On Symmetry Groups and Conservation Laws for Space-Time Fractional Inhomogeneous Nonlinear Diffusion Equation

Vol. 84 (2019)

REPORTS ON MATHEMATICAL PHYSICS

No. 3

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL INHOMOGENEOUS NONLINEAR DIFFUSION EQUATION W EI F ENG Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China (e-mail: [email protected]) (Received April 8, 2019 — Revised September 6, 2019) In this paper, we consider a class of space-time fractional inhomogeneous nonlinear diffusion equations with Riemann–Liouville fractional derivative. Symmetry group method is applied to derive explicit solutions of the governing equation from the reduced fractional ordinary differential equations. Conservation laws admitted by the space-time fractional inhomogeneous nonlinear diffusion equations are obtained with the aid of the nonlinear self-adjointness method. Keywords: space-time fractional inhomogeneous nonlinear diffusion equation, symmetry groups, conservation laws, exact solutions.

1.

Introductions Symmetry groups and conservation laws are important tools for studying nonlinear differential equations. Symmetries are useful for finding group-invariant solutions and play a role in defining invariant Sobolev norms, which was evidenced by the number of research papers, books, and symbolic manipulation software related to it (see e.g. [1–9]). Conservation laws provide conserved quantities used in obtaining estimates for classical solutions, in defining suitable norms for weak solutions, also in studying bi-Hamiltonian structures and recursion operators [3, 10]. Over the past few decades there has been a lot of work related to symmetries for inhomogeneous nonlinear diffusion equation (INDE) x p ut = (x q un ux )x ,

p, q ∈ R,

(1.1)

which can describe diffusion processes in many branches of physics such as plasma physics, kinetic theory of gases, solid state, metallurgy and transport in porous medium [11–13]. In [14], the full classification of point symmetry groups of INDE was shown and the explicit group-invariant solutions were derived by symmetry reduction. The invariant subspaces and conditional Lie–B¨acklund symmetries of INDE were discussed in [15–17]. The author in [18] used the nonclassical symmetry group method to determine exact solutions of INDE. A series of works has concentrated on the discussions of exact solutions and symmetry reductions for the extended forms of INDE [19–24]. [375]

376

W. FENG

Fractional calculus is the differentiation and integration to an arbitrary order, which is originally due to L’Hˆopital and Leibniz in 1695 and was defined by Liouville, Riemann, Gr¨unwald, Letnikov and Caputo [25–27]. Fractional differential equations (FDEs) are viewed as generalizations of classical differential equations of integer order, which can describe many phenomena in fractals, random walk processes, control theory, signal processing, acoustics etc. [27–31]. One of the important features of FDEs is that they depend not only on the instant of time but also on the previous time. Since FDEs appear more and more frequently in real-world problems and engineering applications, they attract considerable interest and many significant theoretical developments have been made. To derive exact solutions of FDEs, several ad hoc methods have been successfully generalized to FDEs, such as Adomian decomposition method [32], homotopy analysis method [33, 34], variational iteration method [35], transform method [36, 37], Lie group method [38–40] and invariant subspace method [41]. The nonlinear self-adjoint method is also extended to FDEs to derive the conservation laws for FDEs [42–44]. The objective of the present paper is to investigate a space-time fractional INDE ∂tα u = x q−p un uxx + nx q−p un−1 u2x + qx q−p−1 un ∂xβ u

(1.2)

β

for u(t, x), where ∂tα and ∂x are, respectively, Riemann–Liouville fractional differential operators of order α with respect to the variable t and of order β with respect to the variable x. The Riemann–Liouville fractional derivative of order α > 0 for a function f (t) is defined as  m d f   m, α = m ∈ N; Dtα f (t) = dt 1 m Z t d   (t − ζ )m−α−1 f (ζ )dζ, m − 1 < α < m, m ∈ N, Ŵ(m − α) dt m 0 (1.3)

and the Riemann–Liouville fractional partial derivative of u(t, x) is defined by  m ∂ u   m, α = m ∈ N; α ∂t u = ∂t 1 m Z t ∂   (t − ζ )m−α−1 u(ζ, x)dζ, m − 1 < α < m, m ∈ N, Ŵ(m − α) ∂t m 0 (1.4) R ∞ z−1 −t where and whereafter Ŵ(z) = 0 t e dt, (Re(z) > 0) is the Euler Gamma function. Many results and properties of those fractional calculus operators are shown in [25, 29, 45]. Note that when α = β = 1, Eq. (1.2) reduces to INDE (1.1). For the case of β = 1, Eq. (1.2) becomes a time-fractional INDE, whose exact solutions were obtained by symmetry group method and invariant subspace method [46]. In [46], the authors also showed the admitted conservation laws of the time-fractional INDE through the nonlinear self-adjoint method.

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

377

The rest of this paper is organized as follows. Section 2 provides the point symmetry groups admitted by Eq. (1.2) and the reduced FODE given by reduction under each point symmetry group. Section 3 applies the symmetry reduction method to derive solutions for the reduced FODE in terms of elementary functions and lists the group-invariant solutions of Eq. (1.2). In Section 4, the nonlinear self-adjoint method is applied to construct conservation laws of equation (1.2). Finally, section 5 has some concluding remarks. 2.

Point symmetries Consider a one-parameter Lie group of infinitesimal transformations in (t, x, u), t¯ = t + ǫτ (t, x, u) + O(ǫ 2 ),

x¯ = x + ǫξ(t, x, u) + O(ǫ 2 ),

u¯ = u + ǫη(t, x, u) + O(ǫ 2 ),

(2.1)

with ǫ being the group parameter. The associated infinitesimal generator X = τ (t, x, u)∂/∂t + ξ(t, x, u)∂/∂x + η(t, x, u)∂/∂u

(2.2)

generates a point symmetry group of Eq. (1.2), provided that the invariance condition
pr(α,β,2) X ∂tα u − x q−p un uxx − nx q−p un−1 u2x − qx q−p−1 un ∂xβ u = 0 (2.3)

satisfied on the solutions of Eq. (1.2). In (2.3), pr(α,β,2) X is the (α, β, 2)-order prolongation of generator X defined by pr(α,β,2) X = X + ηα,t ∂∂tα u + ηβ,x ∂∂ β u + ηx ∂ux + ηxx ∂uxx , x

(2.4)

where the coefficients ηx and ηxx are the integer-order extended infinitesimals [3], ηα,t and ηβ,x are respectively the extended infinitesimals of orders α and β, given by ηα,t = Dtα η + ξ Dtα ux − Dtα (ξ ux ) + Dtα (uDt τ ) − Dtα+1 (τ u) + τ Dtα+1 u,

ηβ,x = Dtα η + ξ Dtα ux − Dtα (ξ ux ) + Dtα (uDt τ ) − Dtα+1 (τ u) + τ Dtα+1 u.

(2.5a) (2.5b)

Here Dt and Dx denote the total derivatives with respect to t and x, respectively. Using the generalized Leibniz rule [27, 45] ∞   X α α Dt (f (t)g(t)) = Dtm f (t)Dtα−m g(t), (2.6) m m=0

where and whereafter

  α Ŵ(α + 1) , = Ŵ(m + 1)Ŵ(α + 1 − m) m

and the generalized chain rule [25]

∞ Um d m f (z) d α f (g(t)) X = , dt α m! dzm m=0

(2.7)

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W. FENG

where

  m X
(α) k n Um = (−1) g k (t) g n−k (t) , k k=0

we can rewrite ηα,t and ηβ,x in (2.5) as η

α,t

=∂tα η + (ηu − αDt τ )∂tα u − u∂tα ηu +

η

=∂xβ η + (ηu − βDx ξ )∂xβ u − u∂xβ ηu +

where

∞   X β − Dxm τ Dxβ−m ut + µβ , m m=1

m

m=1

∞   X α − Dtm ξ Dtα−m ux + µα , m m=1 β,x

∞   X α

∂tm ηu −



  α m+1 D τ Dtα−m u m+1 t (2.8a)

∞   X β

m

m=1

∂xm ηu −



  β m+1 D ξ Dxβ−m u m+1 t (2.8b)

∞ X m X l X k−l     X α m k t m−α uh ∂tl uk−h ∂ m−l+k η µα = , (−1)h k!Ŵ(m + 1 − α) ∂t m−l ∂uk m l h m=2 l=2 k=2 h=0 ∞ X m X l X k−l     X β m k x m−β uh ∂xl uk−h ∂ m−l+k η µβ = . (−1)h m k!Ŵ(m + 1 − β) ∂x m−l ∂uk l h m=2 l=2 k=2 h=0

(2.9a)

(2.9b)

Additionally, to preserve the structure of Riemann–Liouville fractional derivative for space-time fractional differential equations, the constraint on infinitesimals τ (t, x, u)|t=0 = 0,

ξ(t, x, u)|x=0 = 0

(2.10)

is necessary to hold. Splitting Eq. (2.3) with respect to ux , ut , and their x-derivatives and Dtα−m u, α−m Dt ux leads to a system of determining equations ξt = ξu = τx = τu = 0, (β − 2)xξx + ξ = 0, nxη + (αxτt − 2xξx + (q − p)ξ ) u = 0, (2ηxu − ξxx )u + 2nηx = 0,

(2.11a) (2.11b) (2.11c) (2.11d)

xu2 ηuu + nxuηu + n(n − 1)xη + αnxuτt − 2nxuξx + n(q − p)uξ = 0, ∂tα η

u∂tα ηu

q−p n

qx q−p−1 un ∂xβ η

− −x u ηxx −     α m α ∂ ηu − D m+1 τ = 0, m t m+1 t

+

qx q−p−1 un+1 ∂xβ ηu

m = 1, 2, . . . ,

= 0,

(2.11e) (2.11f) (2.11g)

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

    β m β ∂ ηu − D m+1 ξ = 0, m x m+1 t

m = 1, 2, . . . .

