A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

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).

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.

Description of Lie Symmetry Reduction Method for NLF-
PDEs.We present the principal notations and definitions that detecting the symmetries of the NLFPDEs.
Here, the time-fractional NLFPDEs are Suppose that the infinitesimal vector  has the form where  1 ,  2 , and  are considered as the infinitesimals of the transformation's variables (, , ), respectively, and ń1 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 and where   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 Δ =     − (, , ,   ,   , . . . . ..) and  is the second prolongation of the infinitesimal generator .Definition 2. The prolonged vector is demonstrated by (10) where  is the number of dependent variables,  is the number of independent variables, /   1 = /   , and the PDE involves derivatives of up to the order .The condition [21][22][23] is given by
Lemma 4. The  ℎ extended infinitesimal [24,25] for the fractional derivative part utilizing the RL definition with (11) is given by where Remember that

Reduction of Time-Fractional Benjamin-Ono Equation
We use the LSA to find the similarity solution for 1D timefactional BO equation (1).Suppose that ( 2) is an invariant under (5), so that we have Thus, (, ) satisfies (2).Applying the second prolongation to (2), symmetry invariant equation is Substituting 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 get where  1 and  2 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 have Solving this equation,  = ().Putting  = () into (1), we get where  =  1  −1 .
Case 2. For  2 in (20), we have This is the characteristic equation.By solving it, the resulting similarity variable in the form The variables transformation is as follows: where () is a function in one variable .We use (25) to transform (2) into a fractional ODE.

The Explicit Solution for the Time-Fractional Benjamin-Ono Equation by Using PSM
The analytic solutions via PSM [26] are demonstrated.We assume that Differentiating (39) twice regarding , we get and Substituting (39), (40), and (41) into (38), we have Comparing coefficients in (42) when  = 0, we obtain When  ≥ 1, the recurrence relations between the series coefficients are Using (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

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

Conclusions
Lie point symmetry properties of (1 + 1)-dimensional timefractional 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.
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: