A new analytical algorithm for uncertain fractional differential equations in the fuzzy conformable sense

1.   Introduction

In the past few decades, fractional partial differential equations (FPDEs) have received extensive attention in studies of nonlinear fields such as physics, science, biology, chemistry, mathematics, electronics, viscoelasticity, signal processing and soft materials. Obtaining the exact solutions of nonlinear FPDEs is one of the most significant goals of the research process for FPDEs. A large number of papers have presented many effective methods for constructing explicit solutions of FPDEs, such as Lie symmetry analysis ^[1,2,3,4,5], the Laplace transform ^[6], the Adomian decomposition method ^[7], the variational iteration method ^[8], the homotopy analysis method ^[9], the separation of variables method ^[10], the invariant subspace method ^[11], the extended direct algebraic method ^[12], Chebyshev series method ^[13], etc.

Lie symmetry analysis is a very useful tool in constructing exact solutions for nonlinear partial differential equations (PDEs). As a result of applying the Lie symmetry method, many kinds of exact solutions for integer-order PDEs have been obtained, such as similarity solutions, fundamental solutions and traveling wave solutions ^[14,15]. Inspired by the Lie symmetry of integer-order PDEs, researchers have extended and applied the Lie symmetry method to investigate FPDEs and constructed the group invariant solutions for numerous FPDEs. For example, Hu et al. ^[16] studied the exact solutions for the time-fractional KdV-type equation. Leo et al. ^[17] provided a general theoretical framework to extend the classical Lie theory for PDEs to the case of equations of fractional order. Wang et al. ^[18] performed Lie symmetry analysis on the time-fractional Harry Dym equation and constructed group invariant solutions. Lashkarian et al. ^[19] considered the Lie group analysis and obtained the explicit solutions of the time-fractional cylindrical Burgers equation. In addition, Erdélyi-Kober (E-K) fractional differential operators ^[17] can be used to establish the exact solutions for FPDEs; for instance, Cheng and Wang ^[20] constructed the power-series solutions to (2+1)-dimensional time-fractional Navier-Stokes equation by using an E-K fractional differential operator. In ^[21], Zhang and Li achieved explicit solutions to the time-fractional b-family peakon equations, which contain the mixed derivative that encompasses the time-fractional derivative and integer-order $x$-derivative.

Conservation laws play an important role in mathematical physics as a means to study some properties of solutions to nonlinear PDEs, and they can describe the invariable properties of the values of some physical models in nature, such as momentum, energy, mass, potential energy, electric charge and so on. It is well-known that Noether's theorem establishes a close connection between Lie symmetries and the conservation laws of PDEs ^[22]. Ibragimov presented a new conservation theorem in ^[23], and he obtained the conservation laws for PDEs without classical Lagrange equations. In recent years, Lie symmetry analysis and the conservation laws for many PDEs have been discussed, such as those for various wave equations ^[24], reaction-diffusion equations ^[25], the Rosenau-KdV-RLW equation ^[26], time-fractional Cahn-Allen and Klein-Gordon equations ^[27] and the time-fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation ^[28].

Nonlinear equations take the form

with arbitrary coefficients ${a_i(\cdot)}$, ${b_j(\cdot)}$ and annihilating operator $L = I+\frac{d^2}{dx^2}$, and they are called tautological PDEs ^[29], which are also referred to as multi-term time-fractional differential equations. Motivated by ^[21], in this paper, we focus on the study of multi-term time-fractional equations, particularly the time-fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation, respectively. The BBMP equation first proposed by Peregrine ^[30] was derived from the Korteweg-de Vries equation. The BBMP equation is commonly used to study long waves in the shallow water of ocean beaches, drift waves in plasma and the Rossby waves in rotating fluids ^[31]. The BBMP equation is also known as the regularized long-wave equation. This paper focuses on the time-fractional BBMP equation, which has the following form:

where $0 < \alpha < 1$, $u = u(x, t)$ is the unknown function, $x\in\mathbb{R}$ is a space variable and $t > 0$ is the time variable. In particular, when $\alpha = 1$, Eq (1.1) reduces to the classical BBMP equation, which has a wide range of applications in the development of optical equipment, semiconductors ^[32], etc. Khalique ^[33] utilized Lie group analysis to obtain the periodic solutions and soliton solutions and derived the conservation laws for the classical BBMP equation.

