AMPAdvances in Mathematical Physics1687-91391687-9120Hindawi10.1155/2020/31318563131856Research ArticleIterative Analysis of Nonlinear BBM Equations under Nonsingular Fractional Order DerivativeAliGauhar1AhmadIsrar1https://orcid.org/0000-0002-8851-4844ShahKamal1https://orcid.org/0000-0002-8889-3768AbdeljawadThabet234FellahZine El Abiddine1Department of MathematicsUniversity of MalakandChakdara Dir (Lower)Khyber PakhtunkhwaPakistanuom.edu.pk2Department of Mathematics and General SciencesPrince Sultan UniversityRiyadhSaudi Arabiapsu.edu.sa3Department of Medical ResearchChina Medical UniversityTaichung 40402Taiwancmu.edu.cn4Department of Computer Science and Information EngineeringAsia UniversityTaichungTaiwanasia.edu.tw20201382020202013052020160620202306202013820202020Copyright © 2020 Gauhar Ali et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present research work is devoted to investigate fractional order Benjamin-Bona-Mahony (FBBM) as well as modified fractional order FBBM (FMBBM) equations under nonlocal and nonsingular derivative of Caputo-Fabrizio (CF). In this regards, some qualitative results including the existence of at least one solution are established via using some fixed point results of Krasnoselskii and Banach. Further on using an iterative method, some semianalytical results are also studied. The concerned tool is formed when the Adomian decomposition method is coupled with some integral transform like Laplace. Graphical presentations are given for various fractional orders. Also, the concerned method is also compared with some variational-type perturbation method to demonstrate the efficiency of the proposed method.

Prince Sultan UniversityRG-DES-2017-01-17
1. Introduction

Fractional calculus is the generalized form of classical calculus. With the rapid change in science and technology, the aforesaid area has attracted the attention of many researchers. The mentioned branch has many applications in different areas of science like modeling, control theory, physics, signal processing, economics, and chemistry . Different researchers have studied fractional differential equations (FODEs) in their own way, including the stability aspect, qualitative theory, optimization, and numerical simulations. Many real-world problems are nonlinear in nature, and their investigation is important for fruitful information. Therefore, researchers have studied various problems of FODEs by using different techniques and methods. One of the important aspects is the existence theory of solution which has given proper attention in the last years . By using the fixed point theory, the existence theory to numerous problems has been established . The authors in  also studied different aspects of FODEs using a derivative with nonsingular kernel and Laplace transform. Therefore, we intend to establish the aforementioned theory for the following problem with 0,τ=J(1)DCFtγvt,yht,vt,y=0,vy,0=fy,where h:J×RR and fCJ. The existence of at least one solution of (1) has been studied with the help of a fixed point approach, since the differential operator involving fractional order have a great degree of freedom. Therefore, it comprehensively describes many dynamical properties and characteristic of various processes/phenomena [23, 24]. Then, we establish an algorithm to compute the approximate analytical solutions for the following cases of BBM equations with y,tJ,γ0,1 as

Case 1.

(2)DCFtγvt,yvyytt,y+avt,yvyt,y=0,v0,y=fy.

Case 2.

(3)DCFtγvt,y+vyt,y+avt,yvyt,y+vyyyt,y=0,vy,0=fy.

Case 3.

