Abdolamir Karbalaie 1, Hamed Hamid Muhammed 2, Maryam Shabani 3 Mohammad Mehdi Montazeri 4

Transcription

1 ISSN print, online International Journal of Nonlinear Science Vol No.1,pp Exact Solution of Partial Differential Equation Using Homo-Separation of Variables Abdolamir Karbalaie 1, Hamed Hamid Muhammed 2, Maryam Shabani 3 Mohammad Mehdi Montazeri 4 1,2 Division of Informatics, Logistics and Management, School of Technology and Health STH, Royal Institute of Technology KTH, SE , Stockholm, Sweden 3 R&D Division, Naghsh Jahan Intelligent Communication Engineering, Sahin Shahr, Isfahan, Iran 4 Department of Mathematics, Khomeini Shahr Branch, Islamic Azad University, Khomeini Shahr, Isfahan, Iran Received 8 February 2013, accepted 11 July 2013 Abstract: In this study, we find the exact solution of certain partial differential equations PDE by proposing and using the Homo-Separation of Variables method. This novel analytical method is a combination of the homotopy perturbation method HPM with the separation of variables method. The exact solutions are constructed by choosing an appropriate initial approximation in addition to only one term of the series obtained by HPM. The proposed method is introduced an efficient tool for solving a wide class of partial differential equations. It is straight-forward, easy to understand and fast requiring low computational load. Keywords: biological population model; separation of variables; homotopy perturbation method; backward Klomogorov equation; Fornberg Whitham equation 1 Introduction Nonlinear partial differential equations NPDE have been widely studied throughout recent years. These equations play a pivotal role in the modelling of numerous chemical, physical and biological phenomena. The importance of obtaining the exact solutions of NPDEs in mathematics is still a significant problem that needs new methods. Various numerical and analytical methods have been developed and successfully employed to solve these equations such us: the differential transform method DTM [21], the homotopy analysis method HAM [15], the Adomian decomposition method ADM [1], the variational iteration method VIM [3], the homotopy perturbation method HPM [5], the Laplace decomposition method LDM[12,13], the Sumudu decomposition method SDM[14], the tanh method [2], the exp method[7], and the sinh cosh method [19]. Some of these methods use specific transformations and others give the solution as a series which converges to the exact solution. Among these methods, the VIM and ADM are the most used ones to solve differential and integral equations. The HPM was originally proposed by the Chinese mathematician Ji-Huan He [4, 5, 6]. The application of this method in linear and nonlinear problems has been devoted by scientists and engineers, because this method reform the problem at hand into a simpler problem which is easier to solve. However, despite the potential of the HPM, it can not be considered as a universal method to solve all types of nonlinear equations. Overcome this limitation many combination methods have been suggested to improve the homotopy perturbation method s ability for solving nonlinear problems such as the homotopy analysis transform method HATM, the homotopy perturbation Sumudu transform methodhpstm, the variational homotopy perturbation method VHPM, and the homotopy perturbation transformation method HPTM [16]. The HATM is a combination of the homotopy analysis method and the Laplace decomposition method [11]. In the case of the HPSTM, Singh [18] and Karbalaie [10] have combined the homotopy perturbation method and the Sumudu transform method. In the case of the VHPM, Noor and Mohyud-Din [17] have combined the variational iteration method and the homotopy perturbation method. They used this method for solving higher dimensional initial boundary value problems. On the other hand, the HPTM was constructed by combining two powerful methods; namely the homotopy perturbation method and the Laplace transform method. Corresponding author. address: abdolamir.karbalaie@sth.kth.se Copyright c World Academic Press, World Academic Union IJNS /789

