Applications of Homogenous Balance Principles Combined with Fractional Calculus Approach and Separate Variable Method on Investigating Exact Solutions to Multidimensional Fractional Nonlinear PDEs

We investigate the exact solutions of multidimensional time-fractional nonlinear PDEs (fnPDEs) in this paper. In terms of the fractional calculus properties and the separate variable method, we present a new homogenous balance principle (HBP) on the basis of the (1+1)-dimensional time fnPDEs. Taking advantage of the new types of HBP together with fractional calculus formulas that subtly avoid the chain rule, the fnPDEs can be reduced to spatial PDEs, and then we solve these PDEs by the fractional calculus method and the separate variable approach. In this way, some new type exact solutions of the certain time-fractional (2+1)-dimensional KP equation, (3+1)-dimensional Zakharov–Kuznetsov (ZK) equation, and Jimbo–Miwa (JM) equation are explicitly obtained under both Riemann–Liouville derivatives and Caputo derivatives. The dynamical analysis of solutions is shown by numerical simulations of taking property parameters as well.


Introduction
e fractional PDEs (fPDEs) have been more and more widely followed with interest up to now since they can be used to accurately describe many nonlinear stochastic phenomena which depend on both time instant and the previous time history in the real-time problem [1][2][3][4]. Models set up from the fPDEs play very important roles in a range of scientific fields, such as viscoelastic flow [5,6], signal processing [7,8], control systems [9,10], material diffusion including normal diffusion and anomalous diffusion(superdiffusion, subdiffusion, fast diffusion, and slow diffusion) [11][12][13][14][15], biological mathematics [16,17], and magnetohydrodynamics (MHD) [18,19]. Similarly, as some time-fractional PDEs, the space fractional models are also very frequently used in some elastic materials (see [20][21][22][23][24]) in the field of mechanical engineering. erefore, investigating their solutions also draws much attention from the mathematical and physical points of view, and it can help to concisely characterize and well understand the qualitative features of the concerned phenomena and nonlinear processes in various areas of natural science and engineering, which involved the ubiquitous time memory effects [20,21] (including some short memories) and space viscoelastic effects (see [22][23][24][25][26]), in particular many complex random walks and material motions in the microscopic space, such as dynamical behaviors of fractional diffusion, particles spread in heat bath, and soft matter interaction with viscoelasticity.
In the past few years, there were excessive studies on the (1 + 1)-dimensional fractional nonlinear PDEs(fnPDEs), and a lot of excellent tools were used for solving them, which include Adomian decomposition method [27,28], homotopy analysis method (HAM) [29,30], fractional variational method [31,32], Lie symmetry method [33][34][35][36][37][38][39][40], invariant subspace method [40][41][42][43][44], and homogenous balance principle (HBP) [45,46]. Recently, a few of the above methods were also applied to solve several (2 + 1)-dimensional fnPDEs [47][48][49][50]. Although approximate analytic solutions or exact solutions of some (1 + 1)-dimensional fnPDEs and (2 + 1)-dimensional fnPDEs can be successfully obtained by using the above methods, this is far from enough, and these methods still have many limitations in solving more complex multidimensional fnPDEs. In this paper, we suggest a new technique to solve the following type of fnPDEs with certain time-fractional derivatives: , and the index 0 < α ≤ 1. e α-th order time derivative (α ∈ (0, 1]) is well defined as an abnormal derivative in the real applications for explaining short memories of evolutional physical systems; there are several kinds of definitions of fractional derivatives, and the two frequently used, classical, and very widely influential definitions are still Riemann-Liouville definition and Caputo definition (see Definitions 1 and 2). Indeed, with the help of Riemann-Liouville derivative and Caputo derivative, the results derived from the time-fractional PDE models are more precise and more general in nature than those of the integer-order ones, while the smaller the α is, the faster effective time memory enjoys [20,21]. When α � 1, expressions (3) and (4) are exactly in accordance with the classical derivatives (zu/zt); however, compared with the Riemann-Liouville type of derivative, the Caputo type of derivative possesses weaker singularity for handling some fractional initial problems.
Moreover, it is necessary to point out that the two , which were often used to reduce the integer-order PDE to ODE, are actually not valid for the fractional-order one since it had been successfully verified that the compound fractional derivatives disagree with the following chain rule (refer to [45,46,51,52]): thus, we hardly take the fractional derivative of compound function straightly in accordance with the classical chain rule, and the normal chain rule (see [51]) of fractional derivatives was ineffectively applied to solve equation (1) since it contains infinite series. at is to say, the exact solutions of the compound function type of equation (1) were impossible to be obtained by the invalid chain rule (2), and we hardly find the travelling wave solutions (even the exact soliton solutions) of fnPDEs via the two transformations mentioned above or some other complex transformations such as Darboux transformation [53], bilinear method [54], and F-expansion method [55].
To the best of our knowledge, avoiding the invalid fractional chain rule (2), there is no more direct and effective method to obtain the exact solutions for equation (1). e exact solutions of higher-dimensional nonlinear PDEs with time-fractional derivatives were not well obtained. e main difficulty is how to construct solutions to reduce the multidimensional fnPDEs to the classical spatial PDEs. Inspired by the previous homogenous balance principle (HBP) [45][46][47] including fractional calculus method [40][41][42][43][44]56] and separate variable approach, we improve the way [45][46][47] and introduce a new type of HBPs for (1) so that the solutions can be assumed as a general separated variable form and (1) can be reduced to spatial PDEs, and then new type exact solutions of some multidimensional fnPDEs will be successfully obtained by solving these reduced PDEs in the variable separate way under both Riemann-Liouville derivatives and Caputo derivatives.
e main contents of this paper are organized as follows. e definitions and properties of the fractional calculus and fractional Laplace transformation are briefly described in Section 2. In Section 3, based on the fractional derivative formulas and the method of separate variable, the new homogenous balance principle (HBP) is suggested for (1), and in this way, the certain time-fractional (2 + 1)-dimensional KP equation, (3 + 1)-dimension Zakharov-Kuznetsov (ZK) equation, and (3 + 1)-dimension Jimbo-Miwa (JM) equation are reduced to the spatial PDEs and explicitly solved in general separated variable forms; we can see that some of these solutions possess new types including some arbitrary functions which were never attained before by other way, which means more general solutions are obtained. Furthermore, there are real differences between the Riemann-Liouville case and the Caputo case: the singularity occurs in solutions under Riemann-Liouville derivatives but no singularity appears in solutions under Caputo derivatives. e dynamical profiles of these solutions are displayed as can be seen from Figures 1-13 with property parameters, and we also analyze the long time behaviors for some of them as well. e last section is the conclusions of our works.