(4)DCFtγvt,y+vyt,y+av2t,yvyt,y+vyyyt,y=0,vy,0=fy.where a is a real constant. The abovementioned problems are also called regularized long-wave equation which is the improved form of the Korteweg-de Vries equation (KDVE). Such equation has been largely used for modeling of waves of small amplitudes and in the soliton theory of fractals and dynamics. Moreover, KDVE has countless integrals of motion and BBM has only three . For generalized n-dimensional BBM equation and its applications, we refer to [25, 33, 34]. The aforementioned equation has been studied in surface waves of a long period of fluid . Also, for the dynamic aspect of the BBM equation, we refer . The mentioned equation is not only suitable for superficial waves but also for acoustic and hydromagnetic waves; because of this, the BBM equation has upper hand on KDVE. We enrich our study by investigating the modified form of BBM equation abbreviated as MBBM . We use the decomposition method coupled with Laplace transform to establish series solution to our proposed problems (2), (3), and (4). The mentioned problems have been studied by the homotopy perturbation method (HPM), variational method (VHPM), wavelet method, etc., but these studies are limited to fractional order derivative involving the usual Caputo and integer order derivative. To the best of authors’ information, no study exists in the present literature to address the investigation of the aforesaid problems under nonsingular CF derivative. The mentioned derivative was introduced in 2016 and has been found suitable in applications of many thermal problems. The concerned nonlocal integral of CF for a function is the average of the function and its Riemann integral which works as a filter, for various applications of the concerned derivative, we refer to [12, 13, 18, 19]. So far, we know that there is no investigation present in the literature which addresses the study of the mentioned problems under nonlocal and nonsingular kernel derivatives with fractional order. We establish some qualitative results of the existence of at least one solution by Krasnoselskii and Banach fixed point results. Further, by the proposed method of Laplace transform coupled with Adomian decomposition (LADM), we compute the series solution whose convergence is also studied. Also, the results are compared with the results of VHPM. The results reveal that the proposed method can also be used as a powerful tool to find approximate results to many nonlinear problems.

2. PreliminariesDefinition 1 (see [<xref ref-type="bibr" rid="B37">37</xref>]).

Let vH10,a,a>0,γ0,1, then CF derivative is defined below (5)DCFtγvt=γ1γ0texpγts1γvμdη,γ0,1,t0,where the function γ is called normalization.

Definition 2 (see [<xref ref-type="bibr" rid="B38">38</xref>]).

The CF integral with γ0,1 is given below (6)JCFtγvt=1γvtγ+γγ0tvηdη.

Definition 3 (see [<xref ref-type="bibr" rid="B37">37</xref>]).

For the CF derivative of order γ0,1 and n, the Laplace transform is given below (7)LDCFtn+γvts=11γLvn+1tLexpγt1γ=sn+1Lvtsnv0sn1v0vn0s+γ1s.

Definition 4.

The considered method is used to compute the solution in an infinite series form. We consider the solution as (8)vt,y=n=0vny,tand nonlinear term is decompose as (9)Nv=n=0An,where An is given by (10)An=1Γn+1DμnNj=0nμjvjμ=0.

Theorem 5 (Krasnoselskii’s fixed point theorem [<xref ref-type="bibr" rid="B39">39</xref>]).

If DX be a convex and closed nonempty subset, there exist two operators G1 and G2 such that

G1v1+G2v2D for all v1,v2D

G1 is a condensing operator

G2 is continuous and compact

then, there exists at least one solution vD which satisfies G1v+G2v=v.

3. Steps for Existence of Results

In the ongoing section, we discuss the existence of the considered problem.

Lemma 6.

Under Definitions (1) and (2), we have (11)vt,y=fy+1γγht,vt,yh0,v0,yγγ0thθ,vθ,ydθ.

The assumptions needed for our work are

(B1) ht,v is the nonlinear function satisfy the growth condition as (12)ht,vbh+Cvp,p0,1,C0.

(B2) For all v1,v2Rthere exist a positive constant khone can get, (13)ht,v1ht,v2khv1v2,foralltJ.

Furthermore, ht,0=0 holds.

G1,G2:XX are the operators defined as (14)G1t,v=fy+1γγht,vt,yh0,v0,y,G2t,v=γγ0thθ,vθ,ydθ.

Theorem 7.

In light of hypothesis (B1) and (B2), if 1γ/γkh1, then (1) has at least one solution.

Proof.

Using (2.5), and a bounded set defined as D=vX:vXR. The continuity of vt,y implies that G1 and G2 are continuous operators. To show that G1 is a condensing map, considerv1,v2D, under the assumption (B1) (15)G1v1G1v2X=maxtJ1γγht,v1t,y1γγht,v2t,y1γγkhv2v1X.

