{'fraction_non_alphanumeric': 0.0631978999333934, 'fraction_numerical': 0.021705912314383104, 'mean_word_length': 3.74247491638796, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 32, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Symmetry structure of multi-dimensional time-fractional partial differential equations

In this paper, we concentrate on the Lie symmetry structure of a system of multi-dimensional time-fractional partial differential equations (PDEs). Specifically, we first give an explicit prolongation formula of the Lie infinitesimal generator involving time-fractional partial derivative in multi-dimensional case, and then show that the admitted infinitesimal generator has an elegant structure. Furthermore, we present two simple conditions to determine the infinitesimal generator where one is a system of linear time-fractional PDEs, the other is a system of integer-order PDEs and plays the dominant role in the whole procedure. We study three time-fractional PDEs to illustrate the efficiency of the results.


Introduction
The theory of fractional calculus goes back to the Leibniz's letter to L'Hospital [1]. With the rapid development and extensive applications in the last several decades, nowadays fractional PDEs take an important position in describing the phenomena in the fields such as physics, biology and chemistry, where under certain circumstances integer-order PDEs cannot work [2][3][4]. Consequently, considerable attentions have been paid to study fractional PDEs and thus a number of effective techniques and methods have been proposed [5][6][7][8][9][10].
Lie group theory provides widely applicable techniques to study integer-order PDEs, for example, constructing similarity solutions and linearization mappings, investigating integrability, analyzing stability and global behavior of solutions, finding potential variables and nonlocally related systems, etc [11][12][13]. Concerning Lie group theory of fractional PDEs, the fractional derivatives greatly affect the admitted Lie symmetry and related symmetry properties, such as symmetry classification, symmetry reductions and exact solutions [14][15][16][17]. Buckwar and Luchko first established the invariance of a fractional diffusion equation under scaling transformations [18]. Subsequently, application of Lie group theory to fractional PDEs becomes one of the most hottest topics and numerous fractional PDEs are studied from the point of view of Lie symmetry where many significant results are found such as similarity solutions, non-linearization, etc, we refer to [19][20][21][22][23][24][25][26][27][28][29] and references therein.
However, the current main studied objects in the literatures are (1 + 1)-dimensional single time-fractional PDEs, while the extension of Lie group theory and method to the system of multi-dimensional time-fractional PDEs is limited and not many results are found. For example, as the authors known, two (1+N )-dimensional scalar fractional PDEs are considered respectively in [16,24]. Some integrable coupled time-fractional PDEs are studied in [25][26][27]. A generalization of Lie symmetry theory to space-time fractional PDEs is studied in [28] and references therein.
In [29], we study a (1 + 2)-dimensional time-fractional biological population model in view of Lie symmetry theory.
In the procedure of applying Lie symmetry method to either integer-order or fractional PDEs, the prerequisite is the acquirement of affluent symmetry information, thus it is worthy of further studying efficient techniques and simple conditions to determine such Lie symmetries. In particular, knowing the symmetry structure in advance will greatly facilitate the computations [11,12]. In [17], we show that the infinitesimal generators of Lie symmetries of a scalar time-fractional PDE have a simple and unified expression and are completely determined by two elegant conditions which facilitates the whole procedure to be performed with the known solvers on the computer.
In order to facilitate the analysis, according to whether or not the terms in right side independent of u and its x-derivatives, we rearrange system (1) as the following form where F = (F 1 , F 2 , . . . , F q ) is a vector function which collect all terms in E containing at least one u or its x-derivatives, the remainder terms are collected in the vector function H = (H 1 , H 2 , . . . , H q ), which means that all term in H are only functions of t and x. Then for system (2), we start with the prolongation formula of the infinitesimal generator containing fractional partial derivatives in multi-dimensional case and then show the infinitesimal generator of the admitted Lie symmetry to be taken a simple and unified form. Moreover, we present two conditions to determine the Lie symmetries where one is a system of linear time-fractional PDEs and the other is a system of integer-order PDEs which even completely determine the Lie symmetries for some particular time-fractional PDEs. The remainder of the article is outlined as follows: In Section 2, after recall the related definitions and properties, we first give the prolongation formula and the general form of the infinitesimal generator of system (2), and then present two simple determining conditions for the generator. In Section 3, we use the results to study three types of time-fractional PDEs. The last section contains conclusion and future work.

