The two-dimensional nonlinear wave equations are considered. Solution to the problem is approximated by using optimal homotopy asymptotic method (OHAM). The residual and convergence of the proposed method to nonlinear wave equation are presented through graphs. The resultant analytic series solution of the two-dimensional nonlinear wave equation shows the effectiveness of the proposed method. The comparison of results has been made with the existing results available in the literature.
1. Introduction
The wave equations play a vital role in diverse areas of engineering, physics, and scientific applications. An enormous amount of research work is already available in the study of wave equations [1, 2]. This paper deals with the two-dimensional nonlinear wave equation of the form
(1)∂2ux,t∂t2-ux,t∂2ux,t∂x2=1-x2+t22,0≤x,t≤1.
The differential equations (DEs) can be solved analytically by a number of perturbation techniques [3, 4]. These techniques are fairly simple in calculating the solutions, but their limitations are based on the assumption of small parameters. Therefore, the researchers are on the go for some new techniques to overcome these limitations.
The idea of homotopy was pooled with perturbation. Liao [5] proposed homotopy analysis method (HAM) in his Ph.D. dissertation and applied it to various nonlinear engineering problems [6–8]. The homotopy perturbation method (HPM) was initially introduced by He [9–13]. HPM has been extensively used by several researchers successfully for physical models [14–16]. Some useful comparisons between HAM and HPM were done by Domairry and Liang [17, 18].
Recently Marinca and Herişanu [19–21] introduced OHAM for the solution of nonlinear problems which made the perturbation methods independent of the assumption of small parameters, and Ullah et al. [22–26] have extended and applied OHAM successfully for numerous nonlinear phenomena.
The motive of this paper is to apply OHAM for the solution of two-dimensional nonlinear wave equations. In [19–21] OHAM has been proved to be useful for obtaining an approximate solution of nonlinear differential equations. Here, we have proved that OHAM is more useful and reliable for the solution of two-dimensional nonlinear wave equations, hence, showing its validity and greater potential for the solution of transient physical phenomenon in science and engineering.
Section 2 has the basic idea of OHAM formulated for the solution of partial differential equations. In Section 3, the effectiveness of OHAM for two-dimensional nonlinear wave equation has been studied.
2. Basic Formulation of OHAM
Consider the partial differential equation of the following form:
(2)Aux,t+fx,t=0,x∈ΩBu,∂u∂x=0,x∈Γ,
where A is a differential operator, u(x,t) is an unknown function, x and t denote spatial and temporal independent variables, respectively, Γ is the boundary of Ω, and f(x,t) is a known analytic function. A can be divided into two parts: L and N such that
(3)A=L+N.
where L is the simpler part of the partial differential equation which is easier to solve and N contains the remaining part of A.
According to OHAM, one can construct an optimal homotopy ϕx,t;p:Ω×[0,1]→R which satisfies
(4)Hϕx,t;p,p=1-pL(ϕx,t;p)+f(x,t)-H(p)A(ϕx,t;p)+f(x,t)=0.
Here the auxiliary function H(p) is nonzero for p≠0 and H(0)=0. Equation (4) is called optimal homotopy equation. Clearly, we have
(5)p=0⟹Hϕx,t;0,0=Lϕx,t;0+f(x,t)=0,p=1⟹Hϕx,t;1,1=H(1)Aϕx,t;1+fx,t=0.
Obviously, when p=0 and p=1 we obtain ϕ(x,t;0)=u0(x,t) and ϕ(x,t;1)=u(x,t), respectively. Thus, as p varies from 0 to 1, the solution ϕ(x,t;p) approaches from u0(x,t) to u(x,t), where u0(x,t) is obtained from (4) for p=0:
(6)Lu0(x,t)+fx,t=0,Bu0,∂u0∂x=0.
Next, we choose auxiliary function H(p) in the form
(7)Hp=pC1+p2C2+⋯+pmCm.
