Journal of Quality Measurement and Analysis JQMA 4(1) 2008, Jurnal Pengukuran Kualiti dan Analisis

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1 Journal of Quality Measureent and Analysis JQMA 4() 8, Jurnal Pengukuran Kualiti dan Analisis APPROXIMATE ANALYTICAL SOLUTIONS OF THE KLEIN-GORDON EQUATION BY MEANS OF THE HOMOTOPY ANALYSIS METHOD (Penyelesaian Analisis Penghapiran bagi Persaaan Klein-Gordon Menggunakan Kaedah Analisis Hootopi) A.K. ALOMARI, MOHD SALMI MD NOORANI & ROSLINDA MOHD NAZAR ABSTRACT In this paper, the hootopy analysis ethod (HAM) is ipleented to give approxiate and analytical solutions for the Klein Gordon equation. The auxiliary paraeter in the HAM solutions has provided a convenient way of controlling the convergent region of series solutions. This proble shows rapid convergence of the sequence constructed by this ethod to the exact solution. Moreover, this technique reduces the volue of calculations by avoiding discretization of the variables, linearization or sall perturbations. Keywords: Klein Gordon equation; hootopy analysis ethod; analytical solutions ABSTRAK Dala akalah ini, kaedah analisis hootopi telah digunakan untuk enghasilkan penyelesaian analisis dan hapiran bagi persaaan Klein-Gordon. Paraeter bantu dala penyelesaian kaedah analisis hootopi eberikan satu cara yang udah untuk engawal rantau penupuan bagi penyelesaian siri tersebut. Masalah yang dipertibangkan ini enunjukkan penupuan yang pantas kepada penyelesaian tepat bagi jujukan yang dibentuk oleh kaedah ini. Di saping itu, teknik ini juga engurangkan julah pengiraan yang banyak tanpa elakukan pendiskretan peboleh ubah, pelinearan atau usikan kecil. Kata kunci: persaaan Klein-Gordon; kaedah analisis hootopi; penyelesaian analisis. Introduction Nonlinear phenoena that appear in any areas of scientific fields such as solid state physics, plasa physics, fluid dynaics, atheatical biology and cheical kinetics can be odeled by partial differential equations. A broad class of analytical solutions ethods and nuerical solutions ethods were used to handle these probles (Wazwaz 6). In this paper, we consider the Klein Gordon equation u u + bu+ b g() u = f(,) x t, () tt xx u is a function of x and t, g is a nonlinear function, and f is a known analytic function. The Klein Gordon and sine-gordon equations odel any probles in classical and quantu echanics, solitons and condensed atter physics (Abbasbandy 6; Caudrey et al. 975; Cveticanin 5; Deeba & Khuri 5; Dodd et al. 98; El-Sayed 3; Isail & Sarie 989). The approxiate analytical solution to () was presented by Deeba and Khuri (5) using the analytic Adoian decoposition ethod (ADM). Wazwaz (6) presented odified Adoian decoposition ethod (MADM) to solve non-linear Klein-Gordon equation.