Preliminaries
We first review the definition and some related properties of Riemann-Liouville fractional partial derivative [2,3].
where the gamma function is Γ(z) = ∞ 0 e −z t z−1 dt. By Definition 2.1, the Riemann-Liouville fractional partial derivative of a power function t γ is given by Then and only if f = 0, which is different with the usual integer-order derivatives. The Riemann-Liouville fractional partial derivative of the sum and product of two functions u = u(t, x) and v = v(t, x) is formulated as follows [2,3] where the first equality indicates the linear property and the second one is the Leibniz formula, hereinafter, and a, b are two constants, u, v and all their derivatives are continuous in [0, t]. In the equality ∂ n t (∂ α t u) = ∂ n+α t u = ∂ α t (∂ n t u), the former holds identically while the latter works if and only if u (k) = 0 with k ∈ Z + . In particular for v = 1, by Leibniz formula the Riemann-Liouville fractional partial derivative can be expressed as The generalized chain rule for fractional derivative of a composite function f (g(t)) is given by [2] where u i s = ∂u s /∂x i and u θ, i s = ∂u θ s /∂x i , D θ = D θ 1 . . . D θp is the |θ|-order total derivative operator with respect to x and D θ i = D θ i x i . The symbols D t and D x i denote the total derivatives with respect to t and x i respectively, and D 0 t (u) = u, D n+1 t = D t (D n t ) and similar for D x i . Now we give an explicit expression of η α s in the case of p dependent variables and q independent variables.

Lemma 2.2
The α-th (0 < α < 1) extended infinitesimal related to Riemann-Liouville timefractional partial derivative η α s is expressed by where D α t is the total fractional derivative with respect to t. Proof. Extending the transformation group (7) to fractional derivative ∂ α t u s as well as using the operational rules (4) and (5) of fractional partial derivative, we find where η (t, n) s denotes the nth-order extension of η s with respect to t and equals Then substituting it into (13), we have where property (5) is repeatedly used. The proof ends.

Lemma 2.3
An explicit expression of time-fractional total derivative D α t is given by where with the indexes satisfying k 1 + · · · + k q = k ≥ 2 and m 0 + m 1 + m 2 + · · · + m q = n.
Proof. By means of generalized Leibniz rule and generalized chain rule for fractional derivative of a composite function, we obtain where k 1 + · · · + k q = k and m 0 + m 1 + m 2 + · · · + m q = n. Observe that where a j are nonnegative integers and the second sum works on a 1 + · · · + a m i = a ≤ k, a 1 + 2a 2 + · · · + m i a m i = m i . Moreover, each term in (17) is homogeneous in u i and its t-derivatives and the total degree is k i . Thus we isolate the terms in (16) that are linear in u and its derivatives, which is equivalent to search for the terms with the indexes satisfying k = k 1 + · · · + k q ≤ 1. The solutions of this inequality is divided into q + 1 cases where the first one is k i = 0 for each i ∈ N q , the other q cases are k i = 1 for a certain i ∈ N q and k j = 0 for each j( = i) ∈ N q , where N q denotes the set of positive integer not greater than q. Case I. For each i ∈ N q , k i = 0 means k = 0 and requires m i = 0, m 0 = n for all i ∈ N q , otherwise ∂ m i (u k i −r i i )/∂t m i = 0. Thus this case corresponds to the term P I = ∂ α η s /∂t α . Case II. Consider the last q cases. We fix i and analyse k i = 1 and all k j = 0 with j( = i) ∈ N q . Then similar as Case I, for j = i, k j = 0 requires m j = 0 and then /∂t m i = 0 and thus no nonzero terms exist. For the latter case, the terms with the given i are collected as where property (5) and the generalized Leibniz rule are used in the third and last steps respectively.
Therefore, separating the term P I in Case I and the terms q i=1 P i II in Case II from D α t (η s ), we obtain the explicit expression of D α t (η s ) in Lemma 2.3. It completes the proof. By Lemmas 2.2 and 2.3, we give an explicit expression of η α s .
Theorem 2.4 An explicit expression of η α s is given by where µ s is given by (15).
. Using integration by parts and the generalized Leibniz rule in (4), we have Then inserting (14) and (19) into (12) yields (18) x, u (k) ). It completes the proof. In multi-dimensional case, by formula (18) we give an explicit expression of the extended infinitesimal of fractional derivative. It includes the previous prolonged formulas in [24][25][26][27][28] as special cases and also revises the inaccurate expression of µ s . In particular, for the (1 + 1)dimensional case, i.e. x 1 = x, u 1 = u, η s = η, then α-order prolongation formula (18) becomes [17] After obtaining the explicit expression of Pr (α, k) X in (10), the procedure for searching the infinitesimals τ, ξ i and η s in (8) is similar as the one of integer-order PDEs. Therefore, substituting the above formulas into condition (9) and then annihilating to zero first the coefficients of different power of time-fractional partial derivatives of u and followed by the coefficients of integer-order x-derivatives of u, we obtain an over-determined linear system for τ, ξ i and η s which includes integer-order and fractional partial derivative operations. Then solving the system together with the condition τ (t, x, u)| t=0 = 0 gives the generator (8).

