Journal of Computational and Applied Mathematics

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1 Journal of Computational and Applied Mathematics 234 (21) Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: A numerical solution of problems in calculus of variation using direct method and nonclassical parameterization Mohammad Maleki a,, Mahmoud Mashali-Firouzi b a Department of Mathematics, Payame oor University (PU), Isfahan, Iran b Department of Mathematics, School of Mohajer, Hezar Jerib Street, Isfahan, Iran a r t i c l e i n f o a b s t r a c t Article history: Received 28 May 28 Received in revised form 11 December 29 Keywords: onclassical parameterization Calculus of variation onlinear programming umerical methods In this paper a direct method for solving variational problems using nonclassical parameterization is presented. A nonclassical parameterization based on nonclassical orthogonal polynomials is first introduced to reduce a variational problem to a nonlinear mathematical programming problem. Then, using the Lagrange multiplier technique the problem is converted to that of solving a system of algebraic equations. Illustrative examples are given to demonstrate the validity and applicability of the technique. 21 Elsevier B.V. All rights reserved. 1. Introduction In a large number of problems arising in analysis, mechanics, geometry and so forth, it is necessary to determine the maximum or minimum of a certain functional. Because of the important role of this subject in science and engineering, considerable attention has been received by this kind of problems. Such problems are called variational problems. Some popular methods for solving variational problems are direct methods. In [1,2] the direct method of Ritz and Galerkin is investigated for solving variational problems. Chen and Hsiao [3] introduced the Walsh series method to variational problems. Due to the nature of the Walsh functions, the solution obtained was piecewise constant. Some orthogonal polynomials are applied on variational problems to find the continuous solutions for these problems [4 6]. Also the authors of [7 12] introduced the Legendre Wavelets method, rationalized Haar method, Adomian decomposition method, He s variational iteration method and Chebyshev finite difference method for solving variational problems, respectively. More historical comments about variational problems are found in [1,2]. A direct method converts the variational problem into a mathematical programming problem by using either the discretization technique [13,14] or the parameterization technique [15 17]. In classical parameterization, the variables of the problem are estimated using a finite length series of classical orthogonal polynomials such as Chebyshev, Legendre, etc. with unknown parameters [14 17]. The class of solution methods based on orthogonal polynomials has become known as spectral methods. Spectral methods are implemented in various ways. For example, the tau, Galerkin, and collocation methods have all been proposed as implementation strategies [18,19]. The collocation method has established itself as the one that permits the most convenient computer implementation. However, in pseudospectral methods the nodes must correspond to the zeros of the derivatives of classical orthogonal polynomials on the interval [ 1, 1], including the end points. The points are generally based on the Legendre or Chebyshev polynomials. The idea of employing nonclassical weight functions has been used in [2 26]. Corresponding author. address: mm_maleki25@yahoo.com (M. Maleki) /$ see front matter 21 Elsevier B.V. All rights reserved. doi:1.116/j.cam

