Analytical Investigation of Hyperbolic Equations via He s Methods

Transcription

1 American J. of Engineering and Applied Sciences (4): , 8 ISSN Science Pblications Analytical Investigation of Hyperbolic Eqations via He s Methods D.D. Ganji, M. Amini and A. Kolahdooz Department of Mechanical Engineering, Mazandaran University, P.O. Box 484, Babol, Iran Abstract: Pertrbation methods depend on a small parameter which is difficlt to be fond for real-life nonlinear problems. To overcome this shortcoming, two new bt powerfl analytical methods were introdced to solve nonlinear heat transfer problems in this letter, one is He s Variational Iteration Method (VIM) and the other is the Homotopy-Pertrbation Method (HPM). Nonlinear hyperbolic eqations were sed as examples to illstrate the simple soltion procedres. These methods were sefl and practical for solving the nonlinear hyperbolic eqation, which is associated with variable initial condition. Comparison of the reslts has been obtained by both methods with exact soltions reveals that both methods were tremendosly effective. Key words: Hyperbolic eqations, homotopy pertrbation method, variation iteration method INTRODUCTION Hyperbolic partial differential eqations are the sbject of many researches becase of their application in many engineering fields sch as wave eqation and telegraph eqation. In recent years, several sch techniqes have drawn special attention, sch as Hirtoa s bilinear method [], the Adomian s decomposition method [5], the EXP fnction method [- 6], fractional method [7-33], the Homotopy Pertrbation Method (HPM) [3,34-38] and Variational Iteration Method (VIM) [39-47]. Varios methods for obtaining explicit soltions to hyperbolic partial differential eqations have been proposed [-8]. Biazar et al. [] se Adomian decomposition method to solve this eqation. In this work we se homotopy pertrbation method and variational iteration method to solve hyperbolic eqations. Unlike classical techniqes, the nonlinear eqations are solved easily and elegantly withot transforming or linearizing the eqation by sing the Homotopy Pertrbation Method (HPM) [9-]. It provides an efficient explicit soltion with high accracy, minimal calclations and avoidance of physically nrealistic assmptions. The HPM was first proposed by He [-6] and has been shown to solve a large class of nonlinear problems effectively, easily and accrately with approximations converging rapidly to accrate soltions. The HPM was proposed to search for limit cycles or bifrcation crves of nonlinear eqations [7]. In [4], a heristic example was given to illstrate the basic idea of the HPM and its advantages over the d-method, the method was also applied to solve bondary vale problems [8] and heat radiation eqations [9]. Variation iteration method is based on the se of Lagrange mltipliers for identification of optimal vales of parameters in a fnctional. Using this method a rapid convergent seqence is prodced. The variational iteration method is sitable for finding the approximation of the soltion withot discretization of the problem []. The general form of hyperbolic eqation as a kind of second-order qasi-linear partial differential eqation is: a + b + c + e= x xy y Where a, b, c and e may be fnctions of x, y,, and x y bt not of, x xy and (), i.e., The second y order derivatives occr only to the first degree. If b - 4ac>, then Eq. is called hyperbolic eqation. MATERIALS AND METHODS In this stdy, we have applied the Homotopy pertrbation method and variational iteration method to the discssed problems. To illstrate the basic ideas of methods, we have considered the following nonlinear differential eqation. Corresponding Athor: D.D. Ganji, Department of Mechanical Engineering, Mazandaran University, P.O. Box 484, Babol, Iran Fax:

2 Am. J. Engg. & Applied Sci., (4): , 8 Homotopy pertrbation method: To illstrate the basic ideas of this method, we consider the following nonlinear differential eqation: A ()-f (r) =, r Ω () With the bondary conditions of: B(, ) =,r Γ n (3) Where A, B, f (r) and G are a general differential operator, a bondary operator a known analytical fnction and the bondary of the domain Ω, respectively. Generally speaking, the operator A can be divided into a linear part L and a nonlinear part N. Eq. can therefore be rewritten as: L ()+N()-f (r) = (4) By the homotopy techniqe, we constrct a homotopy v(r,p): Ω [, ]? R which satisfies: H(v,p) = ( p)[l(v) L( )] + p[a() f(r)] =,p [,],r Ω (5) Setting p = reslt in the approximate soltion of Eq. to: = lim v = v +v + v + () p? The combination of the pertrbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional pertrbation methods. The series is convergent for most cases. However, the convergent rate depends on the nonlinear operator A (v) (the following opinions are sggested by He [3] ): The second derivative of N (v) with respect to V mst be small becase the parameter may be relatively large, i.e., p N The norm of L mst be smaller than one so v that the series converges Variational iteration method: To clarify the basic ideas of VIM, we consider the following differential eqation: L + F = g (t) () or H(v,p) = L(v) L( ) + pl( ) + p[n(v) f(r)] = (6) Where L is a linear operator, F is a nonlinear operator and g (t) is a heterogeneos term. According to VIM, we can write down a correction fnctional as Where p [, ] is an embedding parameter, while is an initial approximation of Eq. which satisfies the bondary conditions. Obviosly, considering Eq. 5 and 6 we will have: H (v, ) = L(v)-L( ) = (7) H (v, ) = A(v)-f (r) = (8) The changing process of p from zero to nity is jst that of v (r, p) from (r) to (r). In topology, this is called deformation and L (v)-l ( ) and A (v)-f (r) are being called homotopy. According to the HPM, we can first se the embedding parameter p as a small parameter and assme that the soltion of Eq. 5 and 6 can be written as a power series in p, namely: v = v +pv + p v + (9) t (t) = (t) + λ(l () τ+ F () τ g( τ))dτ () n n n Where λ is a general Lagrangian mltiplier [7-] which can be identified optimally via the variational theory. The sbscript n indicates the nth approximation and ũn is considered as a restricted variation [7-], i.e., δ =. n Applications: In order to assess the accracy of VIM and HPM for solving hyperbolic eqations and to compare it with exact soltion, we will consider the three following examples. Example: Consider the partial differential eqation with the initial conditions [7] : + ( x) + (x x ) = x xy y (3) 4

3 Am. J. Engg. & Applied Sci., (4): , 8 (x, ) = x, (x,) = y Homotopy pertrbation method: Now we apply homotopy pertrbation to Eq. 3: v(x,) = x+ y, v(x,) = y (9) v(x,) i =, v(x,) i y= =,i =,,... y The Soltion of Eq. 5-9 may be written as y H(,p): = ( p)(x x ) (x,y) (x,y) + ( x) (x,y) x yx + p = + (x x ) (x,y) y (4) Sbstitting Eq. 9 into Eq. 4 and rearranging based on powers of p-terms, we have: p:x[ (x,y)] x[ y y (x,y)] [ (x,y)] = y x (x,y) [ (x,y)] y x[ (x,y)] + y yx yx p: + (x,y) x[ (x,y)] x[ (x,y)] = y x[ (x,y)] + x[ (x,y)] yx y x y p : + (x,y) [ (x,y)] + (x,y) x[ (x,y)] = yx y (x,y) [ (x,y)] x yx yx y 3 p: + (x,y) [ 3(x,y)] + = y y x[ 3(x,y)] x[ 3(x,y)] (5) (6) (7) (8) To determine, the above eqations shold be solved. Considering the appropriate initial conditions we have: v(x,) = x+ y () In the same manner, the rest of components were obtained sing the maple package. According to the HPM, we can conclde: = limv(x,y) = v(x,y) + p v(x,y) + v(x,y) +... () In this manner three components of the pertrbation series 9 were obtained. So, we have: = x+ y Variational iteration method: Now Before applying this procedre to Eq. 3 following [4,43], we constrct a correction fnctional, as y (x,y) = n(x,y) + λτ ()[ n(x, τ+ ) ( x) x () [ n(x, τ )] + (x x )[ n(x, τ)]]dτ τ x τ Its stationary conditions can be obtained as (x x )[ λ (y)] =, d (x x )[ λτ ( )] =, d dy dτ λ (y)[ n (x,y)] =, λ (y) = x (3) The Lagrangian mltiplier can therefore be identified as: λ( τ ) = τ y x x (4) As a reslt, we obtain the following iteration formla: 4