379

(2.11h)

The above system (2.11) is easily solved to yield the following result: T HEOREM 1. The symmetry group of Eq. (1.2) is spanned by two finitedimensional vector fields scaling inversion

X1 =nt∂t − αu∂u ,

(2.12a)

2

X2 =2x ∂x + xu∂u ,

n = 1,

for

β = 3/2,

q − p = 7/2, (2.12b)

and one infinite-dimensional subalgebra Xf = f (t, x)∂u ,

for

n = 0,

(2.12c)

where f (t, x) satisfies Eq. (1.2). The corresponding transformation groups acting on solution u = f (t, x) are u =λ−α f (λ−n t, x),

(2.13a)

u =(1 + 2ǫx)1/2 f (t, x/(1 + 2ǫx)),

n = 1,

for

β = 3/2,

q − p = 7/2, (2.13b)

with group parameters −∞ < ǫ < ∞, 0 < λ < ∞. Note that the obtained point symmetries admitted by Eq. (1.2) in Theorem 1 are fewer than those for time-fractional INDE, which are shown in [46]. The symmetry group (2.12c) just reflects the linearity of Eq. (1.2). Hence, for finding the group-invariant solutions, the remaining symmetries (2.12a) and (2.12b) need to be considered. 2.1.

Scaling reduction

The symmetry group (2.12a) has invariants s = x,

v = t α/n u.

(2.14)

Eq. (1.2) thereby reduces to a FODE for v = v(s), given by s q−p v n−1 v ′′ + ns q−p v n−2 v ′2 + qs q−p−1 v n−1 Dsβ v = 2.2.

Ŵ(1 − α/n) . Ŵ(1 − α − α/n)

(2.15)

Inversion reduction

The symmetry group (2.12b) has invariants s = t,

v = x −1/2 u.

(2.16)

Eq. (1.2) thereby reduces to a FODE for v = v(s), given by Dsα v = 0,

(2.17)

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W. FENG

where Dsα is the Riemann–Liouville fractional differential operator of order α. It is straightforward to determine that Eq. (2.17) possesses a solution v = 0. Thus, Eq. (1.2) has a trivial solution u = 0. 3. Explicit group-invariant solutions Now the symmetry group method is applied to the reduced FODE (2.15) arising by the reduction of Eq. (1.2) under scaling symmetry (2.12a). The FODE (2.15) has the following symmetry structure. P ROPOSITION 1. The point symmetries admitted by FODE (2.15) are comprised by translation scaling inversion

3.1.

Y1 =∂s , for p = q = 0, Y2 =ns∂s +(p+2)v∂v , for Y3 =2s 2 ∂s +sv∂v ,

for

(3.1a) (3.1b)

q = 0,

q(n−1) = 0,

n=

2−β , β −1

q −p = 2+β. (3.1c)

Translation-symmetry quadratures for FODE (2.15)

The canonical coordinates of the translation symmetry (3.1a) are r = s,

V = v.

(3.2)

Thereby, FODE (2.15) gets transformed into V n−1 V ′′ + nV n−2 V ′2 = K, where and whereafter K=

(3.3)

Ŵ(1 − α/n) . Ŵ(1 − α − α/n)

By virtue of integrating factors 2(n + 2)V n+1 V ′ , we get quadratures Z 1 dV = r + C2 , for n = −2, √ 2 V 2K ln V + C1 Z 1 1 p dV = √ r + C2 , for n 6= −2, n+2 2KV 2−n + C1 V −2n

(3.4) (3.5)

where and whereafter C1 and C2 are arbitrary constants. These integrals (3.4) and (3.5) cannot be evaluated generally to obtain the explicit solutions for V (r). It is easy to see that the quadrature (3.4) provides an implicit solution for equation (3.3). Some explicit solutions in terms of elementary functions can be derived from (3.5) if we make a change of variables W = V m,

(3.6)

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

under which the quadrature (3.5) becomes Z m 1 p dW = √ s + C2 , n+2 2KW 2−n/m + C1 W 2−2(n+1)/m

for n 6= −2.