2 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) In the present paper, we propose a direct method using nonclassical parameterization and nonclassical orthogonal polynomials, for finding the extremal of variational problems. The proposed method requires the definition of collocation points and nonclassical orthogonal polynomials. The application of the method to variational problem leads to a nonlinear mathematical programming problem and using Lagrange multiplier technique, to an algebraic system. In fact, our method is a combination of the pseudospectral and parameterization methods which improves the parameterization technique. The proposed method differs from the traditional parameterization and pseudospectral methods in several ways. First, by using the quasilinearization technique, the presented method could be employed for problems with non quadratic functional. Second, in the presented method the unknown parameters that would be determined are the values of variables x(t), in nodes of interpolation but in traditional parameterizations the unknown parameters are coefficients of a series of orthogonal polynomials. Third, in the proposed method a weighted interpolant is used to approximate the unknown functions as opposed to Lagrange interpolating polynomials. Fourth, the nodes are based on the roots of derivatives of the th order orthogonal polynomials based on arbitrary positive weight function across an arbitrary interval, whereas the traditional pseudospectral methods use nodes based on the roots of the derivatives of th order Jacobi polynomials across the interval [ 1, 1]. The proposed method has the following advantages: it can be employed for time-varying and nonlinear variational problems; the number of unknown parameters to be determined is less than the discretization technique [13,14]; this method converts the problems with non quadratic functional to a sequence of quadratic programming problems and gives the information of quadratic programming problem, the Hessian and the gradient, explicitly, and there is no need to use the differentiation or the finite difference method to construct these information; in this method by taking a fixed degree of weighted interpolation and changing the weight functions the obtained results can be improved. This paper is organized as follows: The following section is devoted to the explanation of the classical and nonclassical estimations of functions and generation of collocation points. In Section 3, we introduce the statement of problems in calculus of variations. In Section 4, we describe the basic formulation of the nonclassical parameterization method. umerical examples are then given in Section 5 to illustrate the applicability of the proposed method. 2. onclassical parameterization 2.1. Expansion in classical and nonclassical orthogonal polynomials The classical parameterization technique [15 17] approximates the function f L 2 [ 1, 1] by a finite length series of classical orthogonal polynomials such as Chebyshev or Legendre polynomials, with unknown parameters f (ˆt) = a j P j (ˆt), ˆt [ 1, 1], j= where P j (ˆt) is the jth order classical orthogonal polynomial and aj s are the unknown parameters. In this paper, we propose to expand the function using weighted interpolations of degree of the form [2,21] f (t) = F (t) = W(t) j= W(t j ) L j(t)f (t j ), t [a, b], (1) where t j, j =, 1,...,, are a set of distinct interpolation nodes in [a, b], W(t) is a positive weight function and L j (t), are a set of interpolating functions satisfies L j (t k ) = δ jk. Thus, F (t), defined by Eq. (1), is an interpolant to the function f (t) in the sense that f (t k ) = F (t k ), k =, 1,...,. The functions L j (t) are often chosen to be sets of interpolating polynomials of degree, in which case they can be represented explicitly by Lagrange s formula Generation of collocation points Here the main idea for the generation of collocation points (interpolation nodes), t j, is to form new sets of polynomials P n (t) orthogonal with respect to a weight function w(t) on the interval [a, b], that is b a w(t)p n (t)p m (t)dt = δ nm. The polynomials satisfy a three-term recurrence relation [22] P k+1 (t) = (t α k )P k (t) β k P k 1 (t), k =, 1, 2,... (3) P 1 (t) =, P (t) = 1. (2)

3 1366 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) It is important to note that in Eq. (1), W(t) is some weight function which adds additional flexibility because by taking a fixed and changing the weight functions, variate weighted interpolations can be employed as approximations of f (t). Also in Eq. (2), w(t) defines the orthogonal polynomials and the collocation points (interpolation nodes). The collocation points t j may be determined by the method outlined in [27]. The approach is based on determining the eigenvalues of a modified tridiagonal Jacobi matrix α β1 β1 α 1 β2 β2 α 2 β3 J = ,.. β 1 α 1 β β α where α, β are obtained from the solution of the linear system of equations ( ) ( ) ( ) P (a) P 1 (a) α ap (a) =. P (b) P 1 (b) bp (b) β The collocation points t j, including the end points, are determined as the eigenvalues of J, and can be determined if the coefficients α k and β k, k =, 1, 2,..., are known. The procedure for calculating the coefficients is discussed in the next subsection Generation of nonclassical orthogonal polynomials The generation of a set of polynomials orthogonal with respect to some weight function w(t) has been discussed in several texts [28 3]. For a nonclassical weight function, the usual Schmidt procedure [31] which involves orthogonalization of a given member of the set to all the functions of lower order, is highly unstable due to roundoff errors, and is not practical. The Schmidt procedure is analogous to the methods based on the moment of the weight function. The best approach is one based on the three term recurrence relation of the polynomials {P k (t)} given in Eq. (3). The recurrence coefficients in Eq. (3) are given in [23] by b a α k = tw(t)p 2 k (t)dt b β = b a a w(t)p 2 k (t)dt, k =, 1, 2,... w(t)p 2 (t)dt, β k = b a w(t)p 2 k (t)dt b. a w(t)p 2 k 1 (t)dt 3. Problem statement The simplest form of a variational problem can be considered as finding the extremum of the functional J[x(t)] = t1 t F[x(t), ẋ(t), t]dt. To find the extreme value of J, the boundary points of the admissible curves are known in the following form x(t ) = α, x(t 1 ) = β. (4) (5) The necessary condition for x(t) to extremize J[x(t)] is that it should satisfy the Euler Lagrange equation F x d ( ) F =, dt ẋ with boundary conditions given in Eq. (5). The above boundary value problem, does not always have a solution and if the solution exists, it may not be unique. ote that if the solution of Euler s equation satisfies the boundary conditions, it is unique. The general form of the variational problem in Eq. (4) is J[x 1, x 2,..., x n ] = t1 t F[x 1, x 2,..., x n, ẋ 1, ẋ 2,..., ẋ n, t]dt, (6)