4 Am. J. Engg. & Applied Sci., (4): , 8 y (x,y) = n(x,y) + λτ ()[ n(x, τ ) + ( x) x (5) [ n(x, τ )] + (x x )[ n(x, τ)]]dτ τ x τ Now we start with an arbitrary initial approximation that satisfies the initial condition: = x+ y (6) Using the above variational formla 5, we have: y = (x,y) + λτ ( )[ (x, τ ) + ( x) x [ (x, τ )] + (x x )[ (x, τ)]]dτ τ x τ (7) Sbstitting Eq. 6 in to Eq. 7 and after simplifications, we have: = x+ y (8) In the same way, we can obtain U (X, Y), U 3 (X, Y) and the rest of the components of the iteration formla. Example : Consider the partial differential eqation with the initial conditions [7] : + + =, x xy y (x,) = x, (x,) = x. y (9) Homotopy pertrbation method: By applying homotopy pertrbation to Eq. 9. We have: (x,y)] + y x p : [ (x,y) + (x,y) = yx (x,y)] + y y 3 p : [ 3 (x,y) + (x,y) = yx (33) (34) To determine, the above eqations shold be solved. Considering the appropriate initial conditions we have: v(x,) = x+ xy, v(x,) = x, y v(x,) i =, v(x,) i y= =,i =,,... y (35) The soltion of Eq may be written as n(x,y) = x + xy (36) n(x,y) = y...andsoon (37) In the same manner, the rest of components was obtained sing the maple package. According to the HPM, we can conclde: = limv(x,y) = v(x,y) p + v(x,y) + v(x,y) +, (38) H(,p) = ( p)[ (x,y)] + p[ (x,y) y x + (x,y) [ (x,y)] + ] = yx y (3) Sbstitting Eq. 9 into Eq. 9 and rearranging based on powers of p-terms, we have: p : [ (x,y)] = y y x p: [ (x,y)] + (x,y) + (x,y) + = yx (3) (3) 4 Therefore the soltion is: : x xy y = + + (39) Variational iteration method: First we constrct a correction fnctional like example, as y (x,y) = (x,y) + λτ ()[ (x, τ ) + n n x n(x, τ) ( n(x, τ )) + ]dτ xτ τ (4) Its stationary conditions can be obtained as

5 Am. J. Engg. & Applied Sci., (4): , 8 d d + ( λ (y)) =, ( λτ ()) =, dy dτ (4) λ (y)[ ] n =, λ (y) = x The Lagrangian mltiplier can therefore be identified as: τ y λ( τ ) = (4) Homotopy pertrbation method: By applying homotopy pertrbation to Eq. 48, we have: y H(,p) = 4( p)x [ (x,y)] + p[ (x,y) 4x [ (x,y)]] = x y (49) Sbstitting Eq. 9 into Eq. 48 and rearranging based on powers of p-terms, we have: As a reslt, we obtain the following iteration formla: p : 4x [ (x,y)] = y (5) y (x,y) = (x,y) + λτ ()[ (x, τ ) + n n x n(x, τ) ( n(x, τ )) + ]dτ xτ τ (43) Now we start with an arbitrary initial approximation that satisfies the initial condition: = xy+ x (44) Using the above variational formla 43, we have y = + λ( τ)[ (x, τ+ ) x (x, τ) ( (x, τ )) + ]dτ xτ τ (45) Sbstitting Eq. 44 in to Eq. 45 and after simplifications, we have: xy x y 4 = + + (46) In the same way, we can obtain U (X, Y) as xy x y = + + (47) And so on. In the same manner the rest of the components of the iteration formla can be obtained. Example 3: Consider the partial differential eqation with the initial conditions [6] : p: 4X [ (x,y)] + (x,y) = y x p : 4x [ (x,y)] + (x,y) = y x 3 p : 4x [ 3(x,y)] + (x,y) = y x 4 p : 4x [ 4(x,y)] + 3(x,y) = y x (5) (5) (53) (54) To determine, the above eqations shold be solved. Considering the appropriate initial conditions we have: v(x,) = x, v(x,) =, y v(x,) i =, v(x,) i y= =,i =,,... y (55) The Soltion of Eq may be written as (x,y) = x (56) y (x,y) = (57) 4x 4 y (x,y) = (58) 6 3x 4x =, x y (x,) = x, (x,) = y (48) y 3(x,y) = 64x (59) 8 y 4(x,y) = 4 48x (6)

6 Am. J. Engg. & Applied Sci., (4): , 8 Therefore the soltion is: y y 7 y y 4x 3x 64x 48x = x (6) 6 4 Variational iteration method: First we constrct a correction fnctional like example, as y (x,y) = (x,y) + λ ( τ )[ x n τ τ τ τ (x, n ) 4x( n(x, ))]d (6) Its stationary conditions can be obtained as d d + 4x ( λ (y)) =, 4x ( λτ ()) = dy dτ λ (y)[ n (x,y)] =, λ (y) = x (63) The Lagrangian mltiplier can therefore be identified as: τ y λτ= ( ) + (64) 4x 4x As a reslt, we obtain the following iteration formla: In the same way, we can obtain U (X, Y), U 3 (X, Y) and U 4 (X, Y) as x + 8x y + 3y = (69) 6 6 x x + 48x y + 36x y + 63y 3 = (7) 64 x y y 7 y y 3 = x (7) 6 4 4x 3x 64x 48x And so on. In the same manner the rest of the components of the iteration formla can be obtained. RESULTS The soltion of hyperbolic eqations has been investigated analytically by Homotopy Pertrbation Method (HPM) and Variational Iteration Method (VIM). U(X, Y) for both Eq. 47 and 7 have been shown in Fig. and respectively. Also in Tables -3, for some vales of x and y, reslts for both methods have been compared with exact soltion. The reslts have shown that they are considerably capable of solving a wide range of hyperbolic eqations. y (x,y) = + λ ()[ τ x n τ (x, n τ))]dτ τ (x, n ) 4x( (65) Now we start with an arbitrary initial approximation that satisfies the initial condition: = x (66) Using the above variational formla 65, we have: Fig. : 3D plot of (x, y) for Eq. 47 y = + λτ ( )[ x (x, τ) 4x ( (x, τ))]dτ τ (67) Sbstitting Eq. 66 in to Eq. 67 and after simplifications, we have: 4x + y = (68) 4 x 44 Fig. : 3D plot of (x, y) for Eq. 7

