Homotopy perturbation method for solving hyperbolic partial differential equations

Transcription

1 Computers and Mathematics with Applications ) wwwelseviercom/locate/camwa Homotopy perturbation method for solving hyperbolic partial differential equations J Biazar a,, H Ghazvini a,b a Department of Mathematics, Faculty of Sciences, The University of Guilan, PO Box 1914, PC 41938, Rasht, Iran b Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, PO Box 316, PC , Shahrood, Iran Received 7 June 2006; received in revised form 20 September 2007; accepted 10 October 2007 Abstract This paper applies the homotopy perturbation method proposed by Ji-Huan He, to obtain approximate analytic solutions of hyperbolic partial differential equations The procedure of the method is systematically illustrated To give an extensive account of the method some examples are provided The results derived by this method will be compared with the results of characteristics method The results of homotopy perturbation method are of high accuracy, verifying that the method is very effective and promising c 2008 Elsevier Ltd All rights reserved Keywords: Hyperbolic partial differential equations; Homotopy perturbation method; Characteristics method; Nonlinear functional equations 1 Introduction Hyperbolic partial differential equations play a dominant role in many branches of science and engineering Wave equation is a typical example, which models problems such as vibrating string, fixed at both the ends, longitudinal vibrations in a bar, propagation of sound waves, electromagnetic waves, they also occur in the field of computational fluid dynamics CFD) It is well known that Euler s equations of gas dynamics are hyperbolic partial differential equations These equations have attracted much attention and investigation of new methods for solving these equations have become an interesting task for the mathematicians Consider the second-order quasi-liner partial differential equation a 2 + b + c + d = 0 1) where a, b, c and d may be functions of x, y, u, u u and but not of, 2 and, ie, the second-order derivatives occur only to the first degree If b 2 4ac > 0 then Eq 1) is called a hyperbolic equation Numerical Corresponding author addresses: biazar@guilanacir J Biazar), hghazvini@guilanacir H Ghazvini) /$ - see front matter c 2008 Elsevier Ltd All rights reserved doi:101016/jcamwa

2 454 J Biazar, H Ghazvini / Computers and Mathematics with Applications ) methods for solving Eq 1), which are commonly, used as characteristics method [1], needed large size of computation work and usually the round-off error causes the loss of accuracy In this article we use He s homotopy perturbation method, well addressed in [2 9], which needs less computations, and leads to higher accuracy Homotopy perturbation method has been used by many mathematicians and engineers to solve various functional equations This method was further developed and improved by He and applied to nonlinear oscillators with discontinuities [3], nonlinear wave equations [4], and boundary value problems [5] It can be said that He s homotopy perturbation method is a universal one and is able to solve various kinds of nonlinear functional equations For examples it was applied to nonlinear Schrödinger equations [10], to nonlinear equations arising in heat transfer [11], to the quadratic Riccati differential equation [12], and to other equations [12 22] In this method the solution is considered as the summation of an infinite series which usually converges rapidly to exact solutions This method continuously deforms the difficult equation under study into a simple equation, easy to solve For the purpose of applications illustration of the methodology of the proposed method, using homotopy perturbation method, we consider the following nonlinear differential equation, Au) f r) = 0, r Ω, Bu, u/ n) = 0, r Γ where A is a general differential operator, f r) is a known analytic function, B is a boundary operator and Γ is the boundary of the domain Ω The operator A can be generally divided into two operators, L and N, where L is a linear and N a nonlinear operator Eq 2) can be, therefore, written as follows: or Lu) + Nu) f r) = 0 Using the homotopy technique, we construct a homotopy vr, p) : Ω [0, 1] R which satisfies: HU, p) = 1 p)[lv) Lu 0 )] + p[av) f r)] = 0, p [0, 1], r Ω, 5) Hv, p) = Lv) Lu 0 ) + plu 0 ) + p[nv) f r)] = 0 6) where p [0, 1], is called homotopy parameter, and u 0 is an initial approximation for the solution of Eq 2), which satisfies the boundary conditions Obviously from Eqs 5) and 6) we will have 2) 3) 4) Hv, 0) = Lv) Lu 0 ) = 0, Hv, 1) = Av) f r) = 0 7) 8) We can assume that the solution of 5) or 6) can be expressed as a series in p, as follows: v = v 0 + pv 1 + p 2 v 2 + Setting p = 1 results in the approximate solution of Eq 2) u = lim p 1 v = v 0 + v 1 + v 2 + 9) 10) 2 The homotopy perturbation method applied to hyperbolic equations Consider hyperbolic equation 1) with the following initial conditions: { ux, 0) = f x) u0, y) = f y) ux, 0) ) or u0, y) ) = gx) = gy) We have applied homotopy perturbation method for special cases in which the coefficients in the Eq 1) do not depend on partial derivatives and u