To get an approximate solution, we expand ϕ(x,t;p,Ci) by Taylor’s series about p in the following manner:
(8)ϕx,t;p,Ci=u0x,t+∑k=1∞ukx,t;Cipk,i=1,2,….
Substituting (8) into (4) and equating the coefficient of like powers of p, we obtain zeroth-order problem, given by (6), the first- and second-order problems are given by (9) and (10), respectively, and the general governing equations for uk(x,t) are given by (11) as follows:
(9)Lu1(x,t)=C1N0u0(x,t),Bu1,∂u1∂x=0,(10)Lu2x,t-Lu1x,t=C2N0u0x,t+C1Lu1x,t+N1u0x,t,u1x,t,Bu2,∂u2∂x=0,(11)Lukx,t-Luk-1x,t=CkN0u0x,t+∑i=1k-1CiLuk-ix,tkkkkkkkkkkk+Nk-iu0x,t,u1x,t,…,uk-ix,t,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkik=2,3,…,Buk,∂uk∂x=0,
where Nk-i(u0(x,t),u1(x,t),…,uk-i(x,t)) are the coefficient of pk-i in the expansion of N(ϕ(x,t;p)) about the embedding parameter p. One has
(12)Nϕx,t;p,Ci=N0u0x,t+∑k≥1Nku0,u1,u2,…,ukpk.
It should be underscored that the uk for k≥0 is governed by the linear equations with linear boundary conditions that come from the original problem, which can be easily solved.
It has been observed that the convergence of the series equation (8) depends on the auxiliary constants C1,C2,…. If it is convergent at p=1, one has
(13)u~x,t;Ci=u0x,t+∑k≥1ukx,t;Ci.
Substituting (13) into (1), it results in the following expression for residual:
(14)Rx,t;Ci=Lu~x,t;Ci+fx,t+Nu~x,t;Ci.
In actual computation k=1,2,3,…,m. If R(x,t;Ci)=0 then u~(x,t;Ci) is the Exact solution of the problem. Generally it does not happen, especially in nonlinear problems.
For determining auxiliary constants, Ci, i=1,2,…,m, there are a number of methods like Galerkin’s method, Ritz method, least squares method, and collocation method. The method of least squares can be applied as follows:
(15)JCi=∫0t∫ΩR2x,t;Cidxdt(16)∂J∂C1=∂J∂C2=⋯=∂J∂Cm=0.
The mth-order approximate solution can be obtained by these optimal constants. The more general auxiliary function Hp is useful for convergence, which depends on constants C1,C2,…,Cm, can be optimally identified by (16), and is useful in error minimization.
3. Application of OHAM to Two-Dimensional Nonlinear Wave Equations
To demonstrate the effectiveness of the formulation of OHAM, we consider two-dimensional nonlinear wave equations of the form (1) with initial conditions
(17)u0,t=t22,∂∂xu0,t=0.
Applying the method formulated in Section 2 leads to
(18)L=∂2ux,t∂x2,N=-ux,t∂2ux,t∂t2,fx,t=1-x2+t22.
Zeroth-Order Problem. Consider
(19)∂u0x,t∂t=1-x2+t22,u00,t=t22,∂∂xu00,t=0.
Its solution is
(20)u0x,t=x22+t22-x2t24-x424.
First-Order Problem. Consider
(21)∂2u1x,t∂x2=1+C1∂2u0x,t∂x2-C1u0x,t∂2u0x,t∂t2+1+C1x22+t22-1,u10,t=0,∂∂xu10,t=0.
Its solution is
(22)u1x,t=-14t2x2-x224+t2x424+7x6720-x82688C1.
Second-Order Problem. Consider
(23)∂2u2x,t∂x2=1+C1∂2u1x,t∂x2+C2∂2u0x,t∂x2kkkkk-C1u0x,t∂2u1x,t∂t2-C1u1x,t∂2u0x,t∂t2kkkkk-C2u0x,t∂2u0x,t∂t2+C2x22+t22-1,u20,t=0,∂∂xu20,t=0.