The Novikov equation is an extremely classical integrable evolution equation, which is thought to be a generalization of the Camassa-Holm equation and originally discovered by Vladimir Novikov ^[34], who has used it to describe fluid motion in shallow water environments. In ^[34], Novikov discovered the first few symmetries and studied a scalar Lax pair of the equation. In this paper, we pay attention to the time-fractional Novikov equation, which is given by

where $0 < \alpha < 1$ and $u = u(x, t)$ is the unknown function of space variable $x$ and time variable $t$.

In view of the non-locality property of fractional derivatives, time-fractional PDEs can be used to describe problems with memory effects and genetic characteristics in real life. Thus, there is practical significance in a study of the time-fractional equations given by Eqs (1.1) and (1.2). To our knowledge, equations containing a time fractional-order mixed derivative and spatial integer-order derivative have rarely been studied, while Eqs (1.1) and (1.2) are such equations. Therefore, in this paper, we will construct the exact solutions of Eqs (1.1) and (1.2) through the use of Lie symmetry analysis and symmetry reduction. Moreover, in view of the difficulty in solving ordinary differential equations (ODEs), we introduce the E-K fractional differential operator to transform Eqs (1.1) and (1.2) into ODEs with only a single variable, and we obtain their power series solutions. In addition, the q-homotopy analysis method and invariant subspace method may be used to discuss the exact solutions of the time-fractional equations given by Eqs (1.1) and (1.2), while, in this paper, we take no account of these methods.

This paper is organized as follows. In Section 2, we will give some basic knowledge and properties of fractional calculus. Section 3 is fully dedicated to obtaining the Lie symmetry groups and exact solutions of the time-fractional BBMP equation and time-fractional Novikov equation. First, we will introduce the Lie symmetry method for general time-fractional PDEs with the Riemann-Liouville derivative. Next, infinitesimal generators for Eqs (1.1) and (1.2) will be constructed, respectively. Then, based on the obtained Lie algebras, we perform symmetry reduction and obtain the group-invariant solutions of the equations under consideration. In addition, by using the E-K fractional differential operator, the power-series solutions for Eqs (1.1) and (1.2) is also constructed. In Section 4, conservation laws for the time-fractional BBMP equation are obtained. At last, we will show the conclusion of this paper.

2.   Preliminaries

The purpose of this section is to review some basic properties of the Riemann-Liouville fractional integral and derivative ^[35], which will be used throughout this paper. Meanwhile, we will give the definition of the E-K fractional differential operator ^[17].

Definition 2.1. ^[35] The $\alpha$th-order Riemann-Liouville integral operator of $v(t)$ is defined by

where $\Gamma(x) = \int_0^{\infty} e^{-z}z^{x-1}{\rm d}z$ is the gamma function.

Definition 2.2. ^[35] The symbol $\partial_t^{\alpha}u$ represents the $\alpha$th-order Riemann-Liouville derivative, denoted as

where $u = u(x, t)$ is the function of the space variable $x$ and time variable $t$, and $\frac{\partial^n}{\partial t^n}$ is the $n$th-order integer derivative. Moreover, the Riemann-Liouville derivative can be expressed as

where $D$ is the differential operator and $n = [\alpha]+1.$

Definition 2.3. ^[35] The adjoint operator $(\partial_t^{\alpha})^{\ast}$ of the Riemann-Liouville fractional differential operator $\partial_t^{\alpha}$ is given by

where $I^{n-\alpha}$ denotes the Riemann-Liouville integral operator.

