − t)α + c2 )eu 2uy ux − ux + uyx

2z + d2 cos

[

−uy

[

√

√

√

2z + d2 cos

2z)(2u2z + uzz ) −

√

√

2z − d2 sin

2z)uz

[

√

[

[√

√

2(d1 cos

√ 2z − d2 sin

]

]

2z)(c1 (T − t)α + c2 )eu (1 − 2uy )(xuy − yux ) − (xuyy − yuxy − ux )

2z + d2 cos

√

√

2(d1 cos

2z)(c1 (T − t)α + c2 )eu (1 − 2ux )(xuy − yux ) − (xuxy − yuxx + uy )

2z + d2 cos

√

√

]

√

2z)(xuy − yux ) − 2uz (xuy − yux )(d1 sin

√

2z + d2 cos

2z) − (xuzy − yuxz )

]

√

2z)(c1 (T − t)α + c2 )eu [(1 − 2ux )(xuz − zux ) − (xuxz − zuxx + uz )]

2z + d2 cos

√

√

2z)(c1 (T − t)α + c2 )eu (1 − 2uy )(xuz − zux ) − (xuyz − zuxy )

2z + d2 cos

[

√

[√

2(d1 cos

√ 2z − d2 sin

√

2z)(xuz − zux ) + (d1 sin

[

√

√

2z − 2uz d2 cos

]

2z)(xuz − zux ) − (xuzz − zuxz − ux )

]

[

[

]

]

√ 2z + d2 cos

Cx

2z)(−2uz (yuz − zuy ) − (yuzz − zuzy − uy )) √ √ √ + 2(d1 cos 2z − d2 sin 2z)(yuz − zuy ) √ √ ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 )eu [(2ux − 1)(α + tut ) + tuxt ]

Cy

ex+y (d1 sin

Cz

ex+y (c1 (T − t)α + c2 )eu (2uz (α + tut ) + tuzt )(d1 sin

Cx

ex+y (d1 sin

Cy

ex+y (d1 sin

Cz

ex+y (c1 (T − t)α + c2 )eu (d1 sin

]

−α − tut

2 − xux −yuy − zuz

√

√ 2z + d2 cos

2z)(c1 (T − t)α + c2 )eu (2uy − 1)(α + tut ) + tuyt

[

[

√

√ 2z + d2 cos

√

[

2z + d2 cos

√ 2z) −

√

2(α + tut )(d1 cos

2z)(c1 (T − t)α + c2 )eu (1 − 2ux )W + xuxx + yuxy + zuxz + ux

[

√

2z + d2 cos

√

]

√

√

2z + d2 cos

2z)(xuxz + yuzy + zuzz + uz − 2Wuz ) +

√

2z − d2 cos

2z)

]

]

2z)(c1 (T − t)α + c2 )eu (1 − 2uy )W + xuxy + yuyy + zuyz + uy

[

√

]

√ 2W (d1 cos

√ 2z − d2 sin

√ 2z)

]

Hence the right-hand side of the equality (34) comes in the form below: Dt (N t L) + Dx (N x L) + Dy (N y L) + Dz (N z L) = 0.

(42)

A comparing between (26) and (42), shows that: C t = N t L,

C x = N x L,

C y = N y L,

C z = N z L.

Also t-component of conservation laws for each of Eqs. (12)–(15) can be arranged by: 1 C t = Dα− Dt (W ) t

∂L ∂ Dαt ut

1 − WDα− Dt T

∂L

( ) ∂L + J D (W ) , D , t t ∂ Dαt ut ∂ Dαt ut

(43)

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

10

E. Lashkarian et al. / Computers and Mathematics with Applications (

)

–

Table 7 W

− ux

− uy

− uz

C x,y,z

Component of conservation laws for Eq. (12) (f = g = h = uA )

Cx

y(az + b)(c1 (T − t)α + c2 )uA

Cy

x(az + b)(c1 (T − t)α + c2 )u

Cz

xy(c1 (T − t)α + c2 )u

Cx

y(az + b)(c1 (T − t)α + c2 )u

Cy

x(az + b)(c1 (T − t)α + c2 )uA

Cz

xy(c1 (T − t)α + c2 )u

Cx

y(az + b)(c1 (T − t)α + c2 )u

C

xuy − yux

y

u

x(az + b)(c1 (T − t)α + c2 )u xy(c1 (T − t)α + c2 )uA (a −

Cz Cx Cy Cz

[ A 2A u

[ 2A u

[ 2A u

[ A 2A u

x(az + b)(c1 (T − t)α + c2 )uA

z

Cy

u

xu2x − ux + xuxx

]

yu2y − ux + yuyx

]

xux uy − uy + xuxy yu2y − uy + yuyy

[ 2A u

]

xu − xuy − uz + xuxz yuy uz − uz + yuyz

]

]

]

(az + b)u2z − auz + (az + b)uzz

[ A

]

]

(az + b)uy uz − auy + (az + b)uzy

Cy

Cx

−( 2tα ut + xux + yuy + zuz )