381

(3.7)

Taking into account the value of C1 , we consider the following two cases. Case 1. C1 = 0. In this case, the quadrature (3.7) can be integrated to get an explicit solution of Eq. (3.3)  2/n p V = (n/2) 2K/(n + 2)r + C2 , for n 6= −2, (3.8) which yields a solution to FODE (2.15), s !2/n Ŵ(1 − α/n) v= n s + C2 , 2(n + 2)Ŵ(1 − α − α/n)

for p = q = 0,

n 6= −2.

(3.9) Case 2. C1 6 = 0. In this case, for the sake of the expression of W in (3.7) being quadratic, we find that conditions 2 − n/m = 0, 1, 2; 2 − 2(n + 1)/m = 0, 1, 2

(3.10a) (3.10b)

are required. Solving the above conditions, we find three subcases. Subcase 2.1. n = −4/3, m = −2/3. In this case, from the quadrature (3.7), we obtain a solution of Eq. (3.3)
−3/2 V = C1 (r + C2 )2 − K/(3C1 ) , for n = −4/3, (3.11) which yields one solution to FODE (2.15),   Ŵ(1 + 3α/4) −3/2 2 v = C1 (s + C2 ) − , 3C1 Ŵ(1 − α/4)

for p = q = 0,

n = −4/3.

(3.12) Subcase 2.2. n = −1, m = −1/2. In this case, from the quadrature (3.7), we obtain three solutions of Eq. (3.3),  p −2 p 2K/C1 sinh −( C1 /2)r +C2 , for n = −1, K > 0, C1 > 0, V= (3.13)

p  p −2 V= −2K/C1 sin −( −C1 /2)r +C2 ,

for n = −1,

K > 0,

 p −2 p V= −2K/C1 cosh −( C1 /2)r +C2 ,

for n = −1,

K < 0,

C1 < 0, (3.14) C1 > 0, (3.15)

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W. FENG

which yields three solutions to FODE (2.15): p  p −2 v= 2Ŵ(1 + α)/C1 sinh −( C1 /2)s + C2 ,

for p = q = 0, n = −1, Ŵ(1 + α) > 0, p  p −2 v= −2Ŵ(1 + α)/C1 sin −( −C1 /2)s + C2 , for p = q = 0, n = −1, Ŵ(1 + α) > 0,  p −2 p −2Ŵ(1 + α)/C1 cosh −( C1 /2)s + C2 , v= for p = q = 0,

Ŵ(1 + α) < 0,

n = −1,

C1 > 0,

(3.16)

C1 < 0,

(3.17)

C1 > 0.

(3.18)

Subcase 2.3. n = m = −1. In this case, from the quadrature (3.7), we obtain two solutions of Eq. (3.3) V =

V =

C1
, √ K cosh(− C1 r + C2 ) − 1

−C1
, √ K sin(− −C1 r + C2 ) + 1

for n = −1,

K > 0,

C1 > 0, (3.19)

for n = −1,

K > 0,

C1 < 0, (3.20)

which yields two solutions to FODE (2.15) v=

v=

3.2.

C1
, √ Ŵ(1+α) cosh(− C1 s +C2 )−1

−C1
, √ Ŵ(1+α) sin(− −C1 s +C2 )+1

for n = −1,

Ŵ(1+α) > 0,

C1 > 0, (3.21)

for n = −1,

Ŵ(1+α) > 0,

C1 < 0. (3.22)

Scaling-symmetry quadratures for FODE (2.15)

The canonical coordinates of the scaling symmetry (3.1b) are r = (ln s)/n,

V = s −(p+2)/n v.

(3.23)

Thereby, FODE (2.15) gets transformed into V n−1 V ′′ + nV n−2 V ′2 + (2np + 3n + 2p + 4)V n−1 V ′ + (p + 2)(np + n + p + 2)V n = n2 K. (3.24) It is readily seen that Eq. (3.24) possesses an invariant solution, namely,
−1/n V = (p + 2)(np + n + p + 2)/(n2 K) . (3.25)

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

383

After changing variables (3.23), we arrive at a solution to FODE (2.15), 1/n  n2 Ŵ(1 − α/n) s (p+2)/n , for q = 0. (3.26) v= (p + 2)(np + n + p + 2)Ŵ(1 − α − α/n) Next, we can obtain several solutions for V (r) in the case of 2np+3n+2p+4 = 0 under which ODE (3.24) becomes V n−1 V ′′ + nV n−2 V ′2 +

n2 V n = n2 K. 4(n + 1)

(3.27)

The integrating factor 2V n+1 V ′ reduces Eq. (3.27) to quadratures Z 1 p dV = r + C2 , for n = −2, 2 −2 V V + 8K ln V + C1 Z 1 q dV = r + C2 , for n 6= −2. 2 2 n 2 + 2Kn V 2−n + C V −2n V 1 2 n+2 4(n+1)

(3.28) (3.29)

Note that (3.28) is an implicit solution of Eq. (3.27). Some explicit solutions in terms of elementary functions can be derived from (3.29) if we change the variable (3.6) to make the expression Z 1 q dW = r + C2 , for n 6= −2, 2 2 n 2 + 2Kn W 2−n/m + C W 2−2/m−2n/m W 1 2 n+2 4(n+1) (3.30) into a quadratic polynomial in W . The required conditions are 2 − n/m = 0, 1, 2, if C1 = 0; 2 − n/m = 0, 1, 2 and 2 − 2(n + 1)/m = 0, 1, 2,

The above conditions yield the following three cases.

if

C1 6= 0.