This show that G1 is a condensing map; further, for the continuity and compactness of G2 for all vD, consider (16)G2vX=maxtJγγ0thθ,vθ,ydθγγmaxtJ0thθ,vθ,ydθγγ0tbh+Cvpdθγγbh+CRPτ.

Therefore, G2 is bounded on D. For continuity considering t1t2>0, one can infer that (17)G2v1t1G2v2t2=γγ0t1hθ,vθ,ydθ0t2hθ,vθ,ydθγγbh+CRpt1t2.

This implies that G2v1t1G2v2t2X0, as t1 tends to t2. So it shows that G2 is compact and equicontinuous; by Theorem 1, the problem (1) has no less than one solution in D.

Theorem 8.

In view of assumption (B2) if 1+γτ1/γkh, then problem (1) has a unique solution.

Proof.

By using (1), we define the operator G as (18)Gvt,y=fy+1γγht,vt,yh0,v0,y+γγ0thθ,vθ,ydθ.

Suppose v1,v2X, we have (19)Gv1Gv2\leqxmaxtJ1γγht,v1t,yht,v2t,y+maxtJ0thθ,v1θ,yhθ,v2θ,ydθ1γγkhv1v2X+γγτkhv1v2X=1+γτ1γkhv1v2X.

Therefore, G is a condensing operator which implies the uniqueness of solution.

4. Main Results

To present the iterative solution of our considered problem, we first give a general procedure for the given problem as (20)DCFtγvt,y=Nvt,y+Rv+gt,y,v0,y=fyy0,1,where N is a nonlinear operator and R is a linear operator and g is external source function. Further, f:JR is a nonlocal, bounded, and continuous function.

Taking Laplace transform of (14) and using the initial condition, we have (21)DCFtγvt,y=fys+s+γ1ssLNvt,y+Rvt,y+gt,y.

Let us consider the solution in terms of a series as (22)v=n=0vn,and decompose the nonlinear term Nvt,y in terms of the Adomian polynomial as (23)Nv=n=0An,where (24)An=1Γn+1DunNj=0nμjvjμ=0.

Using (15) and comparing the terms on both sides, we have (25)v0=fy,v1=L1γ+s1γsLA0t,yRv0t,y+gt,y,vn+1=L1γ+s1γsLAnt,yRvnt,y+gt,y,n0.

After evaluation, the required solution is (26)vt,y=n=0vnt,y=v0t,y+v1t,y+v2t,y+.

Theorem 9.

Let T be a nonlinear contractive operator on a Banach space X, such that for all v,vX, one has (27)TvTvXkvvX,0<k<1.

Then, the unique fixed point v satisfies the relation Tv=v. Let us write the generated series (26) as (28)vn=Tvn1,vn1=i=0n1vi,n=1,2,3,,and assume that v0Srv, where Srv=vX:vvXr,r0. Then, we have (29)A1xnSrv.A2limnvn=v.

Proof.

(A1) By using mathematical induction for n=1, we have (30)v1vX=Tv0TvXkv0vX.

Considering that the result for n1 is true, then (31)vn1vXkn1v0vX.

Now consider (32)vnvX=Tvn1TvXkvn1vXknv0vX.

With the help of (A1), we have (33)vnvXknv0vXknrr,which gives that vnSrv, since (34)vnvXknv0vX,and limnkn=0. Therefore, we have limnvnvX=0 which yields limnvn=v.

4.1. General Procedure for Case <xref rid="casee1" ref-type="statement">1</xref>

Consider the following FBBM equation under the given condition as (35)DCFtγvt,yvyytt,y+avt,yvyt,y=0,v0,y=fy.

Taking Laplace transform of (35), one has (36)Lvt,y=fys+γ+s1γsLvyytt,yavt,yvyt,y.

Let us consider the solution in terms of a series as (37)v=n=0vn,and the decomposition of the nonlinear term is (38)vvy=n=0An,where (39)An=1Γn+1Dμnj=0nμjvjj=0nμjvjyμ=0.

