Operational Matrices for Solving Burgers' Equation by Using Block-Pulse Functions with Error Analysis

Transcription

1 Australian Journal of Basic and Applied Sciences, 5(12): , 2011 ISSN Operational Matrices for Solving Burgers' Equation by Using Block-Pulse Functions with Error Analysis 1 K. Maleknejad, 2 E. babolian, 3 A. Jafari Shaerlar, 3 M. Jahangir 1-3 Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU)-Tehran, Iran. Abstract: This paper is provided to the development of numerical method to deal with Burgers' equation. The technics is based on two dimensional Block Pulse functions under the framework of operational matrices. In this work we apply new ideas to account operational matrix for partial derivative. We show how to convert burgers' equation to nonlinear system of algebra. Error analysis for this method are given by projection operator. The algorithm have been tested for an example. The numerical results obtained by this approach have been compared with the exact solution to show the efficiency of the method. Key words: Nonlinear parabolic equations, Numerical methods, Direct method, Operational matrices, Burgers' equations. The one dimensional nonlinear partial equation. INTRODUCTION,01,0 (1) such that, 0, (2) 0,, (3) 1,, (4) in known as Burgers' equation. Where 0 is the coefficient of the kinematic viscosity,see Polyanin Burgers' equation is a fundamental partial differential equation from fluid mechanics. It occurs in various equation areas of applied mathematics, such as modeling of dynamics, heat conduction, acoustic waves, gas dynamics, traffic flow, and many other application involving the study of nonlinear waves. The mathematical properties of equation (1) have been studied by Cole He also gave an exact solution of Burgers' equation. Benton and Blatzman have demonstrated about 35 distinct exact solution of Burger-like equations and their classifications. It is well known that the exact solution of Burgers' equation can only be computed for restricted values of. Many of the analytical solution involve Fourier series solutions. The convergence of the analytic solution in very slow due to series solution for the smaller velocity constant,see (Aksan, 2006; Asaithambi, 2010). In these cases, numerical solutions may produces results including nonphysical solution. Numerical techniques are tested on the Burgers' equation to make comparison with analytical and exact solutions. Several numerical methods for (1) have been used by many researches, for instance, finite element method (Varoglu, 1980; Caldwell, 1981) finite difference method (Smith, 1989; Ames, 1992) spectral method (Ben, 1997; Babolian, 2009) Adomian decomposition method (Zhu, 2010) direct variational method (Ozas, 1996) and projection method by B-spline (Dag, 2005). In this paper, we present a simple numerical method for solving Burgers' equation by using two dimensional Block-Pulse functions. Block-pulse functions have been studied extensively as a basic set of functions for signal characterization in system science and control (Maleknejad, 2003; Maleknejad, 2005). This set of functions was first introduced to electrical engineering by (Harmuth, 1969; see Babolian, 2008; Jiang, 1992). We solve (1) by two dimensional Block-Pulse functions and apply their operational matrices. We approximate partial derivatives by 2.By using direct method, any partial differential equation is converted to linear or nonlinear algebraic system of equations.the organization of this paper ia as follow:in section 2 we introduce 2 and their propertyes.in section 3 we present operational matrices for partial derivatives. Direct method for solving Burgers' equation are given in section 4. Error analysis for proposed method are investigated. Computational results of the Burgers' equation for a test problem are illustrated. Corresponding Author: K. Maleknejad, Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU)- Tehran, Iran. maleknejad@iust.ac.ir 602

2 Two dimensional Block-Pulse functions: A set of two dimensional Block-Pulse functions Φ,, 0,1,2,..., 1, 0,1,2,..., is defined in the region, and 0, as:,, 1 1, 1 0. (5) where, are arbitrary positive integers, and,. There are some properties for 2 as following: The 2 are disjoint, orthogonal and complete set, We can also expand a two variable function, into series:,,,,,, (6) through determining the block pulse coefficients:,, (7) Also, for vector forms, consider the elements of 2 Φ,,,,,,,,,,,,,,. (8) The two important properties of 2DBPF's are given as: (i) Φ, Φ, Φ,, (9) where is an vector and. Moreover, it can be clearly concluded that for every matrix : (ii): Φ, Φ, Φ,, (10) where is an column vector with elements equal to the diagonal entries of matrix. For simplicity, we use. Let, :, 0, where, and be the parabolic boundary of. If, are finite,,,0,0, If, are infinite,,0 And,, :,,,. (11) without loss of generality, set 0,1 and 1. The inner product.,. and norm. in, are defined as follows:,,,,,,, (12),. (13) Let be the projection operator defined on,, where is finite dimensional, as:,,,,,. (14) 603