(3.31a) (3.31b)

Case 1. C1 = 0 and n = 2m. In this case, the quadrature (3.30) leads to two solutions of Eq. (3.24)  1/n  n2 8(n + 1)2 K 2 , for K(n + 2) > 0, (3.32) V = sinh r + C2 (n + 2) 4|n + 1|   1/n 8(n + 1)2 K n2 2 V = − cosh r + C2 , for K(n + 2) < 0. (3.33) (n + 2) 4|n + 1| With the change of variables (3.23), we arrive at two solutions to FODE (2.15),  1/n   2/n 8(n + 1)2 Ŵ(1 − α/n) n n/(2(n+1)) v= s sinh ln s + C2 , (n + 2)Ŵ(1 − α − α/n) 4|n + 1| for q = 0, (n + 2)Ŵ(1 − α/n)Ŵ(1 − α − α/n) > 0, (3.34)

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W. FENG

1/n   2/n n 8(n + 1)2 Ŵ(1 − α/n) n/(2(n+1)) s cosh ln s + C2 , v= − (n + 2)Ŵ(1 − α − α/n) 4|n + 1| for q = 0, (n + 2)Ŵ(1 − α/n)Ŵ(1 − α − α/n) < 0. (3.35) 

Case 2. C1 = 0 and n = m. In this case, the quadrature (3.30) leads to one solution of Eq. (3.24),     1/n 4(n + 1)2 K n2 V = cosh r + C2 − 1 . (3.36) (n + 2) 2|n + 1|

With the change of variables (3.23), we arrive at a solution to FODE (2.15), 1/n  1/n    n 4(n + 1)2 Ŵ(1 − α/n) n/(2(n+1)) ln s + C2 − 1 , s cosh v= (n + 2)Ŵ(1 − α − α/n) 2|n + 1| for q = 0. (3.37)

Case 3. C1 6 = 0, n = −4/3 and m = −2/3. In this case, the quadrature (3.30) leads to three solutions of Eq. (3.24) −3/2  q 1 2 , for 256K − 3C12 > 0, (256/3)K − C1 sinh(C2 − 4r/3) − C1 /8 V = 8 (3.38)  q −3/2 1 V = , for 256K − 3C12 < 0, C12 − (256/3)K cosh(C2 − 4r/3) − C1 /8 8 (3.39)  −3/2 1p . (3.40) V = C2 exp(−4r/3) ± 256K/3 8 With the change of variables (3.23), we arrive at three solutions to FODE (2.15) s !−3/2 −3/2 1 256Ŵ(1 + 3α/4) 2 v=s , − C1 sinh(ln s + C2 ) − C1 /8 8 3Ŵ(1 − α/4) for q = 0, v = s −3/2

n = −4/3,

256Ŵ(1 + 3α/4) − 3C12 Ŵ(1 − α/4) > 0, (3.41) !−3/2

s 1 256Ŵ(1 + 3α/4) C12 − cosh(ln s + C2 ) − C1 /8 8 3Ŵ(1 − α/4)

,

256Ŵ(1 + 3α/4) − 3C12 Ŵ(1 − α/4) < 0, (3.42)  −3/2 1p v = s −3/2 C2 s ± 256Ŵ(1 + 3α/4)/(3Ŵ(1 − α/4)) , 8 for q = 0, n = −4/3. (3.43) for q = 0,

n = −4/3,

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

3.3.

385

Inversion-symmetry quadratures for FODE (2.15)

The canonical coordinates of the inversion symmetry (3.1c) are r = −1/s,

V = s 1−β v.

