Abstract
We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).
1. Introduction
Lie symmetry method provides an effective tool for deriving the analytic solutions of the nonlinear partial differential equations (NLPDEs) [1–4]. In recent years, many authors have studied the nonlinear fractional differential equations (NLFDEs) because these equations express many nonlinear physical phenomena and dynamic forms in physics, electrochemistry, and viscoelasticity [5–9].
Time-fractional NLDEs arise from classical NLPDEs by replacing its time derivative with the fractional derivative. The methods applied to derive the analytic solutions of NLFPDEs are the exp-function, the expansion, fractional su-equation, Lie symmetry method, and many more [10–19].
The one-dimensional Benjamin-Ono equation is considered here as follows (see [20]):In fact, the BO equation describes one-dimensional internal waves in deep water. We consider LSA for the analytic solutions by using PS expansion for the time-fractional BO equation:In division 2 of this paper, some basic properties of the Riemann–Liouville fractional derivative are shown firstly and then the Lie group method for FPDEs is presented. In division 3, the Lie group to the time-fractional BO equation (FBO) and the symmetry reductions are determined. In division 4, we derive anew arrangement of the FBO equation (2) via the PSM. In division 5, we study the convergence for the series solution. We conclude our work in division 6.
2. Notations and Delineations
2.1. Description of Lie Symmetry Reduction Method for NLFPDEs
We present the principal notations and definitions that detecting the symmetries of the NLFPDEs.
Here, the time-fractional NLFPDEs areSuppose that the infinitesimal vector has the formThe Lie group parameter of infinitesimal transformations [8, 21, 22] has the formulawhere , , and are considered as the infinitesimals of the transformation’s variables , respectively, and is considered as the group parameter; we will take it to be equal to one. The explicit expressions of and , which we consider as the prolongation of the infinitesimals, are given by
andwhere is in [8] assigned as
Theorem 1. Equation (2) coincides with a one-parameter group of transformations (5) with the infinitesimal generator X if and only if the accompanying infinitesimal conditions holds true:where and is the second prolongation of the infinitesimal generator .
Definition 2. The prolonged vector is demonstrated bywhere is the number of dependent variables, is the number of independent variables, , and the PDE involves derivatives of up to the order . The condition [21–23] is given by
Lemma 3. The function is an invariant solution of (3) if and only if (i)
Lemma 4. The extended infinitesimal [24, 25] for the fractional derivative part utilizing the RL definition with (11) is given bywhere
Remember that
3. Reduction of Time-Fractional Benjamin-Ono Equation
We use the LSA to find the similarity solution for 1D time-factional BO equation (1). Suppose that (2) is an invariant under (5), so that we haveThus, satisfies (2). Applying the second prolongation to (2), symmetry invariant equation isSubstituting the values from (6), (7), and (12) into (16) and isolating coefficients in partial derivatives regarding and power of , we have Solving the obtained determining equation, we getwhere and are constants, for simplicity. We take their values equal to one. So, (2) has two vector fields that can generate its infinitesimal symmetry. These Lie vectors are considered as follows:
Case 1. For (19), we haveSolving this equation, Putting into (1), we getwhere .
Case 2. For in (20), we haveThis is the characteristic equation. By solving it, the resulting similarity variable in the formThe variables transformation is as follows:where is a function in one variable . We use (25) to transform (2) into a fractional ODE.
Theorem 5. Transformation (25) reduces (2) to the nonlinear FODE as follows:utilizing the Erdelyi-Kober (EK) fractional derivative operator [20]:whereand
Proof. Utilizing the definition of the RL fractional derivative in (25), we getAssume that , . Thus, (30) becomesApplying the EK fractional integral operator (28) in (31), we haveFor simplicity, we consider , . We thus find that Hence, we haveRepeating times, we haveApplying the EK fractional differential operator (27) in (35), we getSubstituting (36) into (32), we getThus, (2) is reduced to a fractional-order ODE as follows:
4. The Explicit Solution for the Time-Fractional Benjamin-Ono Equation by Using PSM
The analytic solutions via PSM [26] are demonstrated. We assume thatDifferentiating (39) twice regarding , we getandSubstituting (39), (40), and (41) into (38), we haveComparing coefficients in (42) when , we obtainWhen , the recurrence relations between the series coefficients areUsing (44), the series solution for (39) can be represented by substituting (43) and (44) into (39):Upon substitution using similarity variables in (25), the following explicit solutions for (2) are
5. Convergence Analysis
To satisfy the convergence test, there are many kinds of tests as the ratio, the comparison, and the quotient tests. The convergence of the solution equation (46) will be presented as follows. We revamp (46) as follows:Equation (47), utilizing the Gamma function, shows that for arbitrary thatwhere . We now assume another form of the PSM:By comparing the two series, we can observe that , . Hence, the series is the majorant series of (47). So, we find thatConsider an implicit functional system regarding as follows:since is analytic in a neighborhood of , where , and . Then, the series is analytic around and this is verified utilizing [27] and the radius of convergence of this series belongs to a positive domain. This shows that (46) converges around
6. Physical Performance of the Power Series Technique for Eqs. (46)
To have expressed and convenient conception of the physical characteristic of the power series solution, the 3D plots for the explicit solution equations (46) is plotted in Figures 1–4 at by utilizing appropriate parameter forms. The spectacle vision of the real portion of (46) can be visible in the 3D plots proof in Figures 1, 2, 3, and 4, respectively.
7. Conclusions
Lie point symmetry properties of (1 + 1)-dimensional time-fractional Benjamin-Ono equation have been considered with the Riemann–Liouville fractional derivative. These symmetries are used here to transform the FPDEs into NLFODEs. Closed-form solutions are determined by using PSM in the last division. The accuracy exhibits the assembly of the solution. Considerable frames for the acquired explicit solutions were approached.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that they have no conflicts of interest.