3 First, we find an estimation of for arbitrary,. Lemma 1 Let, be defined on, and be projection operator defined by 13 then, (15) where max max, for 0,1. Proof: The integral,φ, is a ramp, on the subinterval,, with average value,. The error in approximating the ramp by this constant value over the subinterval,, is,,,,,, (16) hence, using as least square of the error on, we have,,,, (17),,, (18) and on the interval we have,. (19) Operational matrix for partial derivatives: The expansion of function, over with respect to,,,, 0,1, 1, can be written as.,,,, Φ Φ, (20) where,,,,,,,,,,,,,,, Φ,,,,,,,,,,,,,, and,, 1, 0, (21),,. (22) Now, expressing,,, in terms of the 2 as:,, 0,0,,0,,,,, (23) in which, is th component. Thus Φ, Φ,, (24) where is matrix and is called operational matrix of double integration and can be denoted by, where (25) 604

4 so, the double integral of every function, can be approximated by:, Φ,, (26) by similar method Φ,, in terms of 2 as:, 0,0,...,,0,0,...,0 Φ.,., (27) \ And Φ, Φ.,.. (28) Now, we compute operational matrix for. Lemma 2: Suppose, and is defined on parabolic boundary then operational matrix for by 2DBPFs is approximated as:, Φ, (29) that: Δ, (30) where Δ is the following matrix as: 0, (31) 0 With (32) that is initial boundary vector of. Proof: By applying approximation Φ in 24 instead of we have:, And Φ,, (33),,, 0 Φ, Φ, 0 Φ, Δ Φ,, (34) from 29 and 30 we can conclude: Δ. (35) 605

5 by the same method, operational matrix for, are given as follows. Lemma 3: If, and defined in parabolic boundary then operational matrix for 2DBPFs are approximated as: and Φ,, (36) Φ,, (37) Where Δ Δ, (38) Δ Δ Δ Δ, (39) and Δ, Δ are the following matrices: , (40) , (41) 0 and, are boundary vectors of. Direct method for solving nonlinear : The results obtained in previous section are used to introduce a direct efficient and simple method to solve equations 1 4. We consider equations 1 4 of the form:,, (42), 0, 0,, 1,. Approximating function with respect to 2: by (43), Φ, Φ, Φ,. (44) By substituting the above equations into 36 and using boundary and initial conditions, we obtain a nonlinear system with,, 0,1,...,1 as unknowns: 0. (45) Error analysis: Let the problem be of the form,,, 0, 0,, 1,, (46) where,, belong to 0,1, and is linear. 606

6 By using 13, the discrete approximation of 39 is: \ where, for each,,, belongs to an dimensional subspace., (47) Theorem 1: Let, and, be in, and, be approximate solution by 2DBPFs of 13, (48), then max max max P P (49) Proof: By using properties of projection operators,,,,,,, (50) (51) (52) max max max P P (53) 2 3 where max max max P Pmax for,, so by hypothesis of the theorem, is a finite number and. So, if then tends to zero. Numerical example: We present numerical results to illustrate the effectiveness of the proposed method. Consider Burgers' equation (1) with the following initial and boundary functions., 0,01,1, (54) 0, 1, 0. The exact solution of (1) with above conditions was given in Asaithambi 2010 as., 0, (55) We solve Burgers' equation (1) by direct method and numerical results obtained can be compared with exact solution. The numerical results show that with increasing, the approximate solution gets better. To show the accuracy of method we report norm of the error which is defined by:,,,. (56) Computational results for some, and are given in Tables 1 and 2, Also error surface for some, and are shown in Figures 1, 2, 3. Conclusion: In this paper, we introduced a new numerical scheme for Burgers' equation by two dimension block pulse functions and their operational matrices for partial derivatives. This method can be used for any linear and nonlinear partial differential equations. We can say that this method is feasible and the error is acceptable. the 607

7 implementation of the present method is a very easy acceptable and valid. We can use other piecewise constant functions for example Haar, Walsh and wavelets. Fig. 1: Error surface for m= 16, a =2,.. Fig. 2: Error surface for 16, 2, Fig. 3: Error surface for m=20,a=100,ε= Table 1: Numerical results for,, a=2, m=20,.,.. x Numerical Exact e e e e e e e e-2 Table 2: Numerical results for,, a=2, m=20,.,.. x Numerical Exact e e e e e e e e-5 REFERENCES Aksan, E.N., A. Ozdes and T. Ozis, A numerical solution of burgers' equation based on least squares approximation, App. Math.Comput., 176:

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