4 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) Table 1 Different w(t) and W(t) on [, 1]. Case w(t) W(t) (1 + t) 1 3 e t2 1 4 e 2t 1 5 e 2t e t 6 e 2t (1 + t) t + t cos(t) cos(t) (1 + t) sin(t) 1 with the given boundary conditions for all functions x 1 (t ) = α 1, x 2 (t ) = α 2,..., x n (t ) = α n, (7) x 1 (t 1 ) = β 1, x 2 (t 1 ) = β 2,..., x n (t 1 ) = β n. (8) Here the necessary condition for the extremum of the functional in Eq. (6) is to satisfy the following system of second-order differential equations F d ( ) F =, i = 1, 2,..., n, (9) x i dt ẋ i with boundary conditions given in Eqs. (7) (8). However, the above system of differential equations can be solved easily only for simple cases. Here we consider a direct method for finding the extremum of Eq. (6) using a nonclassical parameterization technique. 4. Direct method using nonclassical parameterization Approximating variables x i (t), i = 1, 2,..., n, using Eq. (1), gives x i (t) = W(t) W(t j ) L j(t)a ij = a T i P(t), i = 1, 2,..., n, (1) j= where a i = [a i, a i1,..., a i ] T, i = 1, 2,..., n, are (+1) 1 vectors of unknown parameters with property that a ij = x i (t j ), and P(t) = [P (t), P 1 (t),..., P (t)] T is ( + 1) 1 vector with P j (t) = W(t) W(t j ) L j(t), j =, 1,...,. Then, ẋ i (t), can be represented as ẋ i (t) = a T i P (t), i = 1, 2,..., n, (11) where, P (t) is derivative vector of P(t). Substituting Eqs. (1) (11) in Eq. (6), the functional J becomes a nonlinear mathematical programming problem of unknown parameters a ij, where i = 1, 2,..., n, and j =, 1,...,. Hence, to find the extremum of J, we solve J a ij =, i = 1, 2,..., n, j =, 1,...,. (12) The above procedure is now used to solve the following variational problems. 5. Illustrative examples In this section, three examples are given to demonstrate the applicability and accuracy of our method. There are many combinations of orthogonal weight functions w(t) and weighted interpolating polynomials W(t) that could be selected [21, 24]. In this paper only a limited number of combinations are used for each example. The different cases are summarized in Table 1, the similar to these weights were also used in [24]. ote that we have computed the numerical results by the well-known symbolic software Mathematica 5.2.

5 1368 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) Example 1 Consider the problem of finding the minimum of the time-varying functional [6] J(x) = [ẋ 2 + tẋ + x 2 ]dt, (13) with boundary conditions x() =, x(1) = 1 4. (14) The exact solution of this problem is x(t) = c 1e t + c 2 e t where c 1 = Using Eqs. (1) (11) and (13), we get J(x) = Eq. (15) can be simplified to 2 e 4(e 2 1) and c 2 = e 2e2 4(e 2 1). [a T P (t)p T (t)a + a T tp (t) + a T P(t)P T (t)a]dt. (15) J(x) = 1 2 at Ha + c T a, (16) where H = 2 [P (t)p T (t) + P(t)P T (t)]dt, c T = tp T (t)dt. Eq. (1) and the boundary conditions in Eq. (14) imply x() = a T P() =, x(1) = a T P(1) = 1 4. (17) The quadratic programming problem in Eqs. (16) (17) can be rewritten as follows: subject to where J(x) = 1 2 at Ha + c T a, F 1 a b 1 =, F 1 = ( ) ( ) P T (), P T b (1) 1 = 1. 4 The optimal values of unknown parameters, a, can be obtained easily using Lagrange multiplier technique as a = H 1 c + H 1 F T 1 (F 1H 1 F T 1 ) 1 (F 1 H 1 c + b 1 ), by substituting these optimal parameters in Eq. (1) the values of x(t) can be calculated. ow we define the maximum errors for x (t) as E = x (t) x exact (t) = max{ x (t) x exact (t), t 1}, where x (t) is the approximation of x exact (t). In Table 2 we give the errors E for = 6 and = 1 and different cases of Table 1. In Table 3 a comparison is made between the approximate values of x(t) using the present method with = 6 and case 8, using the RH functions method [8] for k = 8 and the exact solution. In Table 4 a comparison is made between the approximate values of x(t) using the present method with = 6 and case 8, using the Legendre wavelets method [7] for M = 3 and k = 3 and the exact solution. Table 2 shows that in the present method by taking a fixed and changing the weight functions the obtained results improve.