2 A.K. Aloari, Mohd Sali Md Noorani & Roslinda Mohd Nazar Abbasbandy (6) used variational iteration ethod (VIM) to get the solution of Klein- Gordon equation. Recently, Chowdhury and Hashi (7) used hootopy-perturbation ethod (HPM) to obtain the solution of the equation. Another powerful analytical ethod, called the hootopy analysis ethod (HAM), was first envisioned by Liao (3). Recently this ethod has been successfully eployed to solve any types of probles in science and engineering (Ayub et al. 3; Bataineh et al. 7; Liao 995, 997, 4, 5; Hayat et al. 4; Hayat & Sajid 7). Hootopy analysis ethod contains an auxiliary paraeter which provides us with a siple way to adjust and control the convergence region and the rate of convergence of the series solution. The ai of the present work is to effectively eploy the HAM to establish exact or approxiate solutions for the Klein-Gordon equation. The exaples illustrated in this present paper have not been exactly solved before using HAM. Coparison of the present ethod and the ADM is also presented in this paper.. Basic Idea of HAM In this paper, we apply the HAM to the four probles to be discussed. In order to show the basic idea of HAM, consider thefollowing differential equation: Nuxt [ (, )] =, N is a nonlinear operator, x and t denote the independent variables and u is an unknown function. For siplicity, we ignore all boundary or initial conditions, which can be treated in the siilar way. By eans of the HAM, we first construct the so-called zeroth-order deforation equation ( ql ) [ φ( xtq, ; ) u( xt, )] = q N[ φ( xtq, ; )], () q [,] is the ebedding paraeter, is an auxiliary paraeter, L is an auxiliary linear operator, φ (,; xtq) is an unknown function and u (,) xt is an initial guess of uxt (,). It is obvious that when the ebedding paraeter q = and q =, Eq. () becoes φ( xt, ;) = ux (,), φ( xt, ;) = uxt (, ), respectively. Thus as q increases fro to, the solution φ (,; xtq) varies fro the initial guess u (,) xt to the solution uxt (,). Expanding φ (,; xtq ) in Taylor series with respect to q, one has φ( xtq, ; ) = u( xt, ) + u ( xtq, ), (3) + = φ(,; xtq) u (,) xt =! q q=. (4) 46

3 Analytical solutions of the Klein-Gordon equation by eans of the hootopy analysis ethod The convergence of the above series depends upon the auxiliary paraeter. If it is convergent at q =, one has + uxt (,) = u(,) xt + u (,) xt, = which ust be one of the solutions of the original nonlinear equation, as proven by Liao (3). Define the vectors u(,) xt = { u(,), xt u(,), xt, u(,)} xt. n n Differentiating the zeroth-order deforation equation -ties with respect to q and then dividing the by! and finally setting q =, we get the following th-order deforation equation: Lu [ (,) xt u (,)] xt H(,) xt R( u ), (5) χ = R ( u ) = ( )! Nφ xtq [ (, ; )] q q=, (6) and χ,, =, >. It should be ephasized that u(,) xt for is governed by the linear th-order deforation equation with linear boundary conditions that coe fro the original proble, which can be solved by the sybolic coputation softwares such as Maple and Matheatica. 3. Applications In this part, we will apply the HAM to the linear and nonlinear Klein-Gordon equation. In the first three exaples, we define the linear operator as φ(,; xtq) L[ φ( xtq, ; )] =, (7) with the property LC [ ( xt ) + C( x)] =, Ci ( i=, ) are integral constants. 47

4 A.K. Aloari, Mohd Sali Md Noorani & Roslinda Mohd Nazar Exaple Consider the linear for of Klein-Gordon equation u u = u, (8) tt xx subject to the initial conditions ux (,) = + sin( x), u) =. (9) t To solve Eqs. (8) and (9) by eans of the hootopy analysis ethod, we choose the initial approxiation u( xt, ) = ux (,) = + sin( x). Further, Eq. (8) suggests that we define the nonlinear operator as φ(,; xtq) φ(,; xtq) N[ φ( xtq, ; )] = φ( xtq, ; ). Using the above definition, we construct the zeroth-order deforation equation ( ql ) [ φ( xtq, ; ) u( xt, )] = q N[ φ( xtq, ; )]. () Obviously, when q = and q =, φ( xt, ;) = ux (,), φ( xt, ;) = uxt (, ). Therefore, as the ebedding paraeter q increases fro to, φ (,; xtq) varies fro the initial guess u(,) xt to the solution u(x, t). Expanding φ(x,t;q) in Taylor series with respect to q one has φ( xtq, ; ) = u( xt, ) + u ( xtq, ), + = φ(,; xtq) u (,) xt =! q q=. If the auxiliary linear operator, the initial guess and the auxiliary paraeter h are properly chosen, the above series is convergent at q =, then one has 48

