Homotopy Perturbation Method for Solving Twelfth Order Boundary Value Problems

Transcription

1 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) Homotopy Perturbation Metho for Solving Twelfth Orer Bounary Value Problems H.Mirmorai *, H. Mazaheripour, S.Ghanbarpour, A.Barari Department of Civil Engineering, University of Mazanaran, Babol, Iran ABSTRACT Approximate analytical solutions are obtaine using homotopy perturbation metho (HPM), which is a coupling of traitional perturbation metho an the homotopy metho. Two examples are presente to illustrate the effectiveness of HPM for solving twelfth orer bounary value problems. In orer to achieve aequately precise results with HPM, it is usually require to calculate at least two statements of the p- terms. However, it was shown in the numerical examples that highly accurate results were obtaine by calculating only one p-term of the series, revealing the effectiveness of the HPM solution. It was conclue that HPM is a powerful tool for solving high-orer bounary value problems arising in various fiels of engineering an science. KEYWORDS: Twelfth orer bounary value problems, Approximate analytical solution, Homotopy perturbation metho 1. Introuction Many important phenomena occurring in various fiels of engineering an science are frequently moele through linear an nonlinear ifferential equations. However, it is still very ifficult to obtain close form solutions for most moels of real life problems. Approximate analytical solutions for solving such problems have been gaining popularity in the recent years. The perturbation metho a numerical metho use for solving linear an nonlinear problems - is base on assuming a small parameter. The majority of nonlinear problems however, especially those having strong nonlinearity, have no small parameters at all an the approximate solutions obtaine by the perturbation methos, in most cases, are vali only for small values of the small parameter. Generally, the perturbation solutions are vali as long as the scientific system parameter is small. However, we cannot rely fully on the approximations, because there is no criterion on which the small parameter shoul exist. Thus, it is essential to check the valiity of the approximations numerically an/or experimentally. To overcome these ifficulties, a moifie form of the perturbation metho calle homotopy perturbation metho (HPM) was introuce by He [1], followe by a thorough escription of the metho where it was verifie that the metho is, in fact, a coupling between the traitional perturbation metho an the homotopy metho [2]. HPM has since then been effectively utilize in obtaining approximate analytical solutions to many linear an nonlinear problems arising in engineering an science, such as nonlinear oscillators with iscontinuities [3], nonlinear wave equations [4,5], Burgers an couple Burgers equations [6], general bounary value problems [7], Helmholtz equation an fifth orer KDV equation [8], Axisymmetric flow over a stretching sheet [9], thin film flow of a fourth grae flui own a vertical cyliner [1], nonlinear Volterra Freholm integral equations [11], nonlinear fourth orer bounary value problem [], Ratio-Depenent Preator-Prey System with Constant Effort Harvesting [13]. Twelfth orer ifferential equations have several important applications in engineering. Chanrasekhar [14] showe that when an infinite horizontal layer of flui is put into rotation an simultaneously subjecte to heat from below an a uniform magnetic fiel across the flui in the same 163

2 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) irection as gravity, instability will occur. It was then shown that if instability sets in as overstability, the problem is governe by a twelfth orer ifferential bounary value equation. Such problems arise in geophysics when stuying core flui ajacent to the core-mantle bounary. Fearn an Richarson [15] use a cylinrical moel an a twelfth orer bounary value problem to stuy the combine effects of a stably stratifie layer an a spatially varying magnetic fiel on thermally riven convection. Sachev et al. [16] stuie flow in a paraboloial basin or an ey boune by a free surface using simplifie equations of a twelfth egree nonlinear orinary ifferential equation. In plate eflection theory, several researchers consiere the problem of transverse shear an normal strain an stress effects on antisymmetric bening of isotropic plates by eriving twelfth orer partial ifferential equations [17-2]. Vekua [21] showe that twelfth orer ifferential equations also arise in stuying elastic shells having variable thickness. Several researchers evelope numerical techniques for solving twelfth orer ifferential equations. For example, Twizell et al. [22] evelope finite ifference methos for solving high-orer eigenvalue problems arising in thermal instability; Islam et al. [23] use ifferential transform metho; Siiqi an Twizell [24] use polynomial splines of twelfth egree; Wazwaz [25] use the moifie ecomposition metho. As state above, HPM will be use herein to solve twelfth egree bounary value problem of the general form: () y ( x) + f ( x) y( x) g( x), x [ a, b], y( a) α, y( b) α1, (1) (1) y ( a) γ, y ( b) γ1, (2) (2) y ( a) δ, y ( b) δ1, (1) (3) (3) y ( a) ν, y ( b) ν1, (4) (4) y ( a) ξ, y ( b) ξ1, (5) (5) y ( a) ω, y ( b) ω, 1 where α i, γ i, δ i, ν i, ξ i an ω i ; i, 1 are finite real constants an the functions f(x) an g(x) are continuous on [a,b]. 2. Basic iea of Homotopy-perturbation metho To explain this metho, let us consier the following function: Au ( ) f( r), r Ω (2) with the bounary conitions of: u Bu (, ), r Γ, n where A, B, f ( r ) an Γ are a general ifferential operator, a bounary operator, a known analytical function an the bounary of the omain Ω, respectively. Generally speaking, the operator A can be ivie into a linear part L an a nonlinear part N ( u ). Eq. (2) may therefore be written as: Lu ( ) + N( u) f( r) (4) By the homotopy technique, we construct a homotopy (3) 164