2 Abdolamir Karbalaie, Hamed Hamid Muhammed, etc.: Exact Solution of Partial Differential Equation Using 85 In general, there exists no method that gives an exact solution for FPDEs and most obtained solutions are only approximations. However, Yang [20] used the modified homotopy perturbation method MHPM to obtain the exact solution of the Fokker-Plank equation which is a PDE of integer order. Furthermore Karbalaie et al. [8, 9] used HPM with separation of variables to find the exact solution of one dimensional fractional partial differential equations. In this paper, we extend Yang s method to solve n-dimensional equations, and the resale is called Homo-Separation of Variables. We present an elegant fast approach by designing and utilizing a proper initial approximation which satisfies the initial condition of a PDE as follows u 0 x, t = u x, 0 c 1 t + n c 2 t, i where u x, 0 is the initial condition of the PDE. We use u 0 x, t, which has the form of separation of variables, as an initial condition for HPM. By using this method, the other of the PDE to be solved is reduced into an ODE or a system of ODEs. The structure of the rest of the paper is as follows: In section 2 the basic idea of HPM is presented. In section 3, we describe the Homo-Separation of Variables method. In section 4, we present four examples to show the efficiency of using the HPM to solve PDEs. Finally, relevant conclusions are drawn in section 5. 2 The basic idea of the homotopy perturbation method HPM HPM is a combination of the homotopy technique and the classical perturbation method. HPM is applied to various nonlinear problems as mentioned in the previous section of this paper. In this section, the algorithm of this method is briefly illustrated. To achieve our goal, we consider the nonlinear partial differential equation: L u +N u f r = 0, r Ω, 1 B u, u/ n = 0, r Γ, where L is a linear operator, N is a nonlinear operator, B is a boundary operator, Γ is the boundary of the domain Ω, and f r is a known analytical function. By using the homotopy perturbation technique, we construct a homotopy v r, p : Ω [0, 1] R which satisfies: H v, p = L v L u 0 + pl u 0 + p [N v f r] = 0, 0 p 1, 2 where r Ω and u 0 is an initial approximation for Eq. 1 and p is an embedding parameter. When the value of p is changed from p = 0 to p = 1, we can easily see that H v, 0 = L v L u 0 = 0, 3 H v, 1 = L v + N v f r = 0. 4 This changing process is called deformation, and Eq. 3 and 4 are called homotopic in field of topology. We can assume that the solution of Eq. 2 can be expressed as a power series in p, as given below: v = p i v i = v 0 + pv 1 + p 2 v 2 + p 3 v , 5 In case the p-parameter is considered as small, the best approximation for the solution of Eq. 1 becomes: u = lim p 1 v = v i = v 0 + v 1 + v 2 + v The above solution-series generally converges very rapidly. The convergence of this series has been proved in [4]. IJNS homepage:

3 86 International Journal of Nonlinear Science, Vol , No.1, pp Homo-Separation of variables This approach is a combination of HPM and separation of variables. This can be achieved by substituting Eq. 5 into 2 and equating the terms with identical powers of p. thus a set of equations is obtained as the follows p 0 : L v 0 L u 0 = 0, 7 p 1 : L v 1 + L u 0 + N v 0 = 0, 8. By utilizing the results of the previous equations and substituting them into Eq. 6 we approximate the analytical solution, u x, t, by the truncated series: N u x, t = lim v i. 9 N If in Eq. 8 there exists some v n = 0, where n 1, the exact solution can be written in the following form: n 1 u x, t = v 0 + v v n 1 = v i. 10 For simplicity, we assume that v 1 x, t 0 in Eq. 10, which means that the exact solution in Eq. 1 is And when solving Eq. 7, we obtain the result u x, t = v 0 x, t, 11 u 0 x, t = v 0 x, t, 12 By using Eq.11 and Eq.12 we have: u x, t = v 0 x, t = u 0 x, t, 13 To illustrate our basic idea, we consider the initial approximation of Eq. 1 as follows u x, t = u 0 x, t = u x, 0 c 1 t + n c 2 t, 14 i Our goal in this new method is finding c 1 t and c 2 t. To achieve our goal, we consider the following initial condition of Eq. 1: u x, 0 = gx, 15 Since Eq. 14 satisfies the initial conditions as well as Eq. 15, we get By substituting Eq. 14 into Eq. 8, we obtain L v 1 = L u 0 + N v 0, = [ L u x, 0 c 1 t + n c 1 0 = 1, c 2 0 = 0, 16 n c 2 t + N u x, 0 c 1 t + i ] c 2 t 0, 17 i In this case, the partial differential equation is changed into an ODE or a system of ODEs, which simplifies the problem at hand. The exact solution of the PDE is found when the target unknowns c 1 t and c 2 t are computed; by utilizing Eq. 17 and the initial conditions in Eq. 16. IJNS for contribution: editor@nonlinearscience.org.uk