Preliminaries
In this section, we recall some useful definitions, properties, and theorem.

Definitions and Properties of Two Type Fractional Derivatives
e Riemann-Liouville fractional derivative of order α > 0 is defined by the following expression: e Caputo fractional derivative of order α > 0 is defined by the following expression: Proposition 1. Some properties of the fractional derivative:

Proposition 2.
e rules of the partial fractional derivative of a separate variable form: , (under Riemann-Liouville derivative), 0, (under Caputo derivative).

Definitions of Mittag-Leffler Function and Some Properties
Definition 3. e Mittag-Leffler function is defined by the infinity series expression: with α > 0 and β > 0.

Mathematical Problems in Engineering
e brief proof of (iii) and (iv) is given as follows. For the Caputo derivatives, we have For the Riemann-Liouville derivatives, we obtain These achieve (iii) and (iv) in Proposition 3.

HBP for Some Multidimensional Time-Fractional PDEs
In the beginning of this section, we improve the previous way to construct exact solutions to some multidimensional time-fractional nonlinear PDEs (1) according to ideas of homogenous balance and the fractional properties (5). e main procedures are stated as follows: Step 1. As can be seen from the properties of power function and Mittag-Leffler function in Section 2 and motivated by the HBP of (1 + 1)-dimensional timefractional nonlinear PDE, for equation (1), we suppose the exact solutions as the following two types: (1) For the Riemann-Liouville case: (2) For the Caputo case: Here μ 1 , μ 2 , and λ are undetermined parameters and v(x 1 , . . . , x N ) is a undetermined function.
Step 2. Substituting (9) or (10) into equation (1) by comparing the powers of t or the coefficients of E α,1 (λt α ), we have parameters μ 1 , μ 2 , or λ and the reduced spatial PDE system for v( Step 3. Solving the reduced PDE system (10) by using complex transformation ξ � c 0 + N i�1 c i x i or a separate variable approach (in product or summary form) leads to the exact solutions. Remark 1. When μ 1 , μ 2 � α − 1, singular solutions will appear under the Riemann-Liouville derivative and while solutions appear, no singularity will be obtained in the Caputo sense. t should involve minus power, thereby the assumption (9) fits for the Riemann-Liouville derivative, and since the fractional derivative of Mittag-Leffler function includes an additional term (t − α /Γ(1 − α)) under the Riemann-Liouville derivative, this will complicate the calculation of solutions, and the assumption (10) fits for the Caputo derivative.

HBP for the (2 + 1)-Dimensional Time-Fractional KP Equation.
We first begin with the (2 + 1)-dimensional fractional KP equation: which developed from the classical KdV equation, describes long dispersive wave propagating in two dimension in shallow water.