3 J Biazar, H Ghazvini / Computers and Mathematics with Applications ) To solve Eq 1) with the initial condition ), according to the homotopy perturbation, we construct the following homotopy: 2 ) v 1 p) 0 a 2 v + p c 2 + b 2 v c + 2 v + d ) = 0, c or, v 0 = p a 2 v c 2 b 2 v c d ) c 0 11) With initial approximation u 0 = f x) + gx)y Suppose that the solution of Eq 1) has the form v = v 0 + pv 1 + p 2 v 2 +, 12) Putting 12) into 11), and comparing the coefficients of identical degrees of p, p 0 : 2 v = 0 p 1 : 2 v 1 p 2 : 2 v 2 = a c = a c 2 v v 1 2 b c b c 2 v 0 y d c 0, v 1x, 0) = 0, 2 v 1 y, v 2x, 0) = 0, v 2 x, 0) = 0 v 1 x, 0) = 0 13) For simplicity we take v 0 = u 0 = f x) + gx)y Accordingly we have y y v 1 = a 2 v c 2 b 2 v 0 c ξ d ) dξ dy c y y v 2 = a 2 v c 2 b 2 ) v 1 dξ dy c ξ The approximate solution of 1), therefore, can be obtained by setting p = 1: 14) u = v 0 + v 1 + v 2 + Similarly, to solve Eq 1) with initial conditions ), we construct the following homotopy: v = p c 2 v a b 2 v a d ) a ) With the initial approximation v 0 = u 0 = f y) + gy)x Suppose that the solution of Eq 1) has the form 12), according to the mentioned procedure we have: p 0 : 2 v = 0 p 1 : 2 v 1 2 p 2 : 2 v 2 2 = c a = c a 2 v 0 2 v 1 b 2 v 0 a y d a 0 2, v 10, y) = v 1 0, y) = 0 b 2 v 1 a y, v 20, y) = v 2 0, y) = 0 16)

4 456 J Biazar, H Ghazvini / Computers and Mathematics with Applications ) Table 1 The solution of ux, y) for different values of x and y x y ux, y) homotopy perturbation method) ux, y) characteristics method) So we have v 1 = v 2 = x x 0 0 x x 0 0 c 2 v 0 a b 2 v 0 a ξ d ) dξ dx a ) dξ dx c 2 v 1 a b 2 v 1 a ξ Setting p = 1 results in the approximation solution of Eq 1), 17) v = lim p 1 v = v 0 + v 1 + v Numerical results To illustrate the method some examples are considered here: Example 1 Consider the following equation with initial conditions [1,23]: 2 + u = 0 ux, 0) ux, 0) = x, = x Suppose u = u 0 + pu 1 + p 2 +, according to 11), we have u 0 1 = p y + 1 ) 2 0 Beginning with u 0 x, y) = x + xy, from 13) we have 1 = 1, u 1x, 0) = 0, u 1 x, 0) = 0 18) We can solve Eq 18) easily The solution reads u 1 x, y) = 1 2 y2 So the first-order approximate solution is ux, y) = u 0 x, y) + u 1 x, y) = x + xy y2, which is an exact solution The results are compared with characteristics in Table 1 Example 2 Consider the following equation with initial conditions [1,23]: 2 4x2 = 0 ux, 0) = x 2 ux, 0), = 0