3 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) v( r, p): Ω [,1] R which satisfies: [ ] [ ] Hv (, p) (1 p) Lv ( ) Lu ( ) + pav ( ) f( r), p [,1], r Ω, (5) or [ ] Hv (, p) Lv ( ) Lu ( ) + plu ( ) + pnv ( ) f( r) (6) where p [,1] is an embeing parameter, while u is an initial approximation of Eq.(2), which satisfies the bounary conitions. Obviously, from Eqs. (5) an (6) we will have: Hv (,) Lv ( ) Lu ( ), (7) Hv (,1) Av ( ) f( r), (8) The changing process of p from zero to unity is just that of v( r, p ) from u to ur ( ). In topology, Lv ( ) Lu ( ) an Av ( ) f( r) this is calle eformation, while are calle homotopy. Accoring to the HPM, we can first use the embeing parameter p as a small parameter, an assume that the solutions of Eqs. (5) an (6) can be written as a power series in p: v v + pv + p v + (9) Setting p 1 yiels in the approximate solution of Eq. (9) to: u lim v v 1 + v1+ v 2 + L p (1) The combination of the perturbation metho an the homotopy metho is calle the HPM, which eliminates the rawbacks of the traitional perturbation methos while keeping all its avantages. The series (1) is convergent for most cases. However, the convergent rate epens on the nonlinear operator Av ( ). Moreover, He [1] mae the following suggestions: (1) The secon erivative of N ( v) with respect to v must be small because the parameter may be relatively large, i.e. p 1. (2) The norm of L 1 N v must be smaller than one so that the series converges. 3. Numerical examples In orer to illustrate the ability of HPM in solving twelfth orer problems, we consier the following two examples herein: 165

4 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) Example 1. For x [,1], consier the following bounary value problem ( ) + ( ) ( ), () 3 x y x xy x x x e y(), y(1) (1) (1) y () 1, y (1) e, (2) (2) y (), y (1) 4 e, (3) (3) y () 3, y (1) 9 e, (4) (4) y () 8, y (1) 16e (5) (5) y () 15, y (1) 25 e, (11) The analytic solution of the above ifferential system is x yx ( ) x(1 xe ), () In orer to solve Eq. (11) by the HPM, a homotopy can be constructe as follows: H(, v p) (1 p)( v() x v ( x) ) + 3 p( v( x) + xv( x) + (+ 23 x+ x ) e x ), Substituting v v + pv in to Eq. (13) an rearranging the resultant equation base on powers of p- terms, one has: : ( ), (14) p v x 1 x x 3 x p : ( v 1( x)) + e + xv( x) + e x + 23xe, (15) p 2 :( v 2( x)) + xv1( x), (16) With the following conitions: (13) 166

5 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) v(), v(1), v() 1, v(1) e, 2 2 v 2 (), v 2 (1) 4 e, 3 3 v 3 () 3, v 3 (1) 9e 4 4 v 4 () 8, v 4 (1) 16e 5 5 v 5 () 15, v 5 (1) 25e vi(), vi(1), vi (), vi (1), 2 2 v (), (1), 2 i v 2 i 3 3 v (), (1) 1,2,... 3 i v 3 i i 4 4 v (), (1) 4 i v 4 i 5 5 v (), (1) 5 i v 5 i (17) With the effective initial approximation for v from the conitions (17), solution of Eq. (14) may be written as follows: v ( x) x x x x x.333x.5x 4 3 x x x In the same manner, the rest of components were obtaine using the Maple package. Accoring to the HPM, we can conclue that: (18) ux ( ) lim vx ( ) v( x) + v( x) +..., p 1 1 (19) Therefore, substituting the value of v ( ) x from Eq. (18) in to Eq. (19) yiels: 167