5 Analytical solutions of the Klein-Gordon equation by eans of the hootopy analysis ethod + uxt (,) = u(,) xt + u (,) xt, = which ust be one of the solutions of the original nonlinear equation, as proven by Liao (3). Define the vectors u(,) xt = { u(,), xt u(,), xt, u(,)}. xt n n Differentiating the zeroth-order deforation equation -ties with respect to q and then dividing the by! and finally setting q=, we get the following th-order deforation equation: Lu [ ( xt, ) u ( xt, )] R ( u ), () χ = with the boundary conditions u ) =, u ) =, t u u R ( u ) = u. Now, the solution of the th-order deforation equation () for becoes u ( xt, ) = χ u ( xt, ) + L [ R( u )]. We now successively obtain u (,) xt = t u(,) xt = t t t 4 4 u3(,) xt = t + t + t t t t Then the series solution expression can be written in the for uxt (,) = u(,) xt + u(,) xt + u(,) xt +. The first four ters of the series solution when = are 49

6 A.K. Aloari, Mohd Sali Md Noorani & Roslinda Mohd Nazar u t ) := + sin( x ) u t) := t u t) u 3 t) := := t 4 4 t 6 7 Finally, the approxiate solution in a series for is 4 t t uxt (, ) sin( x ) ,! 4! and this will, in the liit of infinitely any ters, yield the closed-for solution uxt (, ) sin( x) + cosh( t), which is the exact solution. Exaple Consider the linear nonhoogeneous Klein Gordon equation u u u = sin( x)sin( t), (3) tt subject to initial conditions xx ux (, ) =, u ) = sin( x). (4) t To solve Eqs. (3) and (4) by eans of the hootopy analysis ethod, we choose the initial approxiation u( xt, ) = ux (, ) = tsin( x). Further, Eq. (3) suggests that we define the nonlinear operator as φ(,; xtq) φ(,; xtq) N[ φ( xtq, ; )] = φ( xtq, ; ) + sin( x)sin( t). We apply the th-order deforation equation with the boundary conditions u ) =, u ) =, t 5

7 Analytical solutions of the Klein-Gordon equation by eans of the hootopy analysis ethod u u R ( u ) = u + sin( x)sin( t). Now, the solution of the th-order deforation equation for becoes u ( xt, ) = χ u ( xt, ) + L [ R( u )]. We now successively obtain u xt xt x t t x 6 3 (, ) = sin( ) sin( )sin( ) + sin( ), u xt = xt x t 6 xt x t 5 + sin( x) t + tsin( x) + 4 tsin( x), 3 3 (, ) sin( ) sin( )sin( ) sin( ) 4 sin( )sin( ) 3 3 u( xt, ) = sin( xt ) sin( x)sin( t) sin( xt ) 8 sin( x)sin( t) sin( xt ) sin( xt ) 8 sin( x)sin( t) + sin( xt ) sin( x) t + tsin( x) + 8 tsin( x) + 8 tsin( x), Then the series solution expression can be written in the for uxt (,) = u(,) xt + u(,) xt + u(,) xt +. The first four ters of the series solution when = are u t ) := sin( x ) t 3 + sin( x ) sin( t) t sin( x ) 6 u t ) := sin( x ) t 5 sin( x ) sin( t) sin( x ) t 3 + t sin( x ) 3 u 3 t ) := sin( x ) t 7 + sin( x ) sin( t) sin( x ) t 5 + sin( x ) t 3 t sin( x )