(3.44)

When n = 1, by the generalized Leibniz rule (2.6), we can transform FODE (2.15) into an infinite order differential equation for V (r), which is not trackable to solve explicitly. When q = 0, FODE (2.15) then gets transformed into V

3−2β β−1

V ′′ −

4−3β

β−2 V β−1 β−1

V ′2 = K,

(3.45)

which has integrating factor 2βV 1/(β−1) V ′ , so that Eq. (3.45) is reduced to a quadrature Z 1 1 p (3.46) dV = √ r + C2 . β 2(β − 1)V (3β−4)/(β−1) + C1 V 2(β−2)/(β−1)

Analogous to the earlier analysis, we find that quadrature (3.46) can be evaluated explicitly only for C1 = 0. Thus Eq. (3.45) has a solution  2(β−1)/(2−β) p V = (2 − β) K/(2β(β − 1))r + C2 , (3.47) which leads to a solution to FODE (2.15) s !2(β−1)/(2−β) Ŵ(1 − α/n) 1 v =s β−1 (2 − β) + C2 , 2β(β − 1)Ŵ(1 − α − α/n) s for q = 0, 3.4.

p = −(2 + β), n = (2 − β)/(β − 1).

(3.48)

Solutions to Eq. (1.2)

Here we will write out all solutions of Eq. (1.2) arising from Sections 3.1, 3.2 and 3.3, via the invariants (2.14). T HEOREM 2. The space-time fractional inhomogeneous nonlinear diffusion equation (1.2) has the following nontrivial explicit group-invariant solutions 1/n  n2 Ŵ(1−α/n) t −α/n (x +c1 )2/n , for p = q = 0, n 6= −2, u= 2(n+2)Ŵ(1−α−α/n) (3.49)  −3/2 Ŵ(1+3α/4) u = t 3α/4 c1 (x +c2 )2 − , for p = q = 0, n = −4/3, 3c1 Ŵ(1−α/4) (3.50) u=

2c12 t α sinh−2 (c1 x +c2 ) , Ŵ(1+α) for p = q = 0, n = −1,

Ŵ(1+α) > 0,

c1 < 0,

(3.51)

386 u=

u=

W. FENG

2c12 t α sin−2 (c1 x +c2 ) , Ŵ(1+α) for p = q = 0, n = −1,

Ŵ(1+α) > 0,

c1 < 0,

(3.52)

Ŵ(1+α) < 0,

c1 < 0,

(3.53)

−2c12

t α cosh−2 (c1 x +c2 ) , Ŵ(1+α) for p = q = 0, n = −1,

c12 t α , for p = q = 0, n = −1, c1 < 0, (3.54) Ŵ(1+α) (cosh(c1 x +c2 )−1) c12 t α , u= − Ŵ(1+α) (1−sin(c1 x +c2 )) for p = q = 0, n = −1, Ŵ(1+α) > 0, c1 > 0, (3.55)  1/n 2 n Ŵ(1−α/n) u= x (p+2)/n t −α/n , for q = 0, (3.56) (p+2)(np+n+p+2)Ŵ(1−α−α/n) 2/n 1/n    n 8(n+1)2 Ŵ(1−α/n) n/(2(n+1)) −α/n , x t sinh ln x +c u= 1 (n+2)Ŵ(1−α−α/n) 4|n+1| for q = 0, 2np+3n+2p+4 = 0, (n+2)Ŵ(1−α/n)Ŵ(1−α−α/n) > 0, (3.57)      1/n 2/n 8(n+1)2 Ŵ(1−α/n) n u= − , x n/(2(n+1)) t −α/n cosh ln x +c1 (n+2)Ŵ(1−α−α/n) 4|n+1| for q = 0, 2np+3n+2p+4 = 0, (n+2)Ŵ(1−α/n)Ŵ(1−α−α/n) < 0, (3.58)       1/n 1/n 4(n+1)2 Ŵ(1−α/n) n u= x n/(2(n+1)) t −α/n cosh ln x +c1 −1 , (n+2)Ŵ(1−α−α/n) 2|n+1| for q = 0, 2np+3n+2p+4 = 0, (3.59) s !−3/2 √ 256Ŵ(1+3α/4) 2 , u = 16 2t 3α/4 x −3/2 −c1 sinh(ln x +c2 )−c1 3Ŵ(1−α/4) u=

for p = q = 0, √ u = 16 2t 3α/4 x −3/2

256Ŵ(1+3α/4)−3c12 Ŵ(1−α/4) > 0, (3.60) s !−3/2 2 256Ŵ(1+3α/4) c1 − cosh(ln x +c2 )−c1 , 3Ŵ(1−α/4) n = −4/3,

for p = q = 0, n = −4/3, 256Ŵ(1+3α/4)−3c12 Ŵ(1−α/4) < 0, (3.61)  −3/2 p u = t 3α/4 x −3/2 c1 x ±2 Ŵ(1+3α/4)/(3Ŵ(1−α/4)) , for

p = q = 0,

n = −4/3,

(3.62)

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

u=



387

(β−1)/(2−β) (2−β)2 Ŵ(1−α/n) t −α/n x β(β−1)/(2−β) (1+c1 x)2(β−1)/(2−β) , 2β(β −1)Ŵ(1−α−α/n) for q = 0, p = −(2+β), n = (2−β)/(β −1), (3.63)

where c1 and c2 are arbitrary constants.