6 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) ux ( ) x x x (2) x x.333x.5x x x x Comparison of the approximate solutions with exact solution is tabulate in Table. 1 along with the errors of the HPM, showing a remarkable agreement. It is noteworthy that even higher accuracy coul be achieve by calculating secon an thir terms from Eqs. (15) an (16). Table 1 Comparison between exact solution of Eq. (11) an the solution from HPM x Exact solution HPM Error of HPM...E E E E E E E E E E E E-7 Example 2. For x [ 1,1], consier the following bounary value problem () y x y x x x + x ( ) ( ) (2 cos( ) 11sin( )), y( 1) y(1) (1) (1) y ( 1) y (1) 2sin(1), (2) (2) y ( 1) y (1) 4 cos(1) 2sin(1), (3) (3) y ( 1) y (1) 6cos(1) 6sin(1), (4) (4) y ( 1) y (1) 8 cos(1) + sin(1), (5) (5) y ( 1) y (1) 2 cos(1) + 1 sin(1), (21) 168

7 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) The analytic solution of the above ifferential system is 2 yx ( ) ( x 1)sinx (22) A homotopy can be constructe as follows: H(, v p) (1 p)( v() x v ( x) ) + p( v( x) v( x) + 24xcos( x) + 132sin( x)), (23) Substituting v v + pv in to Eq. (23) an rearranging the resultant equation base on powers of p- terms, one has: : ( ), p v x (24) : ( ( )) + 24 cos( ) ( ) + 132sin( ), (25) 1 p v 1 x x x v x x p 2 :( v 2( x)) v1( x), (26) With the following conitions(see next page): 169

8 International Journal of Research an Reviews in Applie Sciences v( 1) v(1), ISSN: X, EISSN: Volume 1, Issue 2(November 29) v( 1) v(1) 2 sin(1), 2 2 v 2 ( 1) v 2 (1) 4 cos(1) 2sin(1) 3 3 v 3 ( 1) v 3 (1) 6 cos(1) 6sin(1) 4 4 v 4 ( 1) v 4 (1) 8 cos(1) sin (1) 5 5 v 5 ( 1) v 5 (1) 2 cos(1) + 1sin(1) (27) vi( 1), vi(1), vi( 1), vi(1), 2 2 v ( 1), (1), 2 i v 2 i 3 3 v ( 1), (1) 1, 2,... 3 i v 3 i i 4 4 v ( 1), (1) 4 i v 4 i 5 5 v ( 1), (1) 5 i v 5 i with the effective initial approximation for v from the conitions (27) an solution of Eq. (24) may be written as follows: v ( x) x x x x x x 4 3 x x x In the same manner, the rest of components were obtaine using the Maple package. Accoring to the HPM, we can conclue that: (28) ux ( ) lim vx ( ) v( x) + v( x) +..., p 1 1 (29) Therefore, substituting the value of v ( ) x from Eq. (28) in to Eq. (29) yiels: ux ( ) x x x x x x (3) Comparison of the approximate solutions with exact solution is tabulate in Table. 2 along with errors of HPM, revealing the high accuracy of the results from HPM. Once again, as state in example 1, it is obvious that higher accuracy coul be obtaine without any ifficulty. 17

9 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) Table 2 Comparison between exact solution of Eq. (21) an the solution from HPM x Exact solution HPM Error of HPM E-9-1.6E E E E E-9...E E E E E E-9 1.6E-9 4. Conclusion In this paper, solution of twelfth orer bounary value problem has been given using homotopy perturbation metho. The numerical examples consiere reveale that homotopy perturbation metho is both accurate an effective for solving twelfth orer bounary value problems, since highly accurate results were obtaine in both examples by calculating a single p-term. It can be conclue that homotopy perturbation metho is a highly efficient metho for solving high-orer bounary value problems arising in various fiels of engineering an science. References [1] He, J.H. (1999). Homotopy Perturbation Technique. Comput. Meth. Applie Mech. Eng., 178(3, 4): [2] He, J.H. (2). A Coupling Metho of a Homotopy Technique an a Perturbation Technique for Non- Linear Problems. Inl. J. Non-Linear Mech., 35(1): [3] He, J.H. (24). The Homotopy Perturbation Metho for Non-Linear Oscillators with Discontinuities. Appl. Math. Comput., 151(1): [4] He, J.H. (25). Application of Homotopy Perturbation Metho to Nonlinear Wave Equations. Chaos, Solitons Fractals., 26(3):