[ A 2A

y(az + b)(c1 (T − t)α + c2 )u

Cz

u

ux uy (az + b) − aux + (az + b)uzx

Cx

Cy

yuz − zuy

u

xy(c1 (T − t)α + c2 )uA

Cx xuz − zux

[ A 2A

Cz

C

[ 2A

[ A 2A

]

2A xux )(xuy u

] − yux ) − x(xuxy − yuxx + uy ) ] (1 − 2A yuy )(xuy − yux ) − y(xuyy − yuxy − ux ) u (1 −

[ A

[

2A u (az b))(xuy u z 2A A xux )(xuz c2 )u (1 u

] − yux ) − (az + b)(xuyz − yuxz ) ] [ − zux ) − x(xuxz − zuxx + uz ) y(az + b)(c1 (T − t)α + − ] [ x(az + b)(c1 (T − t)α + c2 )uA (1 − 2A yuy )(xuz − zux ) − y(xuyz − zuxy ) u [ ] 2A α A xy(c1 (T − t) + c2 )u (a − u uz (az + b))(xuz − zux ) − (az + b)(xuzz − zuxz − ux ) [ ] y(az + b)(c1 (T − t)α + c2 )uA (1 − 2A xux )(yuz − zuy ) − x(yuxz − zuxy ) u ] [ yu x(az + b)(c1 (T − t)α + c2 )uA (1 − 2A y )(yuz − zuy ) − y(yuyz − zuyy + uz ) u [ ] xy(c1 (T − t)α + c2 )uA (a − 2A u (az + b))(yuz − zuy ) − (az + b)(yuzz − zuzy − uy ) u z ] [ xux ) + x( 2t u + ux + xuxx + yuxy + zuxz ) y(az + b)(c1 (T − t)α + c2 )uA W (1 − 2A u α xt ] [ 2A α A x(az + b)(c1 (T − t) + c2 )u W (1 − u yuy ) + y( 2t u + uy + xuxy + yuyy + zuyz ) α yt [ ] xy(c1 (T − t)α + c2 )uA W (a − 2A (az + b)uz ) + (az + b)( 2t u + uz + xuxz + yuzy + zuzz ) u α zt +

Table 8 W

− ux

− uy

− uz

xuy − yux

C x,y,z

Component of conservation laws for Eq. (12) (f = g = h = uA )

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )uA

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 )u

Cz

a1 a2 (c1 (T − t)α + c2 )u

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )u

Cy

a1 a2 (az + b)(c1 (T − t)α + c2 )uA

Cz

a1 a2 (c1 (T − t)α + c2 )u

Cx

a1 a2 (az + b)(c1 (T − t)α + c2 )u

C

y

u

Cy

Cx Cy Cz Cx Cy Cz

[ A 2A u

[ 2A u

[ 2A u

[ A 2A

a1 a2 (az + b)(c1 (T − t)α + c2 )uA a1 a2 (az + b)(c1 (T − t)α + c2 )e

Cy

u

u2x + uxx

]

uy ux + uyx

]

ux uy + uxy u2y + uyy

u

[ 2A u

]

]

]

(az + b)uy uz − auy + (az + b)uzy

Cx

Cz

−( 2tα ut + xux + yuy + zuz )

[ A 2A

u

[ A 2A

ux uz (az + b) − aux + (az + b)uzx

a1 a2 (c1 (T − t)α + c2 )uA

Cx

yuz − zuy

u

Cz

Cz xuz − zux

[ A 2A

[ 2A

ux uz + uxz

]

]

uy uz + yuyz

]

(az + b)u2z − auz + (az + b)uzz

]

[ u

] −2ux (xuy − yux ) − (xuxy − yuxx + uy ) [ ] a1 a2 (az + b)(c1 (T − t)α + c2 )uA −u2A uy (xuy − yux ) − (xuyy − yuxy − ux ) [ ] a1 a2 (c1 (T − t)α + c2 )uA (a − 2A u (az + b))(xuy − yux ) − (az + b)(xuyz − yuxz ) u z [ ] a1 a2 (az + b)(c1 (T − t)α + c2 )uA −u2A ux (xuz − zux ) − (xuxz − zuxx + uz ) [ ] a1 a2 (az + b)(c1 (T − t)α + c2 )uA −u2A uy (xuz − zux ) − (xuyz − zuxy ) [ ] a1 a2 (c1 (T − t)α + c2 )uA (a − 2A u (az + b))(xuz − zux ) − (az + b)(xuzz − zuxz − ux ) u z [ ] a1 a2 (az + b)(c1 (T − t)α + c2 )uA − 2A u (yuz − zuy ) − (yuxz − zuxy ) u x [ ] a1 a2 (az + b)(c1 (T − t)α + c2 )uA − 2A u (yuz − zuy ) − (yuyz − zuyy + uz ) u y [ ] a1 a2 (c1 (T − t)α + c2 )uA (a − 2A u (az + b))(yuz − zuy ) − (az + b)(yuzz − zuzy − uy ) u z [ ] a1 a2 (c1 (T − t)α + c2 )uA −u2A Wux + 2t u + ux + xuxx + yuxy + zuxz α xt [ ] a1 a2 (c1 (T − t)α + c2 )uA −u2A Wuy + 2t u + uy + xuxy + yuyy + zuyz α yt [ ] a1 a2 (c1 (T − t)α + c2 )uA W (a − 2A (az + b)uz ) + (az + b)( 2t u + uz + xuxz + yuzy + zuzz ) u α zt