Next, we apply the full group of point symmetries admitted by Eq. (1.2) to each solutions u = f (t, x) in Theorem 2. The symmetry (2.12a) changes only the constants shown in each solution, nevertheless, symmetry (2.12b) can produce additional new solutions. Taking into account the values of parameters n, p, q and β, we note that transformation (2.13b) can just be applied to solutions (3.56) and (3.63). The results are summarized as follows. T HEOREM 3. When n = 1, β = 3/2, q − p = 7/2, Eq. (1.2) has two additional explicit solutions, given by 3Ŵ(1 − α) −α −3/2 t x (1 + c1 x)2 , u= 8Ŵ(1 − 2α) for q = 0, n = 1, p = −7/2, β = 3/2, (3.64) Ŵ(1 − α) −α −3/2 t x (1 + c2 x)−1/2 (1 + c1 x)2 , u= 6Ŵ(1 − 2α) for q = 0, n = 1, p = −7/2, β = 3/2, (3.65)

where c1 and c2 are arbitrary constants.

Out of the 17 solutions for Eq. (1.2) we have obtained in Theorems 2 and 3, solution (3.49) was derived in recent work [46]. To the best knowledge of the author, the remaining 16 solutions are new. 4.

Conservation laws In this section, the nonlinear self-adjoint method is applied to construct the admitted conservation laws of Eq. (1.2). To begin with, we rewrite Eq. (1.2) into . F = ∂tα u − x q−p un uxx − nx q−p un−1 u2x − qx q−p−1 un ∂xβ u = 0. (4.1) A conservation law of Eq. (4.1) is given by a space-time divergence Dt T (t, x, u, . . .) + Dx X(t, x, u, . . .) = 0

(4.2)

that holds for all solutions of Eq. (4.1), where T and X are, respectively, the conserved density and flux. A formal Lagrangian of Eq. (4.1) is introduced as
L = v(t, x) ∂tα u − x q−p un uxx − nx q−p un−1 u2x − qx q−p−1 un ∂xβ u , (4.3)

where v(t, x) is a new introduced dependent variable. The adjoint equation of (4.1) is defined as the Euler–Lagrange equation δL = 0, (4.4) δu

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W. FENG

where δ/δu is the Euler–Lagrange operator with respect to variable u, given by ∂ ∂ ∂ ∂ ∂ δ = + (Dtα )∗ + (Dxβ )∗ − Dx + Dx2 . β δu ∂u ∂(∂tα u) ∂u ∂u x xx ∂(∂x u)

(4.5)

β

In operator (4.5), (Dtα )∗ and (Dx )∗ are the adjoint operators of Riemann–Liouville β fractional differential operators Dtα , Dx defined by (Dtα )∗ = (−1)m t Iam−α (Dtm ) = Ct Daα ,

k−β

(Dxα )∗ = (−1)k x Ib

β

(Dxk ) = Cx Db ,

(4.6)

k−β

where t Iam−α and x Ib are the right-sided fractional integral operator of order m − α β and k − β, Ct Dbα and Cx Db are the right-sided Caputo fractional derivative operators of order α and β. The right-sided fractional integral operator of order α and the right-sided Caputo fractional derivative operators of order α are, respectively, given by Z a 1 α (ζ − t)m−α−1 f (ζ )dζ, (4.7) t Ib f (t) = Ŵ(m − α) t Z b 1 m−α C α m (t − ζ )m−α−1 f (m) (ζ )dζ, (4.8) (Dt f (t)) = t Db f (t) = t Ib Ŵ(m − α) t where m = [α] + 1. A straightforward calculation in adjoint Eq. (4.4) yields equation (Dtα )∗ v − q(Dxβ )∗ (x q−p−1 un v) − nqx q−p−1 un−1 (∂xβ u)v − x q−p un vxx − 2(q − p)x q−p−1 un vx − (q − p)(q − p − 1)x q−p−2 un v = 0.

(4.9)

Eq. (4.1) is nonlinear self-adjoint if the adjoint equation (4.4) is satisfied for all solutions of Eq. (4.1) upon a substitution v = φ(t, x, u),

(4.10)

where φ(t, x, u) 6 = 0. It implies that condition δ L = λF δu (4.1)

(4.11)

has to be preserved, where λ is an unknown coefficient. It is easy to simplify Eq. (4.11) with (4.10) to equation C α t Db φ

β

− q Cx Db (x q−p−1 un φ) − nqx q−p−1 un−1 (∂xβ u)φ
− x q−p un φu uxx + φuu u2x + 2φxu ux + φxx

+ 2(p − q)x q−p−1 un (φx + ux φu ) − (q − p)(q − p − 1)x q−p−2 un φ
= λ ∂tα u − x q−p un uxx − nx q−p un−1 u2x − qx q−p−1 un ∂xβ u .