6 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) Table 2 The maximum errors of E for = 6, 1 for example E (case 1) E (case 2) E (case 3) E (case 4) E (case 5) E (case 6) E (case 7) E (case 8) E (case 9) E (case 1) Table 3 Estimated and exact values of x(t) for example 1. t RH functions for k = 8 [8] t Present method with = 6 and case 8 Exact t < 1/ /8 t < 2/ /8 t < 3/ /8 t < 4/ /8 t < 5/ /8 t < 5/ /8 t < 6/ /8 t < 7/ /8 t < t = Table 4 Estimated and exact values of x(t) for example 1. t Legendre wavelets method for M = 3 and k = 3 [7] Present method with = 6 and case 8 Exact Example 2 In this example, we consider the following variational problem [2,1]. J = 1 + x 2 (t) dt, ẋ 2 (t) that satisfies the conditions x() =, x(1) =.5. The exact solution of this problem is x(t) = sinh( t). In this problem the functional has not quadratic form, hence, for solving this problem using the presented method, we first (using the quasilinearization method [32]) expand the performance index and boundary conditions around x k (t) and ẋ k (t) up to the second and first order respectively, to get the following sequence of linear-quadratic problems. [ 1 J k+1 = 2 C k (t)x k ] 2 Ek (t)(ẋ k+1 ) 2 + α k (t)x k+1 + β k (t)ẋ k+1 + D k (t)x k+1 ẋ k+1 + h k (t) dt, (2) subject to (18) (19) x k+1 () =, x k+1 (1) =.5, (21)

7 137 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) where h k (t) = f (x k, ẋ k, t) A k (t)x k B k (t)ẋ k C k (t)x k Ek (t)(ẋ k ) 2 + D k (t)x k ẋ k, A k f (x, ẋ, t) (t) =, x x k,ẋ k B k f (x, ẋ, t) (t) =, ẋ x k,ẋ k C k (t) = 2 f (x, ẋ, t), x 2 x k,ẋ k D k (t) = 2 f (x, ẋ, t), x ẋ x k,ẋ k E k (t) = 2 f (x, ẋ, t), ẋ 2 x k,ẋ k α k (t) = A k (t) x k C k (t) ẋ k D k (t), β k (t) = B k (t) x k D k (t) ẋ k E k (t), and in this problem we have f (x, ẋ, t) = (1 + x 2 (t))/ẋ 2 (t). The initial guesses of x (t) and ẋ (t) is requested. We suggest to start from the linear function between the initial value x() = and the final value x(1) =.5 for x (t) and its derivation for ẋ (t). Substituting Eqs. (1) (11) in sequence of problems in Eqs. (2) (21) gives a sequence of problems as follows: [ 1 J k+1 = 2 C k (t) ( ) a T PP T a Ek (t) ( a T P ) P T a + α k (t) ( ) a T P + β k (t) ( a T P ) + D k (t) ( ) ] a T PP T a + h k (t) dt, (22) subject to a T P() =, a T P(1) =.5, where J k+1 is the approximate value of J k+1. The sequence of problems in Eqs. (22) (23) can be rewritten as a sequence of quadratic programming problems as follows: (23) subject to where J k+1 = 1 2 at Ha + c T a + d, F 1 a b 1 =, H = c T = d = h k (t)dt, ( ) P F 1 = T (), P T (1) ( ) b 1 =..5 [ C k (t)pp T + E k (t)p P T + 2D k (t)pp T ] dt, [ α k (t)p T + β k (t)p T ] dt, (24) (25) These quadratic programming problems are solved similarly to Example 1. After obtaining the optimal solution of the unknown parameters a, we substitute these parameters in Eqs. (1) (11) to obtain the new nominal rate x k (t) and ẋ k (t) to be used in the next iteration. These new nominal trajectories have to be substituted in Eqs. (2) (21) to get the next constrained quadratic problem. This procedure has to be repeated until an acceptable convergence is achieved. In this paper the computations are terminated if (J k+1 J k k+1 )/J < 1 1. The convergence is achieved after three iterations for each.