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

E. Lashkarian et al. / Computers and Mathematics with Applications (

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11

Table 9 W

−ux

C x,y,z

Components of conservation laws for Eq. (12) (f = g = h = uA )

Cx

ex+y (d1 sin

Cy

−uy

xuy − yux

[

ex+y (d1 sin ex+y (d1 sin

Cx

ex+y (d1 sin

ex+y (c1 (T − t)α + c2 )uA (d1 sin

Cx

ex+y (d1 sin

Cx Cy Cz

[

2t

ut + xux

Cx

α y +yuy + zuz ) C Cz

ux uy − uy + uxy u2y − uy + uyy

[ 2A u u

ux uz + uxz

uy uz + yuyz

2z)(c1 (T − t)α + c2 )uA (1 −

[

2z)(c1 (T − t)α + c2 )u

]

2z)ux

]

] √

2(d1 cos

√

2z − d2 sin

2z)uy

]

]

2A u )(xuy u x

√

√

2z)( 2A u2 + uzz ) − u z

√

√

2z − d2 sin

]

√

2z + d2 cos

√

√

√

2z + d2 cos

2(d1 cos

√

2z − d2 sin

2z)uz

− yux ) − (xuxy − yuxx + uy )

]

]

2A u )(xuy u y

] − yux ) − (xuyy − yuxy − ux ) [√ ] √ √ √ √ ex+y (c1 (T − t)α + c2 )uA u (xuy − yux )(d1 sin 2z + d2 cos 2z) − (xuzy − yuxz ) 2(d1 cos 2z − d2 sin 2z)(xuy − yux ) − 2A u z √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 )uA (1 − 2A u )(xuz − zux ) − (xuxz − zuxx + uz ) u x √ √ [ ] ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 )uA (1 − 2A u )(xuz − zux ) − (xuyz − zuxy ) u y [√ √ √ √ √ ex+y (c1 (T − t)α + c2 )uA u d cos 2z)(xuz − zux ) 2(d1 cos 2z − d2 sin 2z)(xuz − zux ) + (d1 sin 2z − 2A u z 2 ] −(xuzz − zuxz − ux ) ] [ u )(yuz − zuy ) − (yuxz − zuxy ) ex+y (az + b)(c1 (T − t)α + c2 )uA (1 − 2A u x ] [ ex+y (az + b)(c1 (T − t)α + c2 )uA (1 − 2A u )(yuz − zuy ) − (yuyz − zuyy + uz ) u y ex+y (d1 sin

2z + d2 cos

ex+y (c[1 (T − t)α +

c2 )uA (d1 sin

−(

√

] 2(d1 cos

2z)(2uy uz + uyz ) −

2z)(c1 (T − t)α + c2 )u

2z + d2 cos

√

u

[ A 2A

Cz

Cz

2z + d2 cos

2z)(c1 (T − t)α + c2 )uA

ex+y (d1 sin

√

u

√

√

Cy

Cy

√

√

2z + d2 cos

√

[ 2A

2z)(c1 (T − t)α + c2 )u

ex+y (c1 (T − t)α + c2 )eu (d1 sin

uy ux − ux + uyx

√

[ A 2A

[

]

2z)(2ux uz + uxz ) −

2z)(c1 (T − t)α + c2 )uA

2z + d2 cos

√

u

u2x − ux + uxx

√

2z + d2 cos

√

Cz

Cx

√

√

2z + d2 cos

√

u

2z)(c1 (T − t)α + c2 )u

2z + d2 cos

√

[ 2A

[ A 2A

Cx

Cz

yuz − zuy

√

ex+y (c1 (T − t)α + c2 )eu (d1 sin

Cy

xuz − zux

2z)(c1 (T − t)α + c2 )uA

2z + d2 cos

√

Cz

Cy

−uz

ex+y (d1 sin

√

√

ex+y (d1 sin ex+y (d1 sin

√

[ A

√

2z)(− 2A u (yuz − zuy ) − (yuzz − zuzy − uy )) + u z

2z + d2 cos

√

√

2z)(c1 (T − t)α + c2 )uA (1 −

2z + d2 cos

√

√

2z + d2 cos

ex+y (c1 (T − t)α + c2 )uA (d1 sin

+ 2(d1 cos

√ 2z − d2 sin

√ 2z)

√

√

2z + d2 cos

√ 2(d1 cos

√

√

2z − d2 sin

[

2A u )W u x

+

2t

uxt + ux + xuxx + yuxy + zuxz

[ A

2A u )W u y

+

2t

uyt + uy + xuxy + yuyy + zuyz

2z)(c1 (T − t)α + c2 )u

[

√

(1 −

(1 −

2z)(− 2A u W+ u z

2t

α

α

α

2z)(yuz − zuy )