(4.12)

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

389

Eq. (4.12) can be split with respect to the derivatives of u, leading to an overdetermined system of PDEs on φ, φu − λ = 0, φuu − λn/u = 0, (q − p)φu + xφxu = 0, C α t Db φ

(4.13a) (4.13b) (4.13c)


β − q Cx Db x q−p−1 un φ − x q−p un φxx + 2(p − q)x q−p−1 un φx

− (q − p)(q − p − 1)x q−p−2 un φ − λ∂tα u + q(λu − nφ)x q−p−1 un−1 ∂xβ u = 0, (4.13d) whose solutions are given by φ = x 1−q+p , for λ = 0, n = 0; φ = u, for n = 0, q = p.

(4.14a) (4.14b)

Thus the conserved quantities T and X arising from point symmetry in (2.12) are expressed explicitly by      m−1 X ∂L ∂L m m k α−1−k k − (−1) J1 W, Dt , (4.15) T = (−1) Dt (W )Dt ∂(Dtα u) ∂(Dtα u) k=0      l−1 X ∂L ∂L k β−1−k k l l X= (−1) Dx (W )Dx − (−1) J2 W, Dx , (4.16) β β ∂(Dx u) ∂(Dx u) k=0

where m = [α]+1, l = [β]+1, W = η −ξ ux −τ ut is the corresponding characteristic function of symmetry X = ξ ∂x + τ ∂t + η∂u , the integrals J1 (f, g) and J2 (f, g) are defined by Z tZ a 1 (µ − s)m−α−1 f (s, x)g(µ, x)dµds, (4.17a) J1 (f, g) = Ŵ(m − α) 0 t Z xZ b 1 J2 (f, g) = (µ − s)l−β−1 f (t, s)g(t, µ)dµds. (4.17b) Ŵ(l − β) 0 x Since the solutions of φ, shown in (4.14), exist only for n = 0, conservation laws of Eq. (4.1) can be derived only by point symmetry (2.12a) with n = 0, whose characteristic function is W = u. 4.1. φ = x p

In this case, the corresponding conserved densities derived from (4.15) are given

by Case 0 < α < 1, Case 1 < α < 2,

T1 = x 1−q+p It1−α (u).

(4.18)

T1 = x 1−q+p Dtα−1 (u).

(4.19)

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From (4.16), the corresponding fluxes are of the following forms Case 0 < β < 1, Case 1 < β < 2,

X1 = −qIx1−β (u).

(4.20)

X1 = −qDxβ−1 (u).

(4.21)

4.2. φ = u

In this case, the corresponding conserved densities derived from (4.15) are given

by Case 0 < α < 1, T2 =

uIt1−α (u)

Case 1 < α < 2, T2 =

1 + Ŵ(1 − α)

uDtα−1 (u) − ut Dtα−2 u −

Z tZ 0

1 Ŵ(2 − α)

a

t

(µ − s)−α u(s, x)ut (µ, x)dµds.

Z tZ 0

t

(4.22)

a

(µ − s)1−α u(s, x)utt (µ, x)dµds. (4.23)

From (4.16), the corresponding fluxes are of the following forms: Case 0 < β < 1, X2 = −qx −1 uIx1−β (u) Z xZ b q µ−2 (µ − s)−β u(t, s) (µux (t, µ) − u(t, µ)) dµds. − Ŵ(1 − β) 0 x

(4.24)

Case 1 < β < 2,

X2 = − qx −1 uDxβ−1 (u) + qx −2 (xux − u)
Z xZ b u(t, s) µ3 uxx (t, µ) − 2µ2 ux (t, µ) + 3u(t, µ) q + dµds. Ŵ(2 − β) 0 x µ4 (µ − s)β−1 (4.25) 5. Conclusions In this paper, we have fully classified all point symmetries and conservation laws of the space-time fractional INDE (1.2). The symmetry reduction method is applied to reduce equation (1.2) into a FODE, whose explicit solutions are derived in terms of elementary functions, so that 17 explicit group-invariant solutions are written out. Moreover, the nonlinear adjointness method is used to determine the conserved quantities of Eq. (1.2) in light of the obtained point symmetry groups. For future work, other symmetry-related methods, in particular, the method of group-foliation [6], will be generalized to FDEs for finding more new exact solutions.

ON SYMMETRY GROUPS AND CONSERVATION LAWS FOR SPACE-TIME FRACTIONAL. . .

391

Acknowledgements This project is supported by the Natural Science Foundation of Zhejiang Province (Nos. LY18A010033, LY17A010024) and the National Natural Science Foundation of China (Nos. 11401529). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

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