Exact Solutions under Riemann-Liouville Derivatives.
According to (9), we suppose the exact solutions as where a 0 , a 1 , μ 1 , and μ 2 are the undetermined constants.
Balancing the coefficients of t-power yields the following two subcases: us, μ 1 � μ 2 � − α. Substituting (15) into (14) by comparing the coefficients of t − α and t − 2α yields en, integrating (16a) corresponding to variable x once by selecting an integrate constant 0 leads to is gives rise to the implicit relation: where ϕ(y) is an undetermined function.
Calculating the derivatives of implicit function (18) which leads to the solution of (15) v(x, y) with arbitrary constants a 1 , b 1 , and b 2 and the exact solution of (11) where c 1 and c 2 are the arbitrary constants. Subcase II. μ 2 − α � 2μ 2 and μ 1 + μ 2 � μ 2 . us, μ 1 � 0 and μ 2 � − α. Similarly as above, substituting (13) into (11) by comparing the coefficients of t − α and t − 2α yields en, integrating (23a) corresponding to variable x by choosing an integrate constant 0 once leads to Since v ≠ 0 (only nontrivial solutions are considered), we have Plugging (26) into (23b) yields By rewriting b 1 � a 1 c 1 and b 2 � a 2 c 2 , we have the exact solution where a 0 , b 1 , and b 2 are the arbitrary constants.
Remark 2. We have the same type solutions under the complex transformation ξ � c 1 x + c 2 y + c 0 , so this case is omitted here.

Exact Solutions under Caputo Derivatives.
According to (10), we assume the exact solutions formed as Substituting (29) into (11) by comparing the coefficients of E α,1 (λt α ) leads to Mathematical Problems in Engineering We consider the following two cases: Case I. By using the total differential rule, from (30b), we have Integrating (31) leads to where ϕ(y) and C(y) are the undetermined functions.
en, we have following three type solutions: where a 0 , c, c 1 , and c 2 are the arbitrary constants.
We obtain the exact solution where a 0 , c, c 1 , and c 2 are the arbitrary constants.

Dynamical Analysis of Exact Solution for Fractional KP
Model. Under the Riemann-Liouville case: As can be seen from Figure 1, taking parameters α � (1/3) and α � (2), a 0 � 1, b 1 � 2, b 2 � 1, y � 1, and interval x ∈ [− 10, 10], t ∈ [0, 10], it is shown that solution (28) increases with increase in the spatial variable and decreases with increase in time. When x and y are fixed, the solution tends to 0 at an α decay rate as t ⟶ ∞, the larger the α is, the faster the u decay is.
Under the Caputo case: (i) As can be seen from Figure 2, choosing 10], and x � 2, y ∈ [− 10, 10], t ∈ [0, 10], it is shown that solution (35) decreases with increase in time-space since slope (u x < 0) or (u y < 0); however, it is unbounded as x or y tends to ∞. (ii) As we see from Figure 3, selecting

Exact Solutions under Riemann-Liouville Derivatives.
e (3 + 1)-dimensional fractional ZK equation is investigated in this section.
which represented the acoustic dynamics in a magnetized plasma in three-dimensional space with a low pressure. e exact solution can be assumed as Balancing the coefficients of t-power by plugging (42) into (41) yields the following two cases: By comparing the coefficients of t − α and t − 2α , we have

Substituting (45) into (44b) yields
Subcase I. Assume η � b 1 y + b 2 z + b 0 , then we have Solving (47) leads to the following two solutions: (i) When qb 2 2 + sb 2 1 � 0, (sq < 0), then where a 0 and b 0 are the arbitrary constants and ϕ is an arbitrary function. (ii) When ϕ � c 2 (η 2 /2) + c 1 η + c 0 , then where a � a 0 + c 0 + c 1 b 0 , c 1 , and c 2 are the arbitrary constants. Subcase II. Note that (46) is a linear equation, and if sq < 0, by the linear PDE method, we have which gives rise to exact solution when sq < 0: where a 0 and a 1 are the arbitrary constants and f 1 and f 2 are the two arbitrary functions. Subcase III. According to the linear PDE method, we assume the solution of (46) as a product form of a separate variable ϕ(y, z) � f(y)g(z), thus where c 0 is an arbitrary constant.