Definition 2.4. ^[17] The E-K fractional differential operator $\mathcal{P}_{\beta}^{\tau, \alpha}$ of order $\alpha$ is defined by

where

and

Lemma 2.1. ^[36] Let $0 < \gamma\leq 1$ and $k_1\in\mathbb{R}$. The solution of the fractional ODE

is given by

3.   Lie symmetry and group-invariant solutions

3.1.   Lie symmetry for time-fractional PDEs

Consider the symmetry group for a time-fractional PDE of the following form:

where $u = u(x, t)$ denotes the unknown function of the space variable $x$ and time variable $t$. Suppose that Eq (3.1) remains invariant under the one-parameter Lie group of infinitesimal transformations given by

where $\epsilon\ll1$ is the group parameter and $\xi$, $\tau$ and $\eta$ are smooth functions called infinitesimal generators. The admitted one-parameter Lie group of infinitesimal transformation has the following form:

The Lie invariance criterion for time-fractional PDE (3.1) is

where $\Delta = F(\partial_t^{\alpha}u, \partial_t^{\alpha}(u_{xx}), t, x, u, u_x, u_{xx}, \cdots)$ and $n$ is a nonnegative integer.

The operator ${\rm{Pr}}^{(\alpha, n)}V$ denotes the prolongation of field vector V, and it is defined by

where

In Eq (3.5), $D_t$ and $D_x$ represent the total derivatives, respectively given by

The $\alpha$th-order extended infinitesimal $\eta^{\alpha, t}$ given by Eq (3.5) can be expressed as follows ^[37]:

where

and

Similarly, the expression of $\eta^{\alpha, xx}$ is defined by ^[21]

3.1.1.   Time-fractional BBMP equation

In this subsection, through the Lie symmetry analysis given in Section 3.1, we obtain the Lie symmetry group and infinitesimal generators of the time-fractional BBMP equation given by Eq (1.1).

Theorem 3.1. The Lie point symmetries admitted by Eq (1.1) are

Proof. In view of the Lie symmetry analysis given in Section 3.1, the Lie invariance criterion for Eq (1.1) is

Inserting Eqs (3.5), (3.7) and (3.9) into the invariance condition given by Eq (3.11) and equating the coefficients of the partial derivatives $u_t, \; u_x, \; u_{xx}, \; \cdots,$ fractional derivatives $\partial_t^{\alpha-n}u, \; \partial_t^{\alpha-n}u_x$, $\partial_t^{\alpha}u_{xx}\; \cdots$ and powers of $u$, we can obtain the following over determined equations:

Solving Eq (3.12), the infinitesimal generators can be expressed as follows:

where $C_i\; (i = 1, 2, 3)$ are arbitrary constants. Moreover, noticing that the initial condition

holds for all time-fractional PDEs, the vector fields of Eq (1.1) can be expressed in the form of Eq (3.10). □

3.1.2.   Time-fractional Novikov equation

This subsection is devoted to the investigation of Lie point symmetry of the time-fractional Novikov equation given by Eq (1.2).

Theorem 3.2. Lie algebras admitted by the fractional Novikov equation expressed as Eq (1.2) are spanned by the following vector fields:

Proof. The invariance criterion given by Eq (3.3) of Eq (1.2) can be represented in the following form:

where $\eta^{\alpha, t}, \; \eta^{\alpha, xx}, \; \eta^x, \; \eta^{xx}$ and $\eta^{xxx}$ are defined by Eqs (3.5), (3.7) and (3.9).

Equating the coefficients of fractional derivatives, partial derivatives and various powers of $u$ to be zero, the over determined equations can be constructed as follows:

Solving the above over determined equation given by Eq (3.17), we can deduce that

where $C_1$, $C_2$ and $C_3$ are arbitrary parameters.

In view of the initial condition $\tau(t, x, u)|_{t = 0} = 0$, the infinitesimal generators of Eq (1.2) are given by Eq (3.15). □

3.2.   Symmetry reductions and group-invariant solutions

In the previous section, we obtained the infinitesimal generators for the time-fractional BBMP equation and Novikov equation. In this section, we are ready to perform symmetry reductions and construct their exact solutions. Moreover, the power-series solutions are also obtained through the use of the E-K fractional differential operator.