Symmetry structure
It is well-known that finding solutions of the symmetry determining equations is a rather laborconsuming task for integer-order PDEs [11][12][13], not to mention herein the system involving fractional partial derivative operations. Thus in this section, we analyze the symmetry structure of system (2) and give a simple and unified expression of the infinitesimal generator (8), which will simplify the symmetry determining equations.
Lemma 2.5 Let µ s be given in (15). A necessary and sufficient condition for µ s = 0 is that Proof. By the expression of µ s in (15), if η s is linear in u i then ∂ k η s /∂u k i = 0 with k ≥ 2 and thus µ s = 0. The sufficiency holds. Next we prove necessity, i.e. prove ∂ 2 η s /∂u 2 i = ∂ 2 η s /∂u i ∂u l = 0 (i = l). By the proof of Lemma 2.3, we find that all terms in µ s are nonlinear in u i and their t-derivatives.
Consider the term (∂ t u i ) 2 which occurs uniquely for m i = 2, i = 1, . . . , q. Thus we separate the case m i = 2 from µ s and rewrite µ s as the following form For the coefficient of where property (5) is used. By the uniqueness of (∂ t u i ) 2 , annihilating its coefficient to zero yields . Then further split the case k = k i = 2 from µ s , which implies for j = i, k j = m j = 0, thus we rewrite µ s as the form where we adopt the technique in (21). By the uniqueness of nonlinear terms ∂ s t u i ∂ m i −s t u i with m i = 2 in µ s , we obtain the coefficients C(x, u) = 0 and then ∂ 2 η s /∂u 2 i = 0. Thus in µ s , the terms ∂ k η s /∂u k i (k ≥ 2) or their derivatives vanish while the remaining terms take the form ∂ k η s /∂u k 1 1 ∂u k 2 2 . . . ∂u kq q with nonnegative integer k i ≤ 1 , i = 1, . . . , q.
Next consider the cross terms ∂ 2 η s /∂u i ∂u l (i = l). It corresponds to k = 2, k i = k l = 1 and k d = 0 with d( = i, l) = 1, . . . , q, thus m i ≥ 1, m l ≥ 1 and m d = 0, otherwise µ s vanishes identically. Then one has which are unique in µ s . First consider m i = m l = 1, i.e. the term ∂ t u i ∂ t u l . Then m 0 = n − 2 and the coefficient of where the technique adopted in (21) is used. Vanishing it yields ∂ 2 η s /∂u i ∂u l = B(x, u) t α−3 with an undetermined function B(x, u). Secondly consider m i = 1, m l = 2 which corresponds to the term ∂ t u i ∂ 2 t u l . Then m 0 = n − 3, and by means of (22) Thus by the uniqueness we obtain B(x, u) = 0 and then ∂ 2 η s /∂u i ∂u l = 0. It completes the proof.
Theorem 2.6 If the infinitesimal generator X given in (8) leaves system (2) invariant, then X must take the form X = ( where χ 1 and χ 2 are integral constants, η s = η s (t, x, u) is given by where g s (x), f i (x) and h s (t, x) are undetermined smooth functions of their arguments respectively, and the constant γ s satisfies Proof. We show the theorem by analyzing the structure of Eq.(9) on the space (∂ α t u (k) , ∂ α−1 t u (k) , . . . ). Thus expanding Eq.(9) on the solution space of system (2) yields where η α s takes the form and η θ j (t, x, u (k) ) is given by formula (11). Then substituting (11) and (28) into Eq. (27) and vanishing the coefficients of different order time-fractional partial derivatives of ∂u s /∂x i , one obtains Since Eq.(29) works for k = 1, 2, . . . , then for k = 1 we have . Then Eq.(29) with k ≥ 2 holds identically. Next consider τ = τ (t, x, u). We claim that in system (2), for each x i , there exists at least one u j such that ∂u j /∂x i or its higher order x-derivatives exits. If not, x i can be regarded as a parameter variable and system (2) involves p − 1 independent variables, which is contradictory. Thus we assume such derivatives as the form u ϑ j with ϑ = (ϑ 1 , . . . , ϑ p ) ∈ Z p + satisfying 1 ≤ |ϑ| ≤ k. Observe that in Eq.(27) the derivative ∂u ϑ j /∂t uniquely exists in η θ j (t, x, u (k) ) given by formula (11). More precisely, it appears uniquely in D ϑ (τ ∂u j /∂t) and its coefficient is since ∂ u ϑ j (F) = 0 by the claim. Note that the term τ ∂ t u ϑ j in (11) vanishes identically. Thus we obtain ∂ x i τ = ∂ u j τ = 0, i.e. τ = τ (t). Then the prolonged formula (11) is simplified to With the above simplifications, condition (27) becomes where η θ j (t, x, u (k) ) is given by (30). We now turn to consider η s = η s (t, x, u). Observe that t-derivatives of u i in Eq.(31) uniquely occur in ∂ α−k t (u i ) and µ s where all terms in the former are linear and the ones in the latter are nonlinear. Thus by considering the structure of µ s and Lemma 2.5, we obtain where r i (t, x) and h s (t, x) are undetermined functions.

Determining conditions and algorithm
By means of the symmetry structure of system (2), we show that Lie symmetries of system (2) are determined by two distinct conditions which provide a possibility to use the known computer programs to solve the symmetry determining equations of multi-dimensional timefractional PDEs.
Theorem 2.7 Following the above notations, Lie symmetries of system (2) are completely determined by where I s = {all terms effective in F s } and J s = {the terms which are linear in some u θ j }, then the set I s \ J s = {the terms contained in I s but not in J s }.
On the space (t, x, u), one has Let I s = {all terms effective in F s } and J s = {all terms which are linear in u θ s , where 0 ≤ |θ| ≤ k}, the set I s \ J s = {The terms contained in I s but not in J s }. By considering whether the terms involve u and its x-derivatives or not, we separate Eq.(38) into two parts given in (37). The proof ends. Theorem 2.7 definitely simplifies the computations of Lie symmetry of system (2), which divides the symmetry determining equations into two parts: a system of PDEs in τ, ξ i and η s and a system of linear time-fractional PDEs in h s (t, x). Moreover, for the most of time-fractional PDEs, the second equation in (37) "almost" completely determines the admitted Lie symmetries while the first one either holds automatically or is used to check the final results.
Following the above theoretical preparations, we formulate an algorithm of finding Lie symmetries of system (2) as following three steps: Step 1. Assume system (2) is admitted by the infinitesimal generator (8), then by Theorem 2.6, the infinitesimals τ, ξ i and η j are assumed to be the explicit form (24).
Step 2. Finding the two conditions to determine the Lie symmetry. By Theorem 2.7, first write down the expressions H i , F i and the sets I i , J i , I i \ J i , then obtain the two determining conditions given by (37).
Step 3. Further separation of the second condition in system (37) with respect to u and its x-derivatives to get the symmetry determining system about τ, ξ i and η j , then together with the first condition, solve the system to get the generator (8).
It is worthy of saying that in the third step the separation of the second condition in system (37) is different with the usual one for integer-order PDEs where the former one is divided about u and its derivatives while the latter one is done with respect to x-derivatives of u. Such separation herein results from that the prescribed infinitesimals τ, ξ i and η j in Theorem 2.6 are independent of u and its x-derivatives and thus will generate a more simplified symmetry determining equations.