10 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) [5] Barari, A., Ghotbi, Aboul R., Farrokhza, F. an Ganji, D.D. (28). Variational iteration metho an Homotopy-perturbation metho for solving ifferent types of wave equations. J.Applie Sci., 8(1): - 6. [6] Ghotbi, Aboul R., Avaei, A., Barari, A. an Mohammazae, M. A. (28). Assessment of He s homotopy perturbation metho in Burgers an couple Burgers equations, J. Applie Sci., 8(2): [7] He, J.H. (26). Homotopy Perturbation Metho for Solving Bounary Problems, Phys. Lett. A., 35(1 2): [8] Rafei, M. an Ganji, D.D. (26). Explicit Solutions of Helmholtz Equation an Fifth-Orer KV Equation using Homotopy Perturbation Metho. Int. J. Nonlinear Sci. Numer. Simul., 7(3): [9] Ariel, P.D., Hayat, T. an Asghar, S. (26). Homotopy Perturbation Metho an Axisymmetric Flow over a Stretching Sheet. Int. J. Nonlinear Sci. Numer. Simul., 7(4): [1] Siiqui, A.M., Mahmoo, R. an Ghori, Q.K. (26). Homotopy Perturbation Metho for Thin Film Flow of a Fourth Grae Flui own a Vertical Cyliner. Phys. Lett. A, 352(4 5): [11] Ghasemi, M., Tavassoli Kajani,M. an Babolian, E. (27). Numerical Solution of the Nonlinear Volterra Freholm Integral Equations by using Homotopy Perturbation Metho. Appl. Math. Comput., 188(1): [] S. H. Mirmorai, S. Ghanbarpour, I. Hosseinpour, A. Barari, (29). Application of homotopy perturbation metho an variational iteration metho to a nonlinear fourth orer bounary value problem. Int. J.Math. Analysis., 3(23): [13] Ghotbi, Aboul R., Barari, A. an Ganji, D. D, (28). Solving Ratio-Depenent Preator-Prey System with Constant Effort Harvesting Using Homotopy Perturbation Metho. Math. Prob.Eng., Article ID 94542, 1-8. [14] Chanrasekhar, S. (1961). Hyroynamic an Hyromagnetic Stability. Clarenon Press, Oxfor, Reprinte: Dover Books, New York, [15] Fearn, D.R. an Richarson, L. (1991). Convection in a Non-Uniformly Stratifie Flui Permeate By a Non-Uniform Magnetic Fiel. Geophysical Journal Int., 14(1): [16] Sachev, P.L., Palaniappan. D. an Sarathy, R. (1996). Regular an Chaotic Flows in Paraboloial Basins an Eies. Chaos, Solitons Fractals, 7(3): [17] He, J.F. (1995). A Twelfth-Orer Theory of Antisymmetric Bening of Isotropic Plates. Int. J. Mech. Sci., 38(1): [18] He, J.F. an Ma, B.A. (1994). Vibration Analysis of Laminate Plates using a Refine Shear Deformation Theory. J. Soun Vib., 175(5): [19] Bozhyarnyk, V. (2). On the Problem of Bening of Transversally Isotropic Plates. Eng. Transactions, 48(1). [2] Reissner, A. (1991). A Mixe Variational Equation for a Twelfth-Orer Theory of Bening of Non- Homogeneous Transversely Isotropic Plates. Comput. Mech., 7(5,6): [21] Vekua, I.N. (1965). The Theory of Thin Shallow Shells with Variable Thickness. Tbilisi Metsniereba. 172

11 International Journal of Research an Reviews in Applie Sciences ISSN: X, EISSN: Volume 1, Issue 2(November 29) [22] Twizell, E.H., Boutayeb, A. an Djijeli, K. (1994). Numerical Methos for Eighth-, Tenth- an Twelfth- Orer Eigenvalue Problems Arising in Thermal Instability. Avances Comp. Math., 2(4): [23] Islam, S., Haq, S. an Ali, J. (28). Numerical Solution of Special Twelfth-orer Bounary Value Problems using Differential Transform Metho. Commun. Nonl. Sci. Num. Sim, In Press. [24] Siiqi, S.S. an Twizell, E.H. (1997). Spline Solutions of Linear Twelfth-Orer Bounary-Value Problems. J. Comput. Appl. Math., (78)2: [25] Wazwaz, A.M. (2). The Moifie Decomposition Metho for Solving Linear an Nonlinear Bounary Value Problems of Tenth-Orer an Twelfth Orer. Int. J. Nonlinear Sci. Numer. Simul., 1: