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

Authors: Mohamed R. Ali
Journal: Journal of Applied Mathematics
 (2019)

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## 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).


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## Body

## 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, theG′/G 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]):(1)ut+huxx+uux=0In 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:(2)utα+huxx+uux=0,0<α<1In 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 are(3)∂tαu=Ft,x,u,ux,uxx,……Suppose that the infinitesimal vector X has the form(4)X=ξ1x,t,u∂∂x+ξ2x,t,u∂∂t+ηx,t,u∂∂uThe Lie group parameter of infinitesimal transformations [8, 21, 22] has the formula(5)x-=x+εξ1t,x,u+Oε2,t-=t+εξ2t,x,u+Oε2,u-=u+εηt,x,u+Oε2,∂αu-∂t-α=∂αu∂tα+εηα0t,x,u+Oε2,∂u-∂x-=∂u∂x+εηxt,x,u+Oε2,∂2u-∂x¯2=∂2u∂x2+εηxxt,x,u+Oε2,where ξ1, ξ2, and η are considered as the infinitesimals of the transformation’s variables x,t,u, respectively, and ε«1 is considered as the group parameter; we will take it to be equal to one. The explicit expressions of ηx and ηxx, which we consider as the prolongation of the infinitesimals, are given by(6)ηx=Dxη-uxDxξ1-utDtξ2,and(7)ηxx=Dxηx-uxtDxξ1-uxxDtξ2,where Dx is in [8] assigned as(8)Dx=∂∂x+ux∂∂x+uxx∂∂ux+…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:(9)Prα,2XΔΔ=0=0where Δ=Dtαu-Ft,x,u,ux,uxx,…… and Pr is the second prolongation of the infinitesimal generator X.Definition 2.
The prolonged vector is demonstrated by(10)PrnX=X+∑i=1p∑α=1qξiα∂∂uiα+…+∑j1=1p…∑jn=1p∑α=1qξj1…jnα∂∂uj1…jnαwhere q is the number of dependent variables, p is the number of independent variables, ∂/∂uj1α=∂/∂uxα, and the PDE involves derivatives of up to the order n. The condition [21–23] is given by(11)ξ2t,x,ut=0=0Lemma 3.
The functionu=θ(x,t) is an invariant solution of (3) if and only if (i)
ξ2x,t,θθt+ξ1x,t,θθx=ηx,t,θ.Lemma 4.
Theαth extended infinitesimal [24, 25] for the fractional derivative part utilizing the RL definition with (11) is given by(12)ηα0=∂αη∂tα+ηu-αDtξ2∂αu∂tα-u∂αηu∂tα+μ-∑n=1∞αnDtnξ1Dtα-nux+∑n=1∞αn∂αηu∂tα-αn+1Dtn+1ξ2Dtα-nu,where(13)μ=∑n=2∞∑m=2n∑k=2m∑r=2k-1αnnmkr1k!tn-αΓn+1-α-ur∂m∂tmuk-r∂n-m+k∂tn-m∂ukRemember that (14) α n = - 1 n - 1 α Γ n - α Γ 1 - α Γ n + 1


## 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 are(3)∂tαu=Ft,x,u,ux,uxx,……Suppose that the infinitesimal vector X has the form(4)X=ξ1x,t,u∂∂x+ξ2x,t,u∂∂t+ηx,t,u∂∂uThe Lie group parameter of infinitesimal transformations [8, 21, 22] has the formula(5)x-=x+εξ1t,x,u+Oε2,t-=t+εξ2t,x,u+Oε2,u-=u+εηt,x,u+Oε2,∂αu-∂t-α=∂αu∂tα+εηα0t,x,u+Oε2,∂u-∂x-=∂u∂x+εηxt,x,u+Oε2,∂2u-∂x¯2=∂2u∂x2+εηxxt,x,u+Oε2,where ξ1, ξ2, and η are considered as the infinitesimals of the transformation’s variables x,t,u, respectively, and ε«1 is considered as the group parameter; we will take it to be equal to one. The explicit expressions of ηx and ηxx, which we consider as the prolongation of the infinitesimals, are given by(6)ηx=Dxη-uxDxξ1-utDtξ2,and(7)ηxx=Dxηx-uxtDxξ1-uxxDtξ2,where Dx is in [8] assigned as(8)Dx=∂∂x+ux∂∂x+uxx∂∂ux+…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:(9)Prα,2XΔΔ=0=0where Δ=Dtαu-Ft,x,u,ux,uxx,…… and Pr is the second prolongation of the infinitesimal generator X.Definition 2.
The prolonged vector is demonstrated by(10)PrnX=X+∑i=1p∑α=1qξiα∂∂uiα+…+∑j1=1p…∑jn=1p∑α=1qξj1…jnα∂∂uj1…jnαwhere q is the number of dependent variables, p is the number of independent variables, ∂/∂uj1α=∂/∂uxα, and the PDE involves derivatives of up to the order n. The condition [21–23] is given by(11)ξ2t,x,ut=0=0Lemma 3.
The functionu=θ(x,t) is an invariant solution of (3) if and only if (i)
ξ2x,t,θθt+ξ1x,t,θθx=ηx,t,θ.Lemma 4.
Theαth extended infinitesimal [24, 25] for the fractional derivative part utilizing the RL definition with (11) is given by(12)ηα0=∂αη∂tα+ηu-αDtξ2∂αu∂tα-u∂αηu∂tα+μ-∑n=1∞αnDtnξ1Dtα-nux+∑n=1∞αn∂αηu∂tα-αn+1Dtn+1ξ2Dtα-nu,where(13)μ=∑n=2∞∑m=2n∑k=2m∑r=2k-1αnnmkr1k!tn-αΓn+1-α-ur∂m∂tmuk-r∂n-m+k∂tn-m∂ukRemember that (14) α n = - 1 n - 1 α Γ n - α Γ 1 - α Γ n + 1


## 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 have(15)u-tα+hu-xx+u¯u-x=0Thus, u(x,t) satisfies (2). Applying the second prolongation to (2), symmetry invariant equation is(16)ηα0+uηx+hηxx+uxη=0Substituting the values from (6), (7), and (12) into (16) and isolating coefficients in partial derivatives regarding x and power of u, we have (17)αn∂tnηu-αn+1Dtn+1ξ2=0,n=1,2,3…,ξu2=ξx2=ξu1=ξt1=ηuu=0,αξt2-2ξx1=0,hηxx-u∂tαηu+∂tαη+uηx=0,ξxx1-2ηxu=0.Solving the obtained determining equation, we get(18)ξ1=c2+αxc1,ξ2=2tc1,η=-αuc1,where c1 and c2 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:(19)X1=∂∂x,(20)X2=αx∂∂x+2t∂∂t-αu∂∂u.Case 1.
For (19), we have(21)dx1=dt0=du0Solving this equation, u=ft. Putting u=ft into (1), we get(22)Dtαft=0where u=a1tα-1.Case 2.
ForX2 in (20), we have(23)dxαx=dt2t=-duαuThis is the characteristic equation. By solving it, the resulting similarity variable in the form(24)z1=xt-α/2,z2=ut-α/2.The variables transformation is as follows:(25)u=t-α/2fξ,ξ=xt-α/2,where fξ is a function in one variable ξ. We use (25) to transform (2) into a fractional ODE.Theorem 5.
Transformation (25) reduces (2) to the nonlinear FODE as follows:(26)P3/α1-3α/2,αfξ+hfξξ+ffξ=0utilizing the Erdelyi-Kober (EK) fractional derivative operator [20]:(27)Pβξ2,αfξ=∏j=0n-1ξ2+j-1βddξKβξ2+α,n-αfξ,where(28)Kβξ2,αfξ=1Γα∫1∞u-1α-1u-ξ2+αfξu1/βdu,α>0fξ,α=0and (29)n=α+1,α≠Nα,α∈NProof.
Utilizing the definition of the RL fractional derivative in (25), we get(30)∂αu∂tα=∂n∂tn1Γn-α∫1tt-sn-α-1s-α/2fxs-α/2ds,n-1<α<1,n=1,2,…Assume that v=t/s, ds=-t/v2dv. Thus, (30) becomes(31)∂αu∂tα=∂n∂tntn-3α/21Γn-α∫0∞v-1n-α-1v-n+1-3α/2fξvα/2dvApplying the EK fractional integral operator (28) in (31), we have(32)∂αu∂tα=∂n∂tntn-3α/2K2/α1-α/2,n-αfξFor simplicity, we consider =xt-α/2, ϕ∈(0,∞). We thus find that (33)t∂∂tϕξ=tx-α2t-α/2-1ϕξ=-α2ξ∂∂ξϕξHence, we have(34)∂n∂tntn-3α/2K2/α1-α/2,n-αfξ=∂n-1∂tn-1∂∂ttn-3α/2K2/α1-α/2,n-αfξ=∂n-1∂tn-1tn-3α/2-1n-3α2-α2ξ∂∂ξK2/α1-α/2,n-αfξ.Repeating n-1 times, we have(35)∂n∂tntn-3α/2K2/α1-α/2,n-αfξ=∂n-1∂tn-1∂∂ttn-3α/2K2/α1-α/2,n-αfξ=∂n-1∂tn-1tn-3α/2-1n-3α2-α2ξ∂∂ξK2/α1-α/2,n-αfξ=t-α/2∏j=0n-11-3α2+j-α2ξ∂∂ξK2/α1-α/2,n-αfξApplying the EK fractional differential operator (27) in (35), we get(36)∂n∂tntn-α/3K2/α1-α/2,n-αfξ=t-α/2P2/α1-3α/2,αfξSubstituting (36) into (32), we get(37)∂αu∂tα=t-α/2P2/α1-3α/2,αfξ,Thus, (2) is reduced to a fractional-order ODE as follows:(38)P2/α1-3α/2,αfξ+ffξ+hfξξ=0.


## 4. The Explicit Solution for the Time-Fractional Benjamin-Ono Equation by Using PSM

The analytic solutions via PSM [26] are demonstrated. We assume that(39)fξ=∑n=0∞anϕξn,Differentiating (39) twice regarding ξ, we get(40)f′ξ=∑n=0∞nanϕξn-1,and(41)f′′ξ=∑n=0∞nn-1anϕξn-2,Substituting (39), (40), and (41) into (38), we have(42)∑n=0∞Γ2-3α/2-nα/2Γ2-α/2-nα/2anϕξn+∑n=0∞anϕξn∑n=0∞n+1an+1ϕξn+h∑n=0∞n+2n+1an+2ϕξn=0Comparing coefficients in (42) when n=0, we obtain(43)a2=-12hΓ2-3α/2Γ2-α/2a0+a0a1,When n≥1, the recurrence relations between the series coefficients are(44)an+2=-12hn+2n+1Γ2-3α/2-nα/2Γ2-α/2-nα/2an+n+1anan+1Using (44), the series solution for (39) can be represented by substituting (43) and (44) into (39):(45)fξ=a0+a1ξ+a2ξ2+∑n=1∞an+2ξn+2=a0+a1ξ-12hΓ2-3α/2Γ2-α/2a0+a0a1ξ2-∑n=1∞12hn+2n+1Γ2-3α/2-nα/2Γ2-α/2-nα/2an+n+1anan+1ξn+2.Upon substitution using similarity variables in (25), the following explicit solutions for (2) are(46)ux,t=a0t-α/2+a1xt-α-12hΓ2-3α/2Γ2-α/2a0+a0a1t-α/2xt-α/22-∑n=1∞12hn+2n+1Γ2-3α/2-nα/2Γ2-α/2-nα/2an+∑k=0n∑j=0kaj+ak-jan-kn+1xn+2t-α2n+αβ/2β.


## 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:(47)an+2≤Γ2-3α/2-nα/2Γ2-α/2-nα/2an-∑k=0n∑j=0kajak-jan-k-anEquation (47), utilizing the Gamma function, shows that Γ2-3α/2-nα/2/Γ2-α/2-nα/2<1 for arbitrary n that(48)an+2≤Man-∑k=0n∑j=0kajak-jan-k-anwhere M=max​c1,c2. We now assume another form of the PSM:(49)Bξ=∑n=0∞cnξnBy comparing the two series, we can observe that cn≤an, n=0,1,…. Hence, the series Bξ=∑n=0∞cnξn is the majorant series of (47). So, we find that(50)Bξ=c0+c1ξ+M∑n=0∞cnξ2Bξ+∑n=0∞∑k=0n∑j=0kcjck-jcn-k+∑n=0∞cnξn+2Consider an implicit functional system regarding ξ as follows:(51)βξ,B=B-c0-c1ξ-c2ξ2-Mξ2Bξ+2Bξ2+ξ2-c1ξ-3c0Bsince β is analytic in a neighborhood of (0,c0), where β0,c0=0, and ∂/∂Bβ0,c0=0. Then, the series Bξ=∑n=0∞ξn is analytic around (0,c0) 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 0,c0.


## 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 h=1 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.Figure 1
3D plot of (46) with a0=1.9, a1=1.7, a2=0.77, γ=1, α=0.8, β=2.Figure 2
3D plot of (46) with a0=3.9, a1=1.7, a2=0.87, γ=1, α=0.6, β=1.Figure 3
3D plot of (46) with a0=1, a1=1.1, a2=0.57, γ=1, α=0.5, β=0.7.Figure 4
3D plot of (46) with a0=1, a1=1, a2=0.4, γ=1, α=0.3, β=0.4.


## 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.



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