8 A.K. Aloari, Mohd Sali Md Noorani & Roslinda Mohd Nazar u 4 t) sin( x ) t 9 sin( x ) sin( t) sin( x ) t 7 sin( x ) t 5 := t sin( x ) sin( x ) t 3 Hence, the 5-ter approxiate series solution can be written as t t t t uxt (, ) sin( x) t + +, 3! 5! 7! 9! and this will, in the liit of infinitely any ters, yield the closed-for solution uxt (, ) = sin( x)sin( t). Exaple 3 Consider the nonlinear Klein Gordon equation u u u =, (5) tt xx subject to initial conditions ux (,) = + sin( x), u) =. (6) t To solve Eqs. (5) and (6) by eans of the hootopy analysis ethod, we choose the initial approxiation u( xt, ) = ux (,) = + sin( x). Further, Eq. (5) suggests that we define the nonlinear operator as φ( xtq,; ) φ( xtq,; ) N xtq xtq [ φ(, ; )] = + φ (, ; ). We apply the th-order deforation equation with the boundary conditions u ) =, u ) =, t u u R ( u ) uu. = + j j j= Now, the solution of the th-order deforation equation for becoes u ( xt, ) = χ u ( xt, ) + L [ R( u )]. 5

9 Analytical solutions of the Klein-Gordon equation by eans of the hootopy analysis ethod We now successively obtain u( xt, ) = t (3sin( x) + cos( x) ) u( xt, ) = t( 3sin( xt ) + sin( xt ) cos( x) 36sin( x) 36sin( x) t cos( x) cos( x) cos( x) 4 t ) u( xt, ) = t(sin( xt ) cos( x) + t sin( x) cos( x) sin( ) cos( ) 39 sin( ) 6 sin( ) 78 sin( ) + t x x t x x t x 8 sin( x) 78 sin( xt ) 8 sin( x) + 7 cos( x) + 36 cos( x) 7 cos( ) t x t t t t cos( x) 8t cos( x) 7t cos( x) 36 cos( x) ). So uxt (,) u(,) xt + u(,) xt + u(,) xt + u(,). xt 3 Exaple 4 Finally, we consider the nonlinear nonhoogeneous Klein Gordon equation u u u x t x t tt xx + = cos( ) + cos ( ), (7) subject to initial conditions ux (,) = x, u) =. (8) t To solve Eqs. (7) and (8) by eans of hootopy analysis ethod, fro the freedo to choosing the linear operator in HAM, we choose the linear poperator as φ(,; xtq) L[ φ(,; xtq)] = + φ(,; xtq), (9) with the property LC [ ( x)sin( t) + C( x) cos( t)] =. Further, Eq. (7) suggests that we define the nonlinear operator as 53

10 A.K. Aloari, Mohd Sali Md Noorani & Roslinda Mohd Nazar φ(,; xtq) φ(,; xtq) N xtq xtq x t x t [ φ(, ; )] = + φ (, ; ) + cos( ) cos ( ). Also we can solve the zeroth-order deforation equation () using the linear operator (9) under initial conditions u( xt, ) = xcos( t), u ( xt, ) =. t Now we succefully have u ( xt, ) = x cos( t ). We apply the th-order deforation equation with the boundary conditions u ) =, u ) =, t u u R u uu x t x x ( ) = + cos( ) cos ( ). + j j j= Now, the solution of the th-order deforation equation for becoes u ( xt, ) = χ u ( xt, ) + L [ R( u )]. We now successively obtain u(,) xt =, u(,) xt =, u(,) xt =. 3 Siilarly, higher order solutions are also zero which then yields the exact solution uxt (, ) = xcos( t). 4. Coparison and Discussion In this part we plot the -curves as presented in Figures to 3 for Exaples to 3, respectively. In all of these exaples we show that = - is in the convergent region, also in this case, when = -, we have the ADM and HPM solutions for Exaples and ; which eans the ADM and HPM solutions are special cases of the HAM solution. For Exaple 3, coparison between ADM and HAM is done in Figures 4 and 5. In Exaple 4, we found that HAM is an effective ethod to obtain the initial guess to get the exact solution. 54

11 Analytical solutions of the Klein-Gordon equation by eans of the hootopy analysis ethod Figure : The -curve of -approxiation for Exaple Figure : The -curve of -approxiation for Exaple Figure 3: The -curve of -approxiation for Exaple 3 55

12 A.K. Aloari, Mohd Sali Md Noorani & Roslinda Mohd Nazar Figure 4: The ADM solution for Exaple 3 under 4th-order approxiation Figure 5: The HAM solution for Exaple 3 under 4th-order approxiation 5. Conclusions In this paper, the standard hootopy analysis ethod (HAM) has been successfully eployed to obtain the approxiate analytical solutions of the Klein Gordon equation. In coparison to the Adoian decoposition ethod (ADM), HAM avoids the difficulties arising in finding the Adoian polynoials. In addition, the calculations involved in HAM are siple and straightforward. It is shown that the HAM is a proising tool for both linear and nonlinear partial differential equations. Acknowledgeent The financial supports received fro the Ministry of Science, Technology and Innovation, Malaysia under the grant Sciencefund 4---SF77 and the fundaental research grant (FRGS) UKM-ST--FRGS58-6 fro the Ministry of Higher Education, Malaysia are gratefully acknowledged. 56

13 Analytical solutions of the Klein-Gordon equation by eans of the hootopy analysis ethod References Abbasbandy S. 6. Nuerical solution of non-linear Klein Gordon equations by variational iteration ethod. Int. J. Nuer. Meth. Engng. 7: Ayub M., Rasheed A. & Hayat T. 3. Exact flow of a third grade fluid past a porous plate using hootopy analysis ethod. Int. J. Engng. Sci. 4: 9 3. Bataineh A.S., Noorani M.S.M. & Hashi I. 7. Solving systes of ODEs by hootopy analysis ethod, Co. Nonlinear Sci. Nuer. Siul. doi:.6/j.cnsns (in-press). Caudrey P.J., Eilbeck I.C. & Gibbon J.D The sine-gordon equation as a odel classical field theory. Nuovo Ciento 5: Chowdhury M.S.H. & Hashi I. 7. Application of hootopy-perturbation ethod to Klein Gordon and sine- Gordon equations. Chaos Solitons Fractals doi:.6/j.chaos (in-press). Cveticanin L. 5. The hootopy-perturbation ethod applied for solving coplex-valued differential equations with strong cubic nonlinearity. J Sound Vib. 85: Deeba E.Y. & Khuri S.A. 5. A decoposition ethod for solving the nonlinear Klein Gordon equation. J. Coput. Phys. 4: Dodd R.K., Eilbeck I.C. & Gibbon J.D. 98. Solitons and Nonlinear Wave Equations. London: Acadeic Press. Isail M.S. & Sarie T Spline difference ethod for solving Klein Gordon equations. Dirasat 4: El-Sayed S. 3. The decoposition ethod for studying the Klein Gordon equation. Chaos Solitons Fractals 8: 5 3. Hayat T. & Sajid M. 7. On analytic solution for thin fil flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 36: Hayat T., Khan M. & Asghar S. 4. Hootopy analysis of MHD flows of an Oldroyd 8-constant fluid. Acta Mech. 67: 3 3. Liao S.J An approxiate solution technique which does not depend upon sall paraeters: a special exaple. Int. J. Nonlinear Mech. 3: Liao S.J An approxiate solution technique which does not depend upon sall paraeters (Part ): an application in fluid echanics. Int. J. Nonlinear Mech. 3:85 8. Liao S.J. 3. Beyond Perturbation: Introduction to the Hootopy Analysis Method. Boca Raton: CRC Press. Liao S.J. 4. On the hootopy analysis ethod for nonlinear probles. Appl. Math. Coput. 47: Liao S.J. 5. Coparison between the hootopy analysis ethod and hootopy perturbation ethod. Appl. Math. Coput. 69: Wazwaz A.M. 6. The odified decoposition ethod for analytic treatent of differential equations. Appl. Math. Coput. 73: School of Matheatical Sciences Faculty of Science & Technology Universiti Kebangsaan Malaysia 436 UKM Bangi Selangor D.E. MALAYSIA E-ail: abdoari8@yahoo.co, sn@uk.y, rn@uk.y * * Corresponding author 57