Mathematical Problems in Engineering
By solving them directly, we obtain the following four solutions (67)-(70) where a 0 , a 1 , c 1 , c 2 , c 3 , c 4 , and c 5 are the arbitrary constants and α ≠ (1/2): x + a 1 c 1 sin Subcase IV. We can also assume the solution of (47) as a summary form of a separate variable ϕ(y, z) � f(y) + g(z), then g ′′′ (z) � 0.
(57) us, which leads to the exact solution by rewriting b 2 � a 1 c 2 2 , b 1 � a 1 c 1 and a � a 0 + a 1 c 0 : where a, a 1 , b 2 , and b 1 are the arbitrary constants and f(y) is an arbitrary function.
(ii) Note that (63) is a linear equation, and by the linear PDE method, we have which leads to the exact solution where a 0 and a 1 are constants and f 1 and f 2 are the two arbitrary functions. (iii) Using summary form of the separate variable ϕ � f(y) + g(z) yields Solving (69) directly, we arrive at which gives rise to the exact solution x + a 1 f(y) + a 0 6q where a 0 , a 1 , b 2 , b 1 , and b 0 are the arbitrary constants and f(y) is an arbitrary function. (iv) When a 0 � 0, using the product form of the separate variable ϕ � f(y)g(z) yields qf(y)g ′′′ (z) + sf ′′′ (y)g(z) � 0, which give rises to f ′′ (y) � c 0 f(y), qg ′′′ (z) + sc 0 g(z) � 0.

Remark 6.
In case II, we do not consider the complex transform ξ � ax + by + cz + d for omitting the trivial solution. Remark 8. If taking form (10) into the fractional ZK model (41), we see that the nonlinear term uu x will lead to trivial solutions, thus the Caputo derivative is not considered.

Dynamical Analysis of Exact Solution for Fractional ZK Model
(i) As can be seen from Figure 5, taking α � (3/5), a � 3, a 0 � c 0 � 1, 10], y � 1.5, z � 2.5, x ∈ [− 10, 10], x � − 2.5, z � 2.5, y ∈ [− 10, 10], and x � − 2.5, y � 1.5, z ∈ [− 10, 10], it is shown that solution (49) increases with increase in spatial variables and decreases as time increases. When x, y, and z are fixed, solution (49) tends to 0 at α decay rate as t ⟶ ∞, the larger the α is, the faster the u decay is. (ii) From Figure 6, by taking α � (2/5), 30,30] and t ∈ [0, 10], and x � − 2.5, y � 1.5, z ∈ [− 10, 10], we see solution (54) is periodic bounded and tends to 0 at α decay rate as y and t tend to ∞ if x and z are fixed. e larger the α is, the faster the u decay is. When x and y are fixed, solution (54) decreases to − ∞ as z increases to +∞. (iii) As can be seen from Figure 7 ∞ or decrease to − ∞ as x, y, or z increases to ∞, and ones also tends to 0 at α decay rate as t ⟶ ∞ if x, y, and z are fixed. e larger the α is, the faster the u decay is.     30], the solution (76) is periodic bounded and has α decay rate as t ⟶ ∞ if x and z are fixed, and one has the same dynamical properties as (54) if else.

HBP for the (3 + 1)-Dimensional Fractional JM Equation.
e (3 + 1)-dimensional fractional JM equation will be researched in the following contents: which is developed from the second members of integrable systems of the classical KP hierarchy.
Remark 9. In case of q ≠ − p, from v ′′ (ξ) � 0, we obtain v ′ (ξ) � 0 which only leads to a trivial solution, thus we do not consider this case.
(i) Note that it is a linear equation, and we suppose v(η, z) � f(η)g(z), which arrives at Using the separate variable method, we have Solving (86a) reads that and the solution of (86b) and the exact solution of (78) is described as the following two cases: If rλb ≠ sac, then and the exact solution is where a 0 , a 1 , b 0 , c 0 , a, b, c, c 1 , c 2 , and c 3 are the arbitrary constants.
If c � (rbλ/sa), then  where a 0 , c 0 , a, b, c 1 , c 2 , and c 3 are the arbitrary

Conclusions
In this paper, by taking advantage of the fractional calculus methods, we avoid the invalid chain rule and suggest the new types of HBPs to solve some multidimensional time-fractional PDEs whose exact solutions were hardly obtained before. We get some explicit solutions of the (2 + 1)-dimensional KP equation, (3 + 1)-dimensional ZK equation, and JM equation by solving the reduced PDE system in both Riemann-Liouville and Caputo cases. ese solutions are all in the general separated variable forms of new type which even include arbitrary functions, and the singularity solutions only appear in the Riemann-Liouville case. Furthermore, the dynamical analysis and long-time behaviors of these solutions are also performed. However, to the best of our knowledge, that is far from enough, the HBP is only applicable to solve some special fractional nonlinear PDEs which satisfy the balance conditions (see Section 3.1) and get a few special exact solutions but not more general solutions. For even more multidimensional fractional nonlinear partial differential systems, there is still no better way to acquire their exact solutions generally at present. Nevertheless, finding more effective methods for constructing more exact solutions of more general (N + 1)-dimensional fnPDEs will be quite a meaningful and challenging task in the future.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.