]

] ]

uzt + uz + xuzx + yuzy + zuzz )

]

Table 10 Wi , Si

t-components of conservation laws for Eq. (12)

−ux , S1

[ ] p t 1 ψi (x, y, z) Dαt (ux )(c1 (T − t)α + c2 ) + α c1 ux Dα− (T − t)α−1 + α c1 Γ (n1−α ) T ] [ p t 1 ψi (x, y, z) Dαt (uy )(c1 (T − t)α + c2 ) + α c1 uy Dα− (T − t)α−1 + α c1 Γ (n1−α ) T [ ] p t 1 ψi (x, y, z) Dαt (uz )(c1 (T − t)α + c2 ) + α c1 uz Dα− (T − t)α−1 + α c1 Γ (n1−α ) T [ ] p t p t 1 ψi (x, y, z) Dαt (xuy − yux )(c1 (T − t)α + c2 ) + α c1 (xuy − yux )Dα− (T − t)α−1 + α c1 (x Γ (n1−α ) − y Γ (n2−α ) ) T ] [ p t p t 1 ψi (x, y, z) Dαt (xuz − zux )(c1 (T − t)α + c2 ) + α c1 (xuz − zux )Dα− (T − t)α−1 + α c1 (x Γ (n1−α ) − z Γ (n2−α ) ) T [ ] p t p t 1 ψi (x, y, z) Dαt (yuz − zuy )(c1 (T − t)α + c2 ) + α c1 (yuz − zuy )Dα− (T − t)α−1 + α c1 (y Γ (n1−α ) − z Γ (n2−α ) ) T [ ] ψi (x, y, z) (uxx + uyy + uzz )(c1 (T − t)α + c2 ) + α c1 ut DTα−1 (T − t)α−1 + α c1 I(utt , (T − t)α−1 ) [ ] ψi (x, y, z) Dαt (Wi )(c1 (T − t)α + c2 ) + α c1 Wi DTα−1 (T − t)α−1 + α c1 I(ut + tutt , (T − t)α−1 ) + α 2 c1 2Γ (nt −α) (xp1 + yp2 + zp3 ) ] [ 1 ψi (x, y, z) Dαt (u)(c1 (T − t)α + c2 ) + α c1 uDα− (T − t)α−1 + α c1 I(ut , (T − t)α−1 ) T [ ] p t 1 (T − t)α−1 + α c1 ( Γ (n1−α ) + I(ut + tutt , (T − t)α−1 )) ψi (x, y, z) Dαt (α + tut )(c1 (T − t)α + c2 ) + α c1 (α + tut )Dα− T [ ] ψi (x, y, z) Dαt (Wi )(c1 (T − t)α + c2 ) + α c1 Wi DTα−1 (T − t)α−1 + α c1 Γ (nt−α) (xp1 + yp2 + zp3 + p4 )

−uy , S1 −uz , S1 xuy − yux , S1 xuz − zux , S1 yuz − zuy , S1

−ut , S2 [−tut −

α 2

(xux

+yuy + zuz ), S2 ] u, S2

−(α + tut ), S3 [2 − xux −yuy − zuz , S3 ] [−(

2t

ut + xux α +yuy + zuz ), S4 ]

[ ] ψi (x, y, z) Dαt (Wi )(c1 (T − t)α + c2 ) + α c1 Wi DTα−1 (T − t)α−1 + α c1 Γ (nt−α) (xp1 + yp2 + zp3 ) + 2c1 I(tut , (T − t)α−1 )

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

12

E. Lashkarian et al. / Computers and Mathematics with Applications (

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Table 11 Wi , Si

t-components of conservation laws for Eq. (13) 0 < α < 1

− ux , S 1

[ ] p t 1 −c1 ψi (x, y, z) ux Dα− (T − t)α − Γ (n1−α ) T [ ] p t 1 −c1 ψi (x, y, z) uy Dα− (T − t)α − Γ (n1−α ) T [ ] p t 1 −c1 ψi (x, y, z) uz Dα− (T − t)α − Γ (n1−α ) T [ ] 1 c1 ψi (x, y, z) (xuy − yux )Dα− (T − t)α − Γ (nt−α ) (xp1 − yp2 ) T [ ] 1 c1 ψi (x, y, z) (xuz − zux )Dα− (T − t)α − Γ (nt−α ) (xp1 − zp2 ) T [ ] 1 c1 ψi (x, y, z) (yuz − zuy )Dα− (T − t)α − Γ (nt−α ) (yp1 − zp2 ) T [ ] 1 −c1 ψi (x, y, z) ut Dα− (T − t)α − I(utt , (T − t)α ) T

−uy , S1 −uz , S1 xuy − yux , S1 xuz − zux , S1 yuz − zuy , S1

− ut , S 2 [−tut −

α

(xux 2 +yuy + zuz ), S2 ]

[

1 c1 ψi (x, y, z) Wi Dα− (T − t)α + T

αt (xp1 2Γ (n−α )

+ yp2 + zp3 ) + I(ut + tutt , (T − t)α ) ]

u, S2

1 c1 ψi (x, y, z) uDα− (T − t)α − I(ut , (T − t)α ) T

−(α + tut ), S3

[ ] p t 1 −c1 ψi (x, y, z) (α + tut )Dα− (T − t)α − Γ (n1−α ) − I(ut + tutt , (T − t)α ) T [ ] 1 c1 ψi (x, y, z) Wi Dα− (T − t)α + Γ (nt−α ) (xp1 + yp2 + zp3 + p4 ) T

[

[2 − xux −yuy − zuz , S3 ] 2t

[

ut + xux α +yuy + zuz ), S4 ]

[−(

1 c1 ψi (x, y, z) Wi Dα− (T − t)α + T

t

Γ (n−α )

(xp1 + yp2 + zp3 ) + I(ut + tutt , (T − t)α )

]

]

Table 12 Wi , Si

t-components of conservation laws for Eq. (13) 1 < α < 2

− ux , S 1

[ ] p t 1 2 −c1 ψi (x, y, z) ux Dα− (T − t)α − uxt Dα− (T − t)α − Γ (n1−α ) T T [ ] p t 1 2 −c1 ψi (x, y, z) uy Dα− (T − t)α − uyt Dα− (T − t)α − Γ (n1−α ) T T ] [ p t 1 2 −c1 ψi (x, y, z) uz Dα− (T − t)α − uzt Dα− (T − t)α − Γ (n1−α ) T T [ ] 1 2 c1 ψi (x, y, z) (xuy − yux )Dα− (T − t)α − (xuyt − yuxt )Dα− (T − t)α − Γ (nt−α ) (xp1 − yp2 ) T T [ ] 1 2 c1 ψi (x, y, z) (xuz − zux )Dα− (T − t)α − (xuzt − zuxt )Dα− (T − t)α − Γ (nt−α ) (xp1 − zp2 ) T T ] [ 1 2 c1 ψi (x, y, z) (yuz − zuy )Dα− (T − t)α − (yuzt − zuyt )Dα− (T − t)α − Γ (nt−α ) (yp1 − zp2 ) T T [ ] 1 2 −c1 ψi (x, y, z) ut Dα− (T − t)α + utt Dα− (T − t)α − I(uttt , (T − t)α ) T T ] [ 1 2 αt (xp1 + yp2 + zp3 ) + I(ut + tutt , (T − t)α ) c1 ψi (x, y, z) Wi Dα− (T − t)α − (ut + tutt )Dα− (T − t)α + 2Γ (n T T −α ) [ ] 1 2 c1 ψi (x, y, z) uDα− (T − t)α + ut Dα− (T − t)α − I(utt , (T − t)α ) T T [ ] p t 1 2 −c1 ψi (x, y, z) (α + tut )Dα− (T − t)α + (ut + tutt)Dα− (T − t)α − Γ (n1−α ) − I(ut + tutt , (T − t)α ) T T ] [ 1 c1 ψi (x, y, z) Wi Dα− (T − t)α + Γ (nt−α ) (xp1 + yp2 + zp3 + p4 ) T

−uy , S1 −uz , S1 xuy − yux , S1 xuz − zux , S1 yuz − zuy , S1

− ut , S 2 [−tut −

α 2

(xux

+yuy + zuz ), S2 ]

u, S2

−(α + tut ), S3 [2 − xux −yuy − zuz , S3 ] [−(

2t

[

ut + xux α +yuy + zuz ), S4 ]

Ct =

1 c1 ψi (x, y, z) Wi Dα− (T − t)α − T

n−1 ∑

1−k Dkt (W )Dα− T

k=0

C t = Dαt (W )

∂L ∂ Dt1+α u

∂L ∂ (C Dαt u)

2

α

2 (ut + tutt )Dα− (T − t)α + T

( − J Dnt (W ),

1 − Dα− (W )Dt t

∂L ∂ D1t +α u

∂L

)

∂ (C Dαt u)

( − J W , D2t

t

Γ (n−α )

,

(xp1 + yp2 + zp3 ) +

2

α

I(utt + tuttt , (T − t)α )

]

(44)

∂L

)

∂ Dt1+α u

,

(45)

and t

C =

n−1 ∑

k

α−1−k

(−1) Dt

k=0

(W )Dkt

∂L ∂ (Dαt u)

(

− (−1) J W , n

Dnt

∂L ∂ (Dαt u)

)

.