Three examples 3.1 Time-fractional generalized Zakharov-Kuznetsov equation
The first example is the time-fractional generalized Zakharov-Kuznetsov equation where n is a nonzero constant and u = u(t, x, y). Eq.(39) with α = n = 1 is the Zakharov-Kuznetsov equation which describes weakly nonlinear ion-acoustic wave in a strongly magnetized lossless plasma in two dimensions [30]. We assume that Eq.(39) is admitted by a one-parameter Lie symmetry group with the infinitesimal generator where ξ, τ, ψ and η are smooth functions of t, x, y and u respectively. Note that we will adopt two methods to look for Lie symmetries of Eq.(39) in order to show the efficiency of our algorithm while in next two subsections we directly use our method for the other two examples .
First assume µ = 0 in η α t , then inserting (18) into condition (41) and separating it with respect to different time-fractional derivatives of u yields Then separating the last equation in (42) with respect to t-and x-derivatives of u generates the system of symmetry determining equations which is very huge and complicated since τ, ξ, ψ and η are functions of t, x, y, u, thus clear symmetry structure will definitely drop off the complexity of computations.
The direct role of the infinitesimal operator is to reduce the PDEs into the lower dimensional PDEs. Before the performance, we define the following fractional differential operator in order to present an elegant expression for the similarity reductions Note that in the case of single independent variable operator (46) becomes the classical Erdélyi-Kober fractional differential operator [3] and thus (46) can be called the generalized Erdélyi-Kober fractional differential operator.

Time-fractional Hirota-Satsuma coupled KdV equations
Consider the time-fractional Hirota-Satsuma coupled KdV equations whose Lie symmetry analysis has been performed in [25]. Here we directly use our results to study Lie symmetry of system (47). Assume that a Lie symmetry with the infinitesimal generator where ξ, τ, η and φ are arbitrary smooth functions of t, x, u, v respectively, leaves system (47) invariant, where by Theorem 2.6, τ = χ 2 t 2 + χ 1 t, ξ = ξ(x), By Theorem 2.7, one has H 1 (t, x) = 0, F 1 = uu x + vv x + u xxx , Then the two conditions in (37) for the first equation of system (47) become and for the second equation become , η x , φ x and η xxx , φ xxx , the first-order and third-order extended infinitesimals respectively, are given by Inserting the above prolongations into systems (48) and (49) and annihilating the coefficients of different powers of u, v and their derivatives to zero, we obtain Solving the system yields which is the same as the results in [25]. However, the symmetry determining system (50) is very simple and easy to be solved. Next we use the infinitesimal operators to construct reduced equations. The generator X = ∂ x reduce system (47) to the form where the similarity variables are ζ = t, U (ζ) = u(t, x), V (ζ) = v(t, x). Solving system (51) gives one solution of system (47) u(t, Induced by the operator X = t∂ t + α/3x∂ x − 2α/3u∂ u − 2α/3v∂ v , Eq.(47) is reduced to where the similarity variables are ζ = xt −α/3 , U (ζ) = ut 2α/3 , V (ζ) = vt 2α/3 .