3.2.1.   Similarity reduction and exact solutions of Eq (1.1)

In this subsection, we perform the similarity reduction of the time-fractional BBMP equation given by Eq (1.1), and then the exact solutions are calculated correspondingly.

$Case\; 1.$ $V_1 = \frac{\partial}{\partial x}$. According to the characteristic equation of Eq (1.1), i.e.,

we deduce two independent similarity variables $t$ and $u$. Thus we get the group-invariant solution of Eq (1.1) with the form $u(x, t) = g(t)$, where $g(t)$ satisfies the fractional ODE

By virtue of Lemma 2.1, we obtain

Therefore, the group-invariant solution of Eq (1.1) is

$Case\; 2$. $V_2 = t\frac{\partial}{\partial t}-\alpha u\frac{\partial}{\partial u}$. Considering the characteristic equation

the similarity variables $x$ and $t^{\alpha}u$ are obtained. Thus the group-invariant solution of Eq (1.1) is

where $g(x)$ satisfies the equation

Solving the above equation, we find that

and C is an arbitrary constant.

Plugging Eq (3.22) into Eq (3.21) yields the following exact solution of Eq (1.1):

$Case\; 3.$ $V_3 = \rho V_1+V_2 = \rho\frac{\partial}{\partial x}+t\frac{\partial}{\partial t}-\alpha u\frac{\partial}{\partial u}$, where $\rho$ is a nonzero constant. In this case, we consider the power-series solution with the help of E-K fractional operators for Eq (1.1). Solving the characteristic equation

leads to the similarity variables

where $g$ is an undetermined function with a new independent variable $\xi = e^xt^{-\rho}$. Thus, one can obtain the following theorem.

Theorem 3.3. The similarity transformations given by Eq (3.24) reduce the BBMP equation given by Eq (1.1) to the following form:

Proof. Based on the definition of the Riemann-Liouville derivative given by Eq (2.2), we have

Setting $v = \frac{t}{s}$, and due to Eqs (2.5) and (2.6), Eq (3.26) can be written as

As for $\partial_t^{\alpha}(u_{xx})$, we have

On one hand, similar to Eqs (3.26) and (3.27), one obtains

On the other hand, we obtain

Therefore, combine Eqs (3.28) and (3.29) with Eq (3.30) to get

Substituting Eqs (3.27) and (3.31) into Eq (1.1) yields that Eq (3.25) holds to be true. □

In what follows, we intend to use the power-series method to construct the exact solutions for Eq (3.25). Suppose that Eq (3.25) admits the power-series solution

where $a_n\; (n = 0, 1, 2, \cdots)$ are constants which will be determined later. Then, we have

Next, with the help of Eq (3.33), we consider the following E-K fractional differential operator:

Similar to Eq (3.34), we can also obtain

and

Therefore, inserting Eqs (3.32)–(3.36) into Eq (3.25) yields that

When $n = 1$, by comparing coefficients of $\xi$, we obtain

When $n\geq2$, we have

Consequently, the power series solution for Eq (3.25) can be derived as follows:

Substituting Eq (3.40) into Eq (3.24) leads to the exact solution of the BBMP equation given by Eq (1.1):

3.2.2.   Similarity reduction and exact solution of Eq (1.2)

This subsection derives the similarity variables and their reduction equations and presents the exact solutions of the time-fractional equation given by Eq (1.2).

$Case\; 1$. $V_1 = \frac{\partial}{\partial x}$. We find that the group-invariant solution of Eq (1.2) is $u(x, t) = g(t)$, where $g(t)$ is given by $\partial_t^{\alpha}g(t) = 0$. Then we observe that the exact solution to Eq (1.2) is the same as that presented as Eq (3.20).

$Case\; 2$. $V_2 = 2t\frac{\partial}{\partial t}-\alpha u\frac{\partial}{\partial u}$. The similarity variables of this infinitesimal generator are $x$ and $t^{\frac{\alpha}{2}}u$. Substituting $u(x, t) = t^{-\frac{\alpha}{2}}g(x)$ into Eq (1.2) yields the following reduction equation:

$Case\; 3$. $V_3 = \lambda V_1+V_2 = \lambda\frac{\partial}{\partial x}+2t\frac{\partial}{\partial t}-\alpha u\frac{\partial}{\partial u}$, where $\lambda$ is a nonzero constant. We can also obtain the following reduced equation for the field vector $V_3$.