Its solution is
(24)u2x,t=12661120-115-173+56t2x10C12kkkkkkkkkkk-434x12C12+66x8-15C1kkkkkkkkkkk+-15C1+36t2C12-15C2kkkkkkkkkkk-665280t2x2C1+C2+C12+110880x4kkkkkkkkkkk·-1+t2C1+-1+2t2C12kkkkkkkkkkkkk+-1+t2C2kkkkkkkkkkk-3696x6-7+3t2C1+-14+11t2C12kkkkkkkkkkkkkkkkkkk115+-7+3t2C2.
Third-Order Problem. Consider
(25)∂2u3x,t∂x2=1+C1∂2u2x,t∂x2+C2∂2u1x,t∂x2+C3∂2u0x,t∂x2kkkk-C1u0x,t∂2u2x,t∂t2-C1u2x,t∂2u0x,t∂t2kkkk-C1u1x,t∂2u1x,t∂t2-C2u0x,t∂2u1x,t∂t2kkkk-C2u1x,t∂2u0x,t∂t2-C3u0x,t∂2u0x,t∂t2kkkk+C3x22+t22-1,u30,t=0,∂∂xu30,t=0.
Its solution is
(26)u3x,t=1319334400·-12x2352t237800-12600x2kkkkkkkkkkkkkkkk+2310x4-135x6+7x8kkkkkkkkkkk+x22217600-1034880x2+108240x4kkkkkkkkkkkkkkkk-7612x6+215x8C12kkkk+-13305600x4+9313920x62867240kkkkkkk-1591920x8+195800x10-12586x12kkkkkkk+4922191x14-2867240x16-813x2t2kkkk·129729600-64864800x4-1904760x6kkkkkkki+166452x8-6604x10+231x12C13kkkk-12x2C1660168t260-10x2+x4kkkkkkkkkikkkkkkkk+x21680-392x2+15x4kkkkkkkkkkikkki+352t237800-12600x2kkkkkkkkkkkkkkkkkkkikkk+2310x4-135x6+7x8kkkkkkkkkkkkkkkkkk+x22217600-1034880x2kkkkkkkkkkkkkkkkkkkkkkki+108240x4-7612x6kkkkkkkkkkkkkkkkkkkkkkki+215x8C2kkkk-7920x2168t260-10x2+x4kkkkkkkkkkkkkk+x21680-392x2+15x4kkkk352t237800-12600x2·C2+C3.
Adding (20), (22), (24), and (26), we obtain
(27)ux,t,C1,C2,C3=u0x,t+u1x,t,C1+u2x,t,C1,C2.
The residual can be calculated by using (14). For calculations of the constants C1, C2, and C3, using (27) in (17) and applying the procedure mentioned in (13) and (14), we get
(28)C1=-1.0639118119872306,C2=-9.762978332188523×10-4,C3=1.287603572987478×10-4,ux,t=x20.5+3.12815×10-7x2kkkkkkk-1.0409×10-4x4+2.812×10-4x6kkkkkkk-2.52425×10-5x8+3.37375kkkkkkk×10-5x10-2.03977×10-6x12kkkkkkk+4.50493×10-8x14×t20.5+1.87689×10-3x2kkkkkkk-4.4610×10-4x4+1.53732kkkkkkk×10-3x6-1.38661×10-3x8kkkkkkk+2.28978×10-4x10-1.53259kkkkkkk×10-5x12+5.3608×10-7x14.
The Exact solution is [2]
(29)ux,t=x2+t22,
and HPM solution is [2]
(30)ux,t_HPM=t22+x22-x424+x82688+x6t2240-327x8241920+x8t21120-x10t221600+1038x103628800-x12107520.