7 Table : The soltion of (x, y) for different vales of x and y U(x, y) U(x, y) U(x, y) x y (HPM) (VIM) (exact reslt) Table : The soltion of (x, y) for different vales of x and y U(x, y) U(x, y) U(x, y) x y (HPM) (VIM) (exact reslt) Table 3: The soltion of (x, y) for different vales of x and y U(x, y) U(x, y) U(x, y) x y (HPM) (VIM) (exact reslt) DISCUSSION In this stdy, the athors have intended to show that the two methods, HPM and VIM, are considerably capable of solving a wide range of hyperbolic eqations. The examples given in this stdy reveal that both methods are very effective and have high accracy. In some cases as Examples and, after some steps the remaining terms wold vanish and we derive the exact soltion. In the cases as Example 3, the approximation can be obtained to any desired nmber of terms. VIM and HPM do not need small parameters, the limitations and non-physical assmptions reqired in classical pertrbation methods are eliminated, frthermore, VIM and HPM can overcome the difficlties arising in the calclation of Adomian polynomials. They do not reqire linearization; both methods are very promising tools for hyperbolic eqations. Therefore, both methods will find applications in varios fields. CONCLUSION The methods of HPM and VIM have been sccessflly performed for hyperbolic eqations. In these cases, we obtained excellent performances that might lead to promising approaches for many applications. All the examples showed that the reslts of the present methods were in excellent agreement with exact soltions. It is capable to converge to correct reslts with fewest nmber of iterations or even once, for some cases. Am. J. Engg. & Applied Sci., (4): , 8 45 REFERENCES Biazar, J. and H. Ebrahimi, 5. An approximation to the soltion of hyperbolic eqations by adomian decomposition method and comparison with characteristics method. Appl. Math. Compt., 63: DOI:.6/j.amc Biazar, J., E. Babolian, and R. Islam, 3. Soltion of a system of volterra integral eqations of the first kind by Adomian method. Appl. Math. Compt., 39: DOI:.6/S96-33()73-X 3. He, J.H., 999. Homotopy pertrbation techniqe. Compt. Meth. Appl. Mech. Eng., 78: DOI:.6/S45-785(99) Adomian, G., 989. Nonlinear Stochastic Systems Theory and Applications to Physics. Klwer Academic Pblishers, Dordrecht, pp: 4. ISBN: 97755X. 5. Adomian, G., 994. Solving Frontier Problems of Physics: The Decomposition Method. Klwer Academic Pblishers, Dordrecht, pp: 37. ISBN- : X 6. Evans, G., J. Blackledge and P. Yardley,. Nmerical Methods for Partial Differential Eqations. Springer-Verlag, Berlin, pp: Smith, G.D., 978. Nmerical Soltion of Partial Differential Eqations; Finite Difference Methods. nd Edn., Oxford University Press, Oxford, ISBN- : Wazwaz, A.M.,. A new algorithm for calclation Adomian polynomials for nonlinear operators. Appl. Math. Compt., : DOI:.6/S96-33(99) Ganji, D.D., 6. The application of He's homotopy pertrbation method to nonlinear eqations arising in heat transfer. Phy. Lett., A., 355: DOI:.6/j.physleta Ganji, D.D. and M. Rafei, 6. Solitary wave soltions for a generalized hirota-satsma copled KdV eqation by homotopy pertrbation method. Phys. Lett., A 356:3-37. DOI:.6/j. physleta Rafei, M., D.D. Ganji, 6. Explicit soltions of Helmholtz eqation and fifth-order KdV eqation sing homotopy pertrbation method. Int. J. Nonlinear Sci. Nmer. Simlat., 7: &ETOC=RN&from=searchengine. He, J.H., 4. The homotopy pertrbation method for nonlinear oscillators with discontinities. Applied Math. Compt., 5: DOI:.6/S96-33(3)34-.

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