(46)

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

E. Lashkarian et al. / Computers and Mathematics with Applications (

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13

Table 13 Wi , Si

t-components of conservation laws for Eq. (14) 0 < α < 1 [ ] p t 1 −ψi (x, y, z) Dαt (ux )(c1 t + c2 ) − c1 Dα− (ux ) − Γ (n1−α ) t [ α ] t α−1 −ψi (x, y, z) Dt (uy )(c1 t + c2 ) − c1 Dt (uy ) − Γ (np1−α ) [ α ] p1 t α−1 −ψi (x, y, z) Dt (uz )(c1 t + c2 ) − c1 Dt (uz ) − Γ (n−α) [ α ] 1 ψi (x, y, z) Dt (xuy − yux )(c1 t + c2 ) − c1 Dα− (xuy − yux ) − Γ (nt−α ) (xp1 − yp2 ) t ] [ α 1 ψi (x, y, z) Dt (xuz − zux )(c1 t + c2 ) − c1 Dα− (xuz − zux ) − Γ (nt−α ) (xp1 − zp2 ) t [ α ] α−1 ψi (x, y, z) Dt (yuz − zuy )(c1 t + c2 ) − c1 Dt (yuz − zuy ) − Γ (nt−α) (yp1 − zp2 ) [ ] p t 1 −ψi (x, y, z) Dα+ (u)(c1 t + c2 ) − c1 Dαt (u) − Γ (n1−α ) t [ ] p t 1 ψi (x, y, z) Dαt (Wi )(c1 t + c2 ) − c1 Dα− (Wi ) − Γ (n1−α ) t [ α ] p t 1 ψi (x, y, z) Dt (u)(c1 t + c2 ) − c1 Dα− (u) − Γ (n1−α ) t [ α ] t α−1 −ψi (x, y, z) Dt (α + tut )(c1 t + c2 ) − c1 Dt (α + tut ) − Γ (np1−α ) [ α ] p t 1 ψi (x, y, z) Dt (Wi )(c1 t + c2 ) − c1 Dα− (Wi ) − Γ (n1−α ) t

−ux , S1 −uy , S1 −uz , S1 xuy − yux , S1 xuz − zux , S1 yuz − zuy , S1

−ut , S2 [−tut −

α 2

(xux

+yuy + zuz ), S2 ] u, S2

−(α + tut ), S3 [2 − xux −yuy − zuz , S3 ] [−(

2t

ut + xux α +yuy + zuz ), S4 ]

[ ] 1 ψi (x, y, z) Dαt (Wi )(c1 t + c2 ) − c1 Dα− (Wi ) − t

p1 t

Γ (n−α )

Table 14 Wi , Si

t-components of conservation laws for Eq. (14) 1 < α < 2

−ux , S1

] [ p t 1 −ψi (x, y, z) Dαt (ux )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (ux ) − Γ (n1−α ) t [ ] p t 1 −ψi (x, y, z) Dαt (uy )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (uy ) − Γ (n1−α ) t ] [ p t 1 −ψi (x, y, z) Dαt (uz )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (uz ) − Γ (n1−α ) t [ ] 1 ψi (x, y, z) Dαt (xuy − yux )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (xuy − yux ) − Γ (nt−α ) (xp1 − yp2 ) t [ ] 1 ψi (x, y, z) Dαt (xuz − zux )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (xuz − zux ) − Γ (nt−α ) (xp1 − zp2 ) t ] [ 1 ψi (x, y, z) Dαt (yuz − zuy )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (yuz − zuy ) − Γ (nt−α ) (yp1 − zp2 ) t [ ] 1 −ψi (x, y, z) Dα+ (u)(c1 t 2 + c2 t) − (2c1 t + c2 )Dαt (u) − I(ut , 2c1 ) t [ ] 1 αt ψi (x, y, z) Dαt (Wi )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (Wi ) + 2Γ (n (xp1 + yp2 + zp3 ) + I(tut , 2c1 ) t −α ) [ ] 1 ψi (x, y, z) Dαt (u)(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (u) − I(u, 2c1 ) t ] [ 1 −ψi (x, y, z) Dαt (α + tut )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (α + tut ) − I(α + tut , 2c1 ) t [ ] 1 ψi (x, y, z) Dαt (Wi )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (Wi ) + Γ (nt−α ) (xp1 + yp2 + zp3 − 4c1 p4 ) t

−uy , S1 −uz , S1 xuy − yux , S1 xuz − zux , S1 yuz − zuy , S1

−ut , S2 [−tut −

α 2

(xux

+yuy + zuz ), S2 ] u, S2

−(α + tut ), S3 [2 − xux −yuy − zuz , S3 ] 2t

[ 1 ψi (x, y, z) Dαt (Wi )(c1 t 2 + c2 t) − (2c1 t + c2 )Dα− (Wi ) + t

ut + xux α +yuy + zuz ), S4 ]

[−(

t

Γ (n−α )

(xp1 + yp2 + zp3 ) +

2

α

I(tut , 2c1 )

]

x, y and z −components of conserved vector with symmetries given, for all Eqs. (12)–(15) are as follows: x

(

C =W

y

(

C =W

Cz = W

(

∂L ∂L − Dx ∂ ux ∂ uxx

)

∂L ∂L − Dy ∂ uy ∂ uyy

)

∂L ∂L − Dz ∂ uz ∂ uzz

)

+ Dx (W )

∂L , ∂ uxx

(47)

+ Dy (W )

∂L , ∂ uyy

(48)

+ Dz (W )

∂L . ∂ uzz

(49)

• For Eq. (12) For Case 1: Eq. (16) changes to µ(α )

Dt

− uxx − uyy − uzz = 0.