8 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) Table 5 The maximum errors of E for example E (case 1) E (case 2) E (case 3) E (case 4) E (case 5) E (case 6) E (case 7) E (case 8) E (case 9) E (case 1) Method of Saadatmandi and Dehghan [12] ote that we can employ the method explained in this example for all nonlinear problems that their functional in not quadratic. In Table 5 a comparison is made between the errors E using the present method with = 3, 4, 5 and 6 and the method of Saadatmandi and Dehghan [12] by using Chebyshev finite difference method. ot that their method is not direct and it solves the two point boundary value problem arrives in Eq. (9) Example 3 Consider the extremization of functional [2,1] J(x 1 (t), x 2 (t)) = π 2 with the boundary conditions ( π ) x 1 () =, x 1 = 1, 2 ( π ) x 2 () =, x 2 = 1. 2 [ẋ ẋ x 1x 2 ]dt, (26) The exact solutions of the problem are x 1 (t) = sin(t) and x 2 (t) = sin(t). For this problem the transformation t = π τ is 2 used to change the variational problem in Eqs. (26) (28) to the following form J(x 1 (τ), x 2 (τ)) = π [ 4 2 π 2 ẋ2 + 4 ] 1 π 2 ẋ x 1x 2 dτ, (29) with the boundary conditions x 1 () =, x 1 (1) = 1, x 2 () =, x 2 (1) = 1. We approximate x 1 (τ) and x 2 (τ) and their derivations using Eqs. (1) (11) to get the following nonlinear mathematical programming as approximation to the original problem. subject to where J = a T 1 H 1a 1 + a T 2 H 1a 2 + a T 1 H 2a 2, F 1 a 1 b 1 =, F 1 a 2 b 2 =, H 1 = 2 π P (τ)p T (τ)dτ, H 2 = 2 P(τ)P T (τ)dτ, π ( ) P F 1 = T (), P T (1) (27) (28) (3) (31) (32) (33) (34)

9 1372 M. Maleki, M. Mashali-Firouzi / Journal of Computational and Applied Mathematics 234 (21) Table 6 The maximum errors of E and E for example E (case 1) E E (case 2) E E (case 3) E E (case 4) E E (case 5) E E (case 6) E E (case 7) E E (case 8) E E (case 9) E E (case 1) E Method of Saadatmandi and Dehghan [12] E E ( ) b 1 =, 1 ( ) b 2 =. 1 The nonlinear mathematical programming problem in Eqs. (32) (34) can be solved easily by Lagrange multiplier technique. In Table 6 a comparison is made between the maximum errors E and E for x 1(t) and x 2 (t), respectively, for = 6, 1, 12 and the method of Saadatmandi and Dehghan [12] by using Chebyshev finite difference method. 6. Conclusion This paper describes an efficient method for finding the minimum of a functional over the specified domain. The main objective is to estimate the rate variables using nonclassical parameterization based on nonclassical orthogonal polynomials, to arrive at a nonlinear mathematical programming problem. Applications of the presented method are demonstrated through illustrative examples. The obtained results show that this approach can solve the problem effectively. Acknowledgements The authors are very grateful to the anonymous referees and also Prof. T. Mitsui (editor) for carefully reading the paper and for their comments and suggestions. References [1] I.M. Gelfand, S.V. Fomin, Calculus of Variations, Prentice-Hall, J, 1963 (revised English edition translated and edited by R.A. Silverman). [2] L. Elsgolts, Differential Equations and Calculus of Variations, Mir, Moscow, 1977 (translated from the Russian by G. Yankovsky). [3] C.F. Chen, C.H. Hsiao, A walsh series direct method for solving variational problems, J. Franklin Inst. 3 (1975) [4] R.Y. Chang, M.L. Wang, Shifted Legendre direct method for variational problems, J. Optim. Theory Appl. 39 (1983) [5] I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for solving variational problems, Internat. J. Systems Sci. 16 (1985) [6] C. Hwang, Y.P. Shih, Laguerre series direct method for variational problems, J. Optim. Theory Appl. 39 (1) (1983) [7] M. Razzaghi, S. Yousefi, Legendre wavelets direct method for variational problems, Math. Comput. Simulation 53 (2) [8] M. Razzaghi, Y. Ordokhani, An application of rationalized Haar functions for variational problems, Appl. Math. Comput. 122 (21) [9] M. Dehghan, M. Tatari, The use of Adomian decomposition method for solving problems in calculus of variations, Math. Probl. Eng. 26 (26) [1] M. Tatari, M. Dehghan, Solution of problems in calculus of variations via He s variational iteration method, Phys. Lett. A 362 (27) [11] S.A. Yousefi, M. Dehghan, The use of He s variational iteration method for solving variational problems, Int. J. Comput. Math. (29), doi:1.18/

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