An for different values of n are (40)A0=v0t,yv0yt,y,A1=v0t,yv1yt,y+v0yt,yv1t,y,and so on. Putting these values in (36) and comparing the terms on both sides, we have (41)v0t,y=fy,v1t,y=L1γ+s1γsLv0yytt,yav0t,yv0yt,y,vn+1t,y=L1γ+s1γsLvnyytt,yavnt,yvnyt,y,n0.

After calculation, the solution of the considered problem (35) is obtained in the form of a series.

4.2. General Procedure for Case <xref rid="casee2" ref-type="statement">2</xref>

Consider the following FBBM equation under the given condition as (42)DCFtvvt,y+vyt,y+avt,yvyt,y+vyyyt,y=0,v0,y=fy.

Taking Laplace of (42), one may have (43)Lvt,y=fysγ+s1γsLvt,y+vyyyt,y+avt,yvyt,y.

Here, we consider the unknown solution as (44)v=n=0vn,and the nonlinear term is decomposed as (45)vt,yvy=n=0An,where An is define as (46)An=1Γn+1Dμnj=0nμjvjj=0nμjvjyμ=0.

An for different values of n are (47)A0=v0t,yv0yt,y,A1=v0t,yv1yt,y+v0yt,yv1t,y,and so on. Using these values in equation (43) and equating the corresponding terms on both sides, we have (48)v0t,y=fy,v1=L1γ+s1γsLv0t,y+v0yyyt,y+aA0,v2t,y=L1γ+s1γsLv1t,y+v1yyyt,y+aA1,vn+1t,y=L1γ+s1γsLvnt,y+vnyyyt,y+aAn,n0.

In this way, the series solution of the proposed problem (42) is obtained.

4.3. Procedure for Case <xref rid="casee3" ref-type="statement">3</xref>

Consider the following FMBBM equation under the given condition (49)CFDtγvt,y+vyt,y+av2t,yvyt,y+vyyyt,y=0,v0,y=fy.

Taking Laplace of (49) and after rearranging the terms, we have (50)Lvt,y=fysγ+s1γsLvt,y+vyyyt,y+av2t,yvyt,y.

Here, we consider the unknown vt,y as (51)v=n=0vn,and nonlinear term is decomposed as (52)v2t,yvy=n=0An,where An is “Adomian polynomials” defined as (53)An=1Γn+1Dμnj=0nμjvj2j=0nμjvjyμ=0.

Anfor different values of n are (54)A0=v02t,yv0yt,y,A1=v02t,yv1yt,y+2v0t,yv0yt,yv1t,y,and so on. Putting these values in equation (50) and comparing terms on both sides, we have (55)v0t,y=gy,v1t,y=L1γ+s1γsLv0t,y+v0yyyt,y+aA0,v2t,y=L1γ+s1γsLv1t,y+v1yyyt,y+aA1,vn+1t,y=L1γ+s1γsLvnt,y+vnyyyt,y+aAn,n0.

Hence, in this case, the solution in same way may be computed.

5. Examples

Here, in the ongoing section, we find series solutions for (35), (42), and (49) with the help of LADM using CFFOD.

Example 1.

Consider the following FBBM equation  as (56)CFDtγvt,y=vyytt,yvt,yvyt,y,v0,y=sech2y4.

With the exact solution given below, (57)vt,y=sech2y4t3.

With the help of the procedure discussed in Case 1, one has (58)v0=sech2y4,v1=121+γt1sech4y4tanhy4,v2=γ21+γtγsech6y4tanh3y4+32sech8y4tanhy4+121+γt222γ2t+γ2sech6y4tanh2y4+14sech8y4+12sech6y4tanh2y4.

And hence, the solution of (56) in the form of a series is given by (59)vt,y=sech2y4121+γt,ysech4y4tanhy4γ21+γtγsech6y4tanh3y4+32sech8y4tanhy4+121+γt222γ2t+γ2sech6y4tanh2y4+14sech8y4+12sech6y4tanh2y4.