4 Abdolamir Karbalaie, Hamed Hamid Muhammed, etc.: Exact Solution of Partial Differential Equation Using 87 4 Applications In this section, in order to assess the advantages and the accuracy of the procedure described in the last section, we have applied the method to four different examples with different dimensionalities. Example 1 Consider the following nonlinear Fornberg Whitham equation u t u xxt + u x = uu xxx uu x + 3u x u xx, 18 with the initial condition u x, 0 = e 1 2 x, 19 u 0 x, t = u x, 0 c 1 t + c 2 t, = e 1 2 x c 1 t e 1 2 x c 2 t, 20 D t v 1 = 1 2 e 1 2 x 6ć 1 t + 3ć 2 t + 4c 1 t + 2c 2 t We obtain the ODE system 3 2ć 1 t + ć 2 t + 2 2c 1 t + c 2 t = 0, c 1 0 = 1, c 2 0 = 0. Solving Eq. 22 by using the ODEs properties, we obtain 22 2c 1 t + c 2 t = 2e 2 3 t, 23 u x, 0 = e 1 2 x e 2 3 t = e 1 2x 4 3 t, 24 Example 2 Consider the non-homogenous Backward Klomogorov equation where t > 0, x R, subject to the initial condition We obtain the ODE system u t = x 2 e t u xx + x + 1 u x + tx, 25 u x, 0 = x + 1, 26 u 0 x, t = u x, 0 c 1 t + c 2 t, = x + 1c 1 t + c 2 t, 27 D t v 1 = x ć 1 t c 1 t t + ć 1 t + ć 2 t c 1 t ć 1 t c 1 t t = 0, c 1 0 = 1, ć 1 t + ć 2 t c 1 t = 0, c 2 0 = 0, Solving Eq. 29, 30 by using the ODEs properties, we obtain c 1 t = 2e t t 1, 31 IJNS homepage:

5 88 International Journal of Nonlinear Science, Vol , No.1, pp c 2 t = 1 2 t2, 32 u x, t = x + 1 2e t t t2, 33 Example 3 Consider the generalized biological population model of the form: u x, y, t 2 u 2 x, y, t = t u 2 x, y, t y 2 + u x, y, t 1 ru x, y, t, 34 where t > 0, x, y, r R, subject to the initial condition u x, y, 0 u 0 x, y, t = u x, y, 0 c 1 t + + u x, y, 0 = e 1 2 r 2 x+y. 35 u x, y, 0 c 2 t, y = e 1 2 r 2 x+y c 1 t + r 2 e 1 r x+y 2 2 c 2 t, 36 D t v 1 = e 1 r x+y [ 2 2ć 1 t 2ć 2 t + 2c 1 t + ] 2c 2 t We obtain the ODE system 2ć1 t 2ć 2 t + 2c 1 t + 2c 2 t = 0, c 1 0 = 1, c 2 0 = 0, 38 Solving Eq. 38 by using ODEs properties, we obtain c 1 t r 2 c 2 t = e t, 39 u x, y, t = e 1 2 r 2 x+y e t = e 1 2 r 2 x+y+t, 40 Example 4 Consider the three-dimensional equation u x, y, z, t t = r 2 u x, y, z, t + su x, y, z, t, r, s R, 41 where 2 is the Laplacian operator, t > 0, x, y, z [0, π] [0, π] [0, π], subject to the initial condition ux, y, z, 0 u 0 x, y, z, t = ux, y, z, 0c 1 t + + u x, y, z, 0 = cosx + siny + cosz, 42 ux, y, z, 0 y + ux, y, z, 0 c 2 t z = cosx + siny + cosz c 1 t + sin x + cos y sinz c 2 t, 43 D t v 1 = cosx + siny + cosz ć 1 t + r sc 1 t + sin x + cos y sinz ć 2 t + r sc 2 t 0, 44 IJNS for contribution: editor@nonlinearscience.org.uk