Theorem 3.4. Induced by the general infinitesimal generator $V_3 = \lambda V_1+V_2$, Eq (1.2) can be reduced to the following form:

where $z = e^xt^{-\frac{\lambda}{2}}$, $g = ut^{\frac{\alpha}{2}}$ and $0 < \alpha < 1.$

Proof. For the infinitesimal generator $V_3 = \lambda V_1+V_2 = \lambda\frac{\partial}{\partial x}+2t\frac{\partial}{\partial t}-\alpha u\frac{\partial}{\partial u}$, the characteristic equation is given by

Solving the above equation yields the corresponding similarity variables $e^xt^{-\frac{\lambda}{2}}$ and $t^{\frac{\alpha}{2}}u$; then, we obtain the similarity transformation

where $z = e^xt^{-\frac{\lambda}{2}}$ and $g$ is the function of $z$ to be determined later. Thus, we get

In the sequel, we consider the time-fractional derivatives $\partial_t^{\alpha}u$ and $\partial_t^{\alpha}(u_{xx})$ in Eq (1.2) with $0 < \alpha < 1$, respectively. According to the definition of the Riemann-Liouville fractional derivative, one can obtain

Letting $v = \frac{t}{s}$, and in view of Eq (2.6), Eq (3.45) can be converted to the following form:

Then, utilizing the E-K fractional differential operator given by Eq (2.5), we obtain

As for $\partial_t^{\alpha}(u_{xx})$, similar to Eqs (3.28), (3.29) and (3.30) in the last section, one obtains

Therefore, inserting Eqs (3.44), (3.47) and (3.48) into Eq (1.2), we verify Eq (3.42). □

In what follows, we are ready to obtain the explicit analytic solution for Eq (3.42) by using the power-series method. We first assume that Eq (3.42) has the power-series solution

where $a_n\; (n = 1, 2, 3, \cdots)$ are constants which will be determined later.

Then, we have

Next, we calculate the E-K fractional differential operator. Analogous to Eqs (3.34), (3.35) and (3.36), one respectively obtains

and

Thus, substituting Eqs (3.50)–(3.53) into Eq (3.42), we have the following formula:

Then, in view of the Cauchy product formula, Eq (3.54) can be converted to

Hence, by vanishing the coefficients of different powers of $z$ in Eq (3.55), we deduce that $a_0$ and $a_3$ are arbitrary constants, and that $a_2 = -a_1$. When $n\geq2$, one obtains

Insert Eq (3.56) into Eq (3.49) to obtain the power-series solution of Eq (3.42) as follows:

As a consequence, substituting Eq (3.57) into Eq (3.43), we can construct the explicit solution to Eq (1.2) as follows:

Remark 3.1. In accordance with the method introduced in ^[38] and the explicit function theorem in ^[39], the power-series solutions given by Eqs (3.41) and (3.58) are convergent, and we omit the details of the proof.

4.   Conservation laws for the time-fractional BBMP equation

In this section, the conservation laws for the time-fractional BBMP equation given by Eq (1.1) will be investigated by using the new conservation theorem ^[23,40]. The vector $(C^t, C^x)$ for Eq (1.1) that satisfies the equation

is called the conserved vector, where $C^t = C^t(t, x, u, \cdots)$, $C^x = C^x(t, x, u, \cdots)$.

The formal Lagrangian of Eq (1.1) can be represented as

where $v(x, t)$ is a new dependent variable.

The adjoint equation is given by

here, $\frac{\delta}{\delta u}$ symbolizes the Euler-Lagrange operator defined by

where $(\partial_t^{\alpha})^{\ast}$ is the adjoint operator of $\partial_t^{\alpha}$ that is defined by Eq (2.4).