The approximate solution graphs for various fractional orders are given in Figure 1. We see from graphs as the order γ1, the behavior of the surfaces of the solution tends to the integer order. If we put γ=1 in the approximate solution, we get the solution at the integer order. Now, we compare the four-term LADM solution with the four-term solution of VHPM given in  in Table 1 at γ=1. From Table 1, we see that the absolute error between exact solutions and four-term LADM solutions at the integer order is slightly good than the absolute error for the mentioned four-term solution by using the VHPM. As compared to VHPM, the LADM is simple and easy to use to handle various nonlinear partial differential equations.

Example 2.

Consider the FBBM equation using CFFOD as (60)CFDtγvt,y+vyt,y+vt,yvyt,y+vyyyt,y=0,v0,y=ey.

With the help of procedure discussed for Case 2, one has (61)v0=ey,v1=1+γtγ2ey+e2y,v2=1γ2+2γt1γ+γ2t224ey+14e2y+3e3y,v3=1γ3+3γt1γ2+32γ21γt2+γ3t368ey+136e2y+138e3y+14e4yγ2t221+γt3γ4e2y+6e3y+2e4y,and in the same way, we can find some more terms; therefore, we have (62)vt,y=ey1+γtγ2ey+e2y+1γ2+2γt1γ+γ2t224ey+14e2y+3e3y1γ3+3γt1γ2+32γ21γt2+γ3t368ey+136e2y+138e3y+14e4yγ2t221+γt3γ4e2y+6e3y+2e4y+.

Here, we plot the approximate solution of the FBBM equation up to four terms in Figure 2. The approximate solution graphs for various fractional orders are given in Figure 2. We see from graphs as the order γ1, the behavior of the surfaces of the solution tends to the integer order. If we put γ=1 in the approximate solution, we get the approximate solution at integer order.

Example 3.

Consider the FBBM equation using CFFOD as (63)CFDtγvt,y+vyt,y+vt,yvyt,y+vyyyt,y=0,v0,y=y2.and in the same way, we can find some more terms; therefore, we have (64)v0=y2,v1=1+γtγ2y+2y3,v2=1γ2+2γt1γ+γ2t22142y2+4y3+6y4,v3=1γ3+3γt1γ2+32γ21γt2+γ3t3624+200y+12y2+88y3+20y4+48y5γ2t221+γt3γ4y+16y3+12y5.

Here, we plot the approximate solution of FBBM equation up to four terms in Figure 3. The approximate solution graphs for various fractional orders are given in Figure 3. We see from graphs as the order γ1, the behavior of the surfaces of the solution tends to the integer order solution. If we put γ=1 in the approximate solution, we get the approximate solution at the integer order for the same problem.

Example 4.

Consider the modified FBBM equation using CFFOD as (65)CFDtγvt,y+vyt,y+v2t,yvyt,y+vyyyt,y=0,v0,y=ey.

With the help of the procedure mentioned in Case 3, we have (66)v0=ey,v1=1+γtγ2ey+e3y,v2=1γ2+2γt1γ+γ2t224ey+36e3y+5e5y,v3=1γ3+3γt1γ2+32γ21γt2+γ3t368ey+1104e3y+778e5y+22e7yγ2t221+γt3γ12e3y+20e5y+7e7y,and in the same way, we can find some more terms; therefore, we have (67)vt,y=ey1+γtγ2ey+e3y+1γ2+2γt1γ+γ2t224ey+36e3y+5e5y1γ3+3γt1γ2+32γ21γt2+γ3t368ey+1104e3y+778e5y+22e7yγ2t221+γt3γ12e3y+20e5y+7e7y+.

Here, we plot the approximate solution of the FBBM equation up to four terms in Figure 4. The approximate solution graphs for various fractional orders are given in Figure 4. We see from graphs as the order γ1, the behavior of the surfaces of the solution tends to the integer order solution. Also, if we put γ=1 in the approximate solution, we get the approximate solution at integer order for the same problem.

Example 5.

Consider the modified FBBM equation using CFFOD as (68)CFDtγvt,y+vyt,y+v2t,yvyt,y+vyyyt,y=0,v0,y=y2.