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

14

E. Lashkarian et al. / Computers and Mathematics with Applications (

)

–

Table 15 Wi , Si

t-components of conservation laws for Eq. (15) 0 < α < 1

−ux , S1

1 −c1 ψi (x, y, z)Dα− (ux ) + t

α−1

−uy , S1

−c1 ψi (x, y, z)Dt

α−1

(uy ) +

p1 t

Γ (n−α ) p1 t

Γ (n−α )

−uz , S1

−c1 ψi (x, y, z)Dt

xuy − yux , S1

1 c1 ψi (x, y, z)Dα− (xuy − yux ) + t

α−1

(uz ) +

p1 t Γ (n−α ) p1 t

Γ (n−α )

xuz − zux , S1

c1 ψi (x, y, z)Dt

yuz − zuy , S1

1 c1 ψi (x, y, z)Dα− (yuz − zuy ) + t

− ut , S 2

−c1 ψi (x, y, z)(uxx + uyy + uzz ) +

[−tut −

α 2

(xux

+yuy + zuz ), S2 ] u, S2

(xuz − zux ) +

p1 t Γ (n−α ) p1 t Γ (n−α ) p1 t

Γ (n−α )

1 −c1 ψi (x, y, z)Dα− (tut + α2 (xux + yuy + zuz )) + t 1 c1 ψi (x, y, z)Dα− (u) + t

α−1

p1 t

Γ (n−α )

−(α + tut ), S3

−c1 ψi (x, y, z)Dt

[2 − xux −yuy − zuz , S3 ]

1 c1 ψi (x, y, z)Dα− (2 − xux − yuy − zuz , S3 ) + t

[−(

2t

ut + xux α +yuy + zuz ), S4 ]

p1 t

Γ (n−α )

(α + tut ) +

p1 t

Γ (n−α ) p1 t

Γ (n−α )

1 2t −c1 ψi (x, y, z)Dα− ( α ut + xux + yuy + zuz ) + t

p1 t

Γ (n−α )

Table 16 Wi , Si

t-components of conservation laws for Eq. (15) 1 < α < 2

−ux , S1

[ ] p t 1 2 −c1 ψi (x, y, z) tDα− (ux ) − Dα− (ux ) − Γ (n1−α ) t t [ α−1 ] t α−2 −c1 ψi (x, y, z) tDt (uy ) − Dt (uy ) − Γ (np1−α ) [ α−1 ] p1 t α−2 −c1 ψi (x, y, z) tDt (uz ) − Dt (uz ) − Γ (n−α) [ ] p t 1 2 c1 ψi (x, y, z) tDα− (xuy − yux ) − Dα− (xuy − yux ) − Γ (n1−α ) t t [ α−1 ] p1 t α−2 c1 ψi (x, y, z) tDt (xuz − zux ) − Dt (xuz − zux ) − Γ (n−α ) [ ] p t 1 2 c1 ψi (x, y, z) tDα− (yuz − zuy ) − Dα− (yuz − zuy ) − Γ (n1−α ) t t [ α ] t −c1 ψi (x, y, z) tDt (u) − Dtα−1 (u) − Γ (np1−α ) [ α−1 ] p1 t α−2 c1 ψi (x, y, z) tDt (Wi ) + Dt (Wi ) − Γ (n−α ) [ ] p t 1 2 c1 ψi (x, y, z) tDα− (u) − Dα− (u) − Γ (n1−α ) t t ] [ α−1 t α−2 −c1 ψi (x, y, z) tDt (α + tut ) − Dt (α + tut ) − Γ (np1−α ) [ α−1 ] p t 2 c1 ψi (x, y, z) tDt (Wi ) + Dα− (Wi ) − Γ (n1−α ) t

−uy , S1 −uz , S1 xuy − yux , S1 xuz − zux , S1 yuz − zuy , S1

−ut , S2 [−tut −

α 2

(xux

+yuy + zuz ), S2 ] u, S2

−(α + tut ), S3 [2 − xux −yuy − zuz , S3 ] 2t

ut + xux α +yuy + zuz ), S4 ]

[−(

1 2 c1 ψi (x, y, z) tDα− (Wi ) − Dα− (Wi ) − t t

[

]

p1 t

Γ (n−α )

Take

v1 = xy(az + b)(c1 (T − t)α + c2 ), v2 = a1 a2 (az + b)(c1 (T − t)α + c2 ), √ √ v3 = ex+y (d1 sin 2z + d2 cos 2z)(c1 (T − t)α + c2 ). C x , C y and C z are components of conservation laws which are shown in Tables 1–3 respectively. For Case 2: In this case, Eq. (16) gets the following form: µ(α )

Dt

( ) − eu u2x + uxx + u2y + uyy + u2z + uzz = 0.

x, y and z components of conservation laws for translated equation and vi 4–6.

1 ≤ i ≤ 3 are calculated in Tables

For Case 3: According to case 3, we have µ(α )

Dt

( ) ( ) + AuA−1 u2x + u2y + u2z − uA uxx + uyy + uzz = 0.

Similarly, the previous cases, the components of conservation laws are summarized in Tables 7–9. Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.

E. Lashkarian et al. / Computers and Mathematics with Applications (

)

–

15

• For Eq. (13)

Substituting φ (t) = c1 (T − t)α−1 into Tables 1–9, components of conservation laws for Eq. (13) will be obtained, similarly. • For Eq. (14) By replacing φ (t) = c1 t + c2 and φ (t) = c1 t 2 + c2 t instead of φ (t) = c1 (T − t)α + c2 into Tables 1–9, we can find components of conservation laws for Eq. (14). • For Eq. (15) In a similar way, for Eq. (15), x, y and z components of conservation laws are obtained by changing φ (t) with φ (t) = c1 for α ∈ (0, 1) and φ (t) = c1 t for α ∈ (1, 2). To obtain t-component of conservation laws, each of Eqs. (12)–(15) should be considered separately. For all cases, we consider the following general definitions. S 1 = f = g = h = 1 , f = g = h = e u , f = g = h = uA ,

}

{

S2 = {f = g = h = 1} ,

S 3 = f = g = h = eu ,

{

}

S 4 = f = g = h = uA ,

{

}

and ψi (x, y, z) = {ψ1 (x, y, z), ψ2 (x, y, z), ψ3 (x, y, z)} such that ψ1 , ψ2 and ψ3 are defined in (35). t-component of conservation laws Eq. (12) for different Wi is shown in Table 10. and t-component of conservation laws Eqs. (13), (14) and (15) for 0 < α < 1 , 1 < α < 2 are summarized in Tables 11–16, respectively. 5. Conclusion The main goal of this work was to introduce fractional conservation laws of (1) using fractional symmetries. Fractional calculus is an old mathematical topic, but the fractional symmetries and fractional conservation laws have been introduced, recently. We used formal Lagrangian and Ibragimov’s extension to find conservation laws of Eq. (1). Because there is not Lagrangian for Eq. (1). Therefore, a generalized fractional version of Ibragimov’s theorem which setup a relation between fractional symmetries and fractional conservation laws is presented in this paper. Finally we use fractional relationship for analysing the conservation laws of Eq. (1). References [1] G.W. Bluman, S.C. Anco, Symmetry and Integrating Methods for Differential Equation, in: Appl. Math. Sci., vol. 154, Springer-Verlag, New York, 2002. [2] G.W. Bluman, A.F. Cheviakov, S.C. Anco, Application of Symmetry Methods to Partial Differential Equations, Springer, New York, NY, USA, 2009. [3] R.K. Gazizov, A.A. Kasatkin, S. Yu Lukashchuk, Continuous transformation groups of fractional differential equations, Vestnik, USATU 9 (2007) 125–135 (in Russian). [4] R.K. Gazizov, A.A. Kasatkin, S. Yu Lukashchu, Symmetry properties of fractional diffusion equations, Phys. Scr. T136 (2009) 014016. [5] H.Z. Liu, Complete group classifications and symmetry reductions of the fractional fifth-order KdV types of equations, Stud. Appl. Math. 131 (2013) 317–330. [6] R. Sahadevan, T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg–deVries equations, J. Math. Anal. Appl. 393 (2012) 341–347 (399). [7] G.W. Wang, X.Q. Liu, Y.Y. Zhang, Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 2321–2326. [8] Q.W. Wang, T.Z. Xu, T. Feng, Lie symmetry analysis and explicit solutions of the time fractional fifth-Order KdV equation, PLoS One 9 (2) (2014) e88336. [9] Q.W. Wang, T.Z. Xu, Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis, Nonlinear Dynam. 76 (2014) 571–580. [10] S. Anco, G. Bluman, Direct construction of conservation laws, Phys. Rev. Lett. 78 (1997) 2873. [11] S.C. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations part II: General treatment, European J. Appl. Math. 13 (2002) 567–585. [12] G.W. Bluman, . Temuerchaolu, Comparing symmetries and conservation laws of nonlinear telegraph equations, J. Math. Phys. (2005) 073513. [13] N.H. Ibragimov, Nonlinear self-adjointness in constructing conservation laws, Arch. ALGA 7/8 (2011) 1–99. [14] N.H. Ibragimov, A new covservation theorem, J. Math. Anal. Appl. 333 (1) (2007) 311–328. [15] R.K. Gazizov, N.H. Ibragimov, S. Yu Lukashchuk, Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations, Commun. Nonlinear Sci. Numer. Simul. (2014). http://dx.doi.org/10.1016/j.cnsns.2014.11.010. [16] S. Yu Lukashchuk, Conservation laws for time-fractional subdiffusion and diffusion-wave equations, Nonlinear Dynam. (2015). http://dx.doi.org/10. 1007/s11071-015-1906-7. [17] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, Sons Inc., New York, 1993. [18] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, 1974, p. 234. [19] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Academic Press, New York, 1999.

Please cite this article in press as: E. Lashkarian, et al., Conservation laws of (3 + α )-dimensional time-fractional diffusion equation, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.10.001.