5 J Biazar, H Ghazvini / Computers and Mathematics with Applications ) Table 2 The solution of ux, y) for different values of x and y x y ux, y) homotopy perturbation method) ux, y) characteristics method) Its homotopy can be written down as follows: u 0 1 = p 2 ) u 4x Beginning with u 0 x, y) = x 2, from 13) we have 1 = 1 4x 2, u 1x, 0) = 0, u 1 x, 0) = 0 The solution reads u 1 x, y) = 4 1 So, in order to, u 2 x, y) = 1 y 4 32 u 3 x, y) = x 6 u 4 x, y) = y 6 x 10 y 8 x 14 y 2 x 2 So the fourth-order approximation solution is: ux, y) x y 2 x y 4 32 x y x y 8 x 14 The results are compared with characteristics in Table 2 Example 3 Consider the following equation with initial conditions [1,23]: x) + x4 x 2) = 0 u0, y) u0, y) = y, = 1 Its homotopy can be expressed, as follows: = p 2x 1) y x2 x 2) 0 2 By imposing with initial approximation, u 0 x, y) = x + y, from 16), we have 1 = 0, u 2 1 0, y) = u 1 0, y) = 0, the solution reads u 1 x, y) = 0, therefore, the first-order approximation solution is u = u 0 + u 1 = x + y, which is an exact solution The results are compared with characteristics in Table 3 )

6 458 J Biazar, H Ghazvini / Computers and Mathematics with Applications ) Table 3 The solution of ux, y) for different values of x and y x y ux, y) homotopy perturbation method) ux, y) characteristics method) Conclusions In this work, we have achieved an approximation for solution of hyperbolic partial differential equation by applying homotopy perturbation method The small size of computations in comparison with the computational size required in characteristics method and the rapid convergence show that the homotopy perturbation method is more reliable and introduces a significant improvement in solving the hyperbolic equations It is worth pointing out that, the results in Examples 1 3 are exactly the same as the results of Adomian decomposition method [23] References [1] Smith, Numerical Methods for Partial Differential Equations, Oxford Press, 1978 [2] JH He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B ) [3] JH He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation ) [4] JH He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals ) [5] JH He, Homotopy perturbation method for solving boundary value problems, Physics Letters A ) [6] JH He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering ) [7] JH He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics 35 1) 2000) [8] JH He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation ) [9] JH He, Homotopy perturbation method: A new nonlinear analytical technique, Applied Mathematics and Computation ) [10] J Biazar, H Ghazvini, Exact solutions for nonlinear Schrödinger equations by He s homotopy perturbation method, Physics Letters A ) [11] DD Ganji, The application of He s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A ) [12] Z Odibat, S Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos, Solitons and Fractals 36 1) 2008) [13] AM Siddiqui, R Mahmood, QK Ghori, Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane, Chaos, Solitons and Fractals 35 1) 2008) [14] L Cveticanin, Homotopy perturbation method for pure nonlinear differential equation, Chaos, Solitons and Fractals ) [15] J Biazar, M Eslami, H Ghazvini, Homotopy perturbation method for systems of partial differential equations, International Journal of Nonlinear Science and Numerical Simulation 8 3) 2007) [16] J Biazar, H Ghazvini, He s homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Chaos, Solitons and Fractals in press) [17] J Biazar, H Ghazvini, Numerical solution for special non-linear Fredholm integral equation by HPM, Applied Mathematics and Computation 195 2) 2008) [18] OK Ghori, M Ahmed, AM Siddiqui, Application of homotopy perturbation method to squeezing flow of a Newtonian fluid, International Journal of Nonlinear Science and Numerical Simulation 8 2) 2007) [19] MA Rana, AM Siddiqui, QK Ghori, Application of He s homotopy perturbation method to Sumudu transform, International Journal of Nonlinear Science and Numerical Simulation 8 2) 2007) [20] H Tari, DD Ganji, M Rostamian, Approximate solutions of K 2, 2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, International Journal of Nonlinear Science and Numerical Simulation 8 2) 2007) [21] A Ghorbani, J Saberi-Nadjafi, He s homotopy perturbation method for calculating Adomian polynomials, International Journal of Nonlinear Science and Numerical Simulation 8 2) 2007) [22] T Ozis, A Yildirim, Traveling wave solution of Korteweg-de Vries equation using He s homotopy perturbation method, International Journal of Nonlinear Science and Numerical Simulation 8 2) 2007) [23] J Biazar, H Ebrahimi, An approximation to the solution of hyperbolic equations by Adomian decomposition method and comparison with characteristics method, Applied Mathematics and Computation )