With the help of the procedure discussed for Case 3, one may have (69)v0=y2,v1=1+γtγ2y+2y5,v2=1γ2+2γt1γ+γ2t222+120y2+20y4+18y8,v3=1γ3+3γt1γ2+32γ21γt2+γ3t36160+240y+100y3+6786y5+344y7+244y11γ2t221+γt3γ12y3+40y7+28y11,and in the same way, we can find the other terms. Therefore, we get (70)vt,y=y21+γtγ2y+2y51γ2+2γt1γ+γ2t222+120y2+20y4+18y81γ3+3γt1γ2+32γ21γt2+γ3t36160+240y+100y3+6786y5+344y7+244y11γ2t221+γt3γ12y3+40y7+28y11+.

Here, we plot the approximate solution of FBBM equation up to four terms in Figure 5. The approximate solution graphs for various fractional orders are given in Figure 5. We see from graphs as the order γ1, the behavior of the surfaces of the solution tends to the integer order solution. Also, if we put γ=1 in the approximate solution, we get the approximate solution at integer order for the same problem.

Surface plots of the required solution up to four terms at different values of γ for Example 1.

Comparison between the absolute error at VHPM  and the present method (LADM) with the exact solution of Example 1.

y0.030.040.05
0.011.1543×1041.0534×1041.4926×1041.0067×1041.8307×1041.0546×104
0.022.5862×1041.0987×1043.2626×1042.4321×1043.9387×1042.0088×104
0.034.2956×1042.2345×1045.3101×1043.7054×1046.3239×1044.1234×104
0.046.2827×1043.1033×1047.6350×1043.4034×1048.9864×1046.6523×104
0.058.5474×1044.5643×1041.0237×1033.9876×1041.1926×1041.0195×104

Surface plots of the resultant solution up to four terms at different values of γ for Example 2.

Surface plots of the resultant solution up to four terms at different values of γ for Example 3.

Surface plots of the resultant solution up to four terms at different values of γ for Example 4.

Surface plots of the resultant solution up to four terms at different values of γ for Example 5.

6. Conclusion

In our work, some existence results about the solution to the nonlinear problem of BBM equations under nonsingular kernel-type derivative have been developed successfully. We have discussed different cases of the concerned equations for semianalytical results. For approximate analytical results, a novel iterative method of Laplace transform coupled with Adomian polynomials has been used. Further, by providing an example, we have computed the absolute errors in comparison with VHPM for first four-term solutions at different values of variables t and y against γ=1. We observed that the absolute error is slightly good than the mentioned VHPM. Therefore, the concerned method of LADM can be used as a powerful tool to handle many nonlinear problems of FODEs. Since, the aforementioned equations are increasingly used to model numerous phenomena of physics including the propagation of heat or sound waves, fluid flow, elasticity, electrostatics, and electrodynamics, and population dynamics in biology. A large numbers of the aforementioned equations may be used in fluid mechanics and hydrodynamics. Since fractional derivatives have a greater degree of freedom and produce the complete spectrum of the physical phenomenon which include the ordinary derivative as particular case, global dynamics of the aforesaid physical phenomenon may be investigated. Since the BBM equation can also be used to model various physical systems like acoustic-gravity waves in compressible fluids, acoustic waves in enharmonic crystals, the hydromagnetic waves in cold plasma, (see ), investigation of the BBM equation and its various cases under different fractional order derivatives may be lead us to investigate some more comprehensive results by using various fractional orders which will include the classical order solution as a special case. The nonlocal behaviors of such problems can be well studied by using nonsingular fractional order derivative. In the future, the concerned BBM equation can be investigated by using more general fractional order derivative with nonsingular kernel of the Mittag-Leffler function.

Data Availability

Data availability is not applicable in this manuscript.

Conflicts of Interest

There is no competing interest regarding this work.

Authors’ Contributions

An equal contribution has been done by all the authors.

Acknowledgments

Prince Sultan University provided support through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.