Time-fractional nonlinear telegraph equations
The third example is the time-fractional nonlinear telegraph equations with variable coefficients where P (u) and Q(u) are two smooth nonzero functions of u which make system (52) nonlinear. In what follows, we will perform a complete Lie symmetry classification of system (52) which means to classify the functions P (u) and Q(u) making system (52) admit the extended symmetries and then to determine the symmetries. An equivalent transformation of system (52) is given by where nonzero constants β i satisfy β 1 β 3 = β 2 = 0. It implies that in the procedure of symmetry classification for system (52), scalings of P (u) and G(u) do not affect the final classified results. For example, if P (u) = β 1 u 2 + β 2 , G(u) = β 3 u, we assume P (u) = u 2 +β 2 , G(u) = u.
Assume that a Lie symmetry group with the infinitesimal generator where the infinitesimals ξ, τ, η and φ are undetermined smooth functions of t, x, u and v respectively, leaves system (52) invariant, by Theorem 2.6, τ = χ 2 t 2 + χ 1 t, ξ = ξ(x), where, hereinafter, f i (x), g i (x), h i (t, x) are undetermined functions and c j are arbitrary constants, i = 1, 2 and j = 1, 2, 3. Moreover, by Theorem 2.7 for system (52), one has H 1 (t, We first consider the first equation in system (52). By Theorem 2.7, I 1 = {v x } and J 1 = {v x }, then I 1 \ J 1 = ∅ and we obtain where φ x is the first order extended infinitesimal of φ with respect to x and expressed by Inserting (56) into the second equation in system (55) and separating it with respect to v, v x and u x , we obtain g 2 (x) = c 1 , and Observe that if f 1 (x)f 2 (x) = 0, P (u) is a constant and G(u) is either linear in u or a constant, which is contradict with that system (52) is nonlinear. Thus f 1 (x)f 2 (x) = 0. Since f 2 (x)P (u) = f 1 (x) and P (u) = 0, thus f 1 (x) = 0 implies f 2 (x) = 0 and vice versa, i.e. f 1 (x) = f 2 (x) = 0. Then by dividing the last equation in system (57) with respect to t, we obtain the symmetry determining equations for the first equation in system (52) We claim that χ 2 = 0. If χ 2 = 0, by (26) one has γ 1 = γ 2 = (α − 1)/2, then the second equation in system (58) becomes −αχ 2 = 0, which does not hold since α = 0. Thus system (58) becomes Then by the above analysis the infinitesimals in (54) are simplified to Now we turn to the second equation in system (52). Since system (52) is nonlinear, then either P (u) is a nonconstant function or Q(u) is a nonlinear function or both of them. In order to facilitate symmetry classification, we divide P (u) and G(u) into two parts respectively, set P (u) = P n + β, G(u) = λ u + G n , where λ and β are two constants, P n = P n (u) is a function without containing constant term and G n = G n (u) is a nonlinear function without containing the linear term λ u.
With the above assumptions, I 2 = {u, u x , P n u x , G n }, J 2 = {u, u x }, I 2 \ J 2 = {P n u x , G n } and F 2 = P n u x + G n + βu x + λ u. By Theorem 2.7, the two determining conditions for the second equation in system (52) are where η x is give by We claim that if h 1 (t, x) is a constant, then h 1 (t, x) = 0. Assume h 1 (t, x) = C is a constant, then solving the first equation in system (59) gives h 2 (t, Equating the coefficient of x to zero yields C = 0 and then k(t) = c 2 t α−1 . Thus h 1 (t, x) = 0, h 2 (t, x) = c 2 t α−1 .
Inserting η and η x into Eq.(60b) and separating it with respect to u x yields where system (59) is used. We start with Eq.(61a) to classify the pairs (P (u), G(u)) and to determine the corresponding Lie symmetries. Since P n = 0, Eq.(61a) implies g 1 (x) and h 1 (t, x) are constants respectively, and thus h 1 (t, x) = 0, h 2 (t, x) = c 2 t α−1 by the claim. Let g 1 (x) = ω.

Conclusions
We further investigate Lie symmetry structure of a system of multi-dimensional time-fractional PDEs and give an explicit formula of the extended infinitesimal of fractional partial derivative. Moreover, we show that the infinitesimal generator of the system has an elegant structure and uniquely determined by two simple conditions. Applications to three examples demonstrate the efficiency of the results.
It is worthy of saying that the whole procedure to search for Lie symmetries of the multidimensional system is mechanic and by means of our results quite possible to be performed on the computer with lower complexity of computation. In addition, whether the above results hold for time-fractional PDEs with Caputo fractional derivative are still unknown. We will proceed in this way and report the related results.