4. Results and Discussions
The formulation presented in Section 2 provides accurate solutions for the problems demonstrated in Section 3. We have used Mathematica 7 for most of our computational work. In Table 1 and Figures 1, 2, 3, we have compared the OHAM results with the results obtained by HPM and Exact for various values of x and t at spatial domain [0,1] for Table 1 and at different values of x and fixed value of t=1 for Figures 1–3. In Table 2, we have presented absolute errors at different values of x and t. Figure 4 presents the residual at a spatial domain [0,1] at t=1. The convergence of OHAM is presented in Figure 5 at t=1.
Comparison of Exact, HPM, and OHAM solutions for different values of x and t.
x
t
Exact solution
HPM solution
OHAM solution
0
0
0.00000
0.000000
0.00000
0.062
0.162
0.015044
0.015044
0.015044
0.156
0.156
0.024336
0.024336
0.024336
0.225
0.128
0.033504
0.033505
0.033504
0.187
0.275
0.055297
0.055297
0.055297
0.281
0.281
0.078961
0.078961
0.0789608
0.343
0.343
0.117649
0.117650
0.117648
0.350
0.263
0.095834
0.095835
0.095834
0.468
0.468
0.219024
0.219032
0.219022
0.369
0.384
0.141801
0.141810
0.141808
0.350
0.870
0.439700
0.439706
0.439697
0.539
0.403
0.226465
0.226476
0.226463
0.780
0.111
0.310360
0.310263
0.310358
0.593
0.593
0.351649
0.351704
0.351647
0.620
0.685
0.426813
0.426913
0.426808
0.656
0.656
0.430336
0.430462
0.430332
0.781
0.781
0.609961
0.610499
0.609963
0.843
0.843
0.710649
0.711668
0.710652
0.968
0.968
0.937024
0.940295
0.936958
0.975
0.692
0.714744
0.716206
0.714690
1.000
1.000
1.000000
1.004310
0.999877
Comparison of absolute errors of HPM and OHAM solutions for different values of x and t.
x
t
E*
E**
0
0
0.00000000
0.00000000
0.062
0.162
6.003×10-12
1.74057×10-12
0.156
0.156
1.128×10-9
6.00903×10-9
0.225
0.128
2.613×10-9
2.49951×10-9
0.187
0.275
1.213×10-8
3.5136×10-8
0.281
0.281
1.275×10-7
1.91487×10-7
0.343
0.343
6.370×10-7
5.53487×10-7
0.350
0.263
3.309×10-7
4.03006×10-7
0.468
0.468
7.922×10-7
2.28841×10-7
0.369
0.384
1.273×10-6
8.50245×10-7
0.350
0.870
5.736×10-6
3.11566×10-6
0.539
0.403
1.122×10-6
2.45830×10-6
0.780
0.111
9.873×10-5
4.00013×10-6
0.593
0.593
5.497×10-5
4.1745×10-6
0.620
0.685
1.010×10-4
4.68453×10-6
0.656
0.656
1.263×10-4
3.64289×10-6
0.781
0.781
5.378×10-4
2.2877×10-6
0.843
0.843
1.019×10-3
2.6668×10-6
0.968
0.968
3.271×10-3
6.56401×10-5
0.975
0.692
1.461×10-3
5.41072×10-5
1.000
1.000
4.310×10-3
1.22591×10-4
E*=HPM-Exact, E**=OHAM-Exact.
3D plot for the OHAM solution at t=1.
3D plot for the Exact solution at t=1.
2D plot for the OHAM and Exact solutions at t=1.
2D plot for the residual of OHAM at t=1.
2D plot for the convergence of orders of OHAM solution at t=1.