In view of Eqs (4.2) and (4.3), we can derive the adjoint equation of Eq (1.1) with the following form:

Regarding the nonlinear self-adjointness of Eq (1.1), Eq (4.5) must be satisfied for all solutions of Eq (1.1) with the substitution

The derivatives of Eq (4.6) are given as

Substituting Eqs (4.6) and (4.7) into Eq (4.5), we obtain the following nonlinear self-adjointness expression:

Solving the above Eq (4.8), we conclude that

where $C$ is a constant. Then, taking $C = 1$, according to the vector fields described by Eq (3.10), as admitted by Eq (1.1), and the formulas of Lie characteristic functions $W_i = \eta_i-\xi_iu_x-\tau_iu_t\; (i = 1, \; 2)$, the characteristic functions can be expressed as follows

The fractional Noether operator for the variable $t$ is defined by ^[21]

where the operator $\mathcal{J}(f, g)$ is given by

Equivalently, the fractional Noether operator for the component $x$ of the conserved vector is introduced as follows:

Therefore, with the help of Eqs (4.11) and (4.13), we construct the conservation laws for Eq (1.1), whose components of the conserved vectors are as follows:

and

5.   Discussion on exact solutions

In this section, we provide the 3-D graphical interpretations of the exact solutions given by Eqs (3.23), (3.41) and (3.58) for the time-fractional BBMP equation and Novikov equation, respectively. Moreover, we will discuss the influence of the parameter $\alpha$ in Eqs (1.1) and (1.2) on solutions.

Figure 1 shows the influence of fractional parameter $\alpha$ on $u(x, t)$ in Eq (3.23) (i.e., a solution) of Eq (1.1). From Figure 1, we can observe that the maximum value of solution $u(x, t)$ decreases as $\alpha$ increases when the constant $C = 1$ and the points $(x, t)$ are fixed. Particularly, the solution $u(x, t)$ approaches zero with the increase of the parameter $\alpha$ from 0.25 to 0.9 when the points $(x, t)$ are near the point (0, 0).

Figure 2 displays the effect of fractional parameter $\alpha$ on $u(x, t)$ in Eq (3.23) (i.e., a solution) of Eq (1.1). The graphs show that the curvature of the graphs decreases as the fractional parameter $\alpha$ increases from 0.1 to 0.9.

Figure 3 shows the influence of fractional parameter $\alpha$ on $u(x, t)$ in Eq (3.41) (i.e., a solution) of Eq (1.1). From Figure 3, we can find that the value of the solution $u(x, t)$ increases as the fractional parameter $\alpha$ increases. In particular, the curvature of the plots increases as the fractional parameter $\alpha$ increases from 0.1 to 0.9 near the point (0, 0).

Figure 4 displays the effect of fractional parameter $\alpha$ on $u(x, t)$ in Eq (3.58) (i.e., a solution) of Eq (1.2). The plots show that the value of the solution $u(x, t)$ decreases as the fractional parameter $\alpha$ increases from 0.1 to 0.9.

6.   Conclusions

In this paper, in accordance with Lie symmetry analysis and the power-series method, the explicit solutions of the time-fractional BBMP equation and time-fractional Novikov equation have been successfully constructed. More specifically, we have established the Lie algebra admitted by Eqs (1.1) and (1.2) and obtained the group-invariant solutions by means of symmetry reductions. By using the fractional E-K operator, we have transformed Eqs (1.1) and (1.2) into ODEs (3.25) and (3.42) with only one variable, respectively, and constructed their power-series solutions. Moreover, the conservation laws for the time-fractional BBMP equation have been obtained by using the new Noether's theorem. In addition, we have constructed the 3-D graphs of corresponding exact solutions by using Maple software. In the future, we will devote ourselves to constructing exact solutions of Eqs (1.1) and (1.2) with the help of other methods, such as the q-homotopy analysis method, invariant subspace method, extended direct algebraic method, Chebyshev series method, etc.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to thank the editor and the anonymous referees for their valuable suggestions and comments, which have improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 12271433).

Conflict of interest

All authors declare no conflict of interest.