6 Abdolamir Karbalaie, Hamed Hamid Muhammed, etc.: Exact Solution of Partial Differential Equation Using 89 We obtain the ODE system ć 1 t + r sc 1 t = 0, c 1 0 = 1, ć 2 t + r sc 2 t = 0, c 2 0 = 0, Solving Eq. 45 and 46 by using ODEs properties, we obtain c 1 t = e ts r, 47 c 2 t = 0, 48 u x, y, z, t = cosx + siny + cosz e ts r, 49 5 Conclusions The fundamental goal of this work is to propose a simple method for the solution of partial differential equations. The goal has been achieved by applying Homo-Separation of Variables in addition to using the initial conditions only. This method is by using the modified homotopy perturbation method and a proper initial approximation which satisfies the initial condition of the PDE at hand. The current method combines the following three advantages: 1. The method is straight-forward and is used in a direct way without using linearization or restrictive assumptions. 2. Converting partial differential equations into two or more ordinary differential equations that are often mach easier to solve. 3. This method is able to find the exact solutions of PDEs. We have applied the method to four examples with one, two and three dimensions. All the results were calculated by using the symbolic calculus software Maple 14. The considered method provides a very efficient, succinct and powerful mathematical tool for solving many other partial differential equations in mathematical physics. Acknowledgments The authors would like to thank Prof. Bjorn-Erik Erlandsson, Mohammad Karbalaie and Mehran Shafaiee for their help and support as well as the inspiration we got through friendly scientific discussions. References [1] G. Adomian. Solving frontier problems of physics: The decomposition method. Academic Publishers, Boston and London [2] E. Fan. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A., 2000: [3] J.H. He. A new approach to nonlinear partial differential equations. Commun Nonlinear Sci Numer Simul., 21997: [4] J.H. He. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics., : [5] J.H. He. Homotopy perturbation technique. Computational, Methods in Applied Mechanics and Engineering., : [6] J.H. He. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation., : [7] J.-H. He and X.-H. Wu. Exp-function method for nonlinear wave equations. Chaos, Solitons and Fractals, : IJNS homepage:

7 90 International Journal of Nonlinear Science, Vol , No.1, pp [8] A. Karbalaie, M. M. Montazeri, H. H. Muhammed. New Approach to Find the Exact Solution of Fractional Partial Differential Equation. WSEAS Transactions on Mathematics, : [9] A. Karbalaie, H. H. Muhammed, B. E. Erlandsson. Using Homo-Separation of variables for solving systems of nonlinear fractional partial differential equations. International Journal of Mathematics and Mathematical Sciences, [10] A. Karbalaie, M. M. Montazeri, H. H. Muhammed. Exact solution of time-fractional partial differential equations using Sumudu transform. WSEAS Transactions on Mathematics, [11] M. Khan, M.A. Gondal, I. Hussain, S. Karimi Vanani. A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi-infinite domain. Mathematical and Computer Modeling, : [12] M. Khan. An effective modification of the Laplace decomposition method for nonlinear equations. International Journal of Nonlinear Sciences and Numerical Simulation, : [13] M. Khan. A Laplace decomposition algorithm applied to a class of non- linear differential equations. Journal of Applied Mathematics, 12001: [14] D. Kumar, S. Jagdev, R. Sushila Sumudu Decomposition Method for Nonlinear Equations. In International Mathematical Forum, 72012: [15] L. Shijun. On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, : [16] Y. Liu. Approximate solutions of fractional nonlinear equations using homotopy perturbation transformation method. Mathematical Problems in Engineering, [17] M. A. Noor and S.T. Mohyud-Din. Variational homotopy perturbation method for solving higher dimensional initial boundary value problems. Mathematical Problems in Engineering., [18] J. Singh, D. Kumar, Sushila. Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations. Adv. Theor. Appl., [19] A. M. Wazwaz. A reliable technique for solving the wave equation in an infinite one-dimensional medium. Applied Mathematics and Computation, :1 7. [20] G. Yang, R. Chen, L. Yao. On exact solutions to partial differential equations by the modified homotopy perturbation method. Acta Mathemathicae Applicatae Sinica., : [21] M.J. Jang, C.L. Chen, Y.C. Liu. Two-dimensional differential transform for partial differential equations. Applied Mathematics and Computation, : IJNS for contribution: editor@nonlinearscience.org.uk