5. Conclusion
In this paper, we studied the two-dimensional nonlinear wave equation. We have used the OHAM to approximate the solution of the titled problem. It is clear from the present work that OHAM can successfully be applied to nonlinear phenomena like the one we had. The obtained results are accurate which shows the effectiveness and validity of the proposed method. It is observed that OHAM is simpler in applicability and more convenient to control the convergence and involve less computational work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
HemedaA. A.Variational iteration method for solving wave equation2008568194819532-s2.0-5104908974610.1016/j.camwa.2008.04.010MR2466697GhasemiM.KajaniM. T.DavariA.Numerical solution of two-dimensional nonlinear differential equation by homotopy perturbation method2007189134134510.1016/j.amc.2006.11.1642-s2.0-34248392690MR2330212ColeJ. D.1968Waltham, Mass, USABlaisdell PublishingMR0246537NayfehA. H.MookD. T.1979New York, NY, USAJohn Wiley & SonsMR549322LaioS. J.1992Shanghai Jiao Tong UniversityLiaoS. J.A kind of approximate solution technique which does not depend upon small parameters199732815822LiaoS. J.A new branch of solutions of boundary-layer flows over a stretching flat plate20054925292539LiaoS.-J.A new branch of solutions of boundary-layer flows over a permeable stretching plate200742681983010.1016/j.ijnonlinmec.2007.03.007MR23287352-s2.0-34547506935HeJ.-H.Homotopy perturbation technique19991783-425726210.1016/S0045-7825(99)00018-3MR1711041HeJ.-H.Homotopy perturbation method for bifurcation of nonlinear problems20056220720810.1515/IJNSNS.2005.6.2.207MR3110161HeJ.-H.Asymptotology by homotopy perturbation method2004156359159610.1016/j.amc.2003.08.011MR20881252-s2.0-4344696077HeJ.-H.Application of homotopy perturbation method to nonlinear wave equations200526369570010.1016/j.chaos.2005.03.0062-s2.0-18844426016HeJ.-H.The homotopy perturbation method for nonlinear oscillators with discontinuities20041511287292MR20379672-s2.0-124228758710.1016/S0096-3003(03)00341-2GanjiD. D.SadighiA.Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations2006744114182-s2.0-33748919061GanjiD. D.RajabiA.Assessment of homotopy-perturbation and perturbation methods in heat radiation equations200633339140010.1016/j.icheatmasstransfer.2005.11.0012-s2.0-33144478462GanjiD. D.NourollahiM.MohseniE.Application of He's methods to nonlinear chemistry problems2007547-81122113210.1016/j.camwa.2006.12.0782-s2.0-34748831014MR2398134DomairryG.NadimN.Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation2008351931022-s2.0-3704902840210.1016/j.icheatmasstransfer.2007.06.007LiangS. X.JeffreyD. J.Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation2009141240574064MR25375662-s2.0-6734916253810.1016/j.cnsns.2009.02.016MarincaV.HerişanuN.NemeşI.Optimal homotopy asymptotic method with application to thin film flow20086364865310.2478/s11534-008-0061-x2-s2.0-48349119574MarincaV.HerişanuN.Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method2010606160716152-s2.0-7795605881310.1016/j.camwa.2010.06.042MR2679127MarincaV.HerişanuN.Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method201032991450145910.1016/j.jsv.2009.11.0052-s2.0-74149085248UllahH.IslamS.IdreesM.ArifM.Solution of boundary layer problems with heat transfer by optimal homotopy asymptotic method2013201310MR3102722NawazR.UllahH.IslamS.IdreesM.Application of optimal homotopy asymptotic method to Burger equations20132013838747810.1155/2013/387478MR3082034UllahH.IslamS.IdreesM.NawazR.Application of optimal homotopy asymptotic method to doubly wave solutions of the coupled Drinfel'd-Sokolov-Wilson equations20132013836281610.1155/2013/362816MR3139223UllahH.IslamS.IdreesM.FizaM.Solution of the differential-difference equations by optimal homotopy asymptotic method20142014752046710.1155/2014/520467MR3216056UllahH.IslamS.KhanI.ShardianS.FizaM.AbdelhameedT. N.Approximate solution of the generalized coupled Hirota-Satsuma coupled KdV equation by extended optimal homotopy asymptotic method20142730223036