Numerical Evaluation of Integrals with weight function x k Using Gauss Legendre Quadrature Rules

Transcription

1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: 29-76X. Volume, Issue Ver. V (May - Jun. 2), PP Numerical Evaluation of Integrals with weight function k Using Gauss Legendre Quadrature Rules Shivaram. K. T, Manjula. G 2, Umesha. V, Prakasha. H.T, Department of Mathematics, Dayananda Sagar College of Engineering, Bangalore, Karnataka, India. 2 Department of Mathematics, Government P. U. College, Thippasandra. Karnataka, India. Department of Mathematics, A.S.C. Independent P. U. College, Bangalore, Karnataka, India. Abstract : In this paper. Gauss Legendre quadrature have been applied for numerical solution of the integral k of the form f d, where k is real number. We compare the numerical solutions with J. Ma, V. Rokhlin, S. Wandzura. et al. [8]. The performance of the method is illustrated with numerical eamples. Key words: Finite element method, Numerical Integration, Gauss Legendre Quadrature I. Introduction Integration is of fundamental importance in both pure and applied mathematics as well as in several areas of science and engineering. Most such integrals cannot be evaluated eplicitly or analytically and, with many others, it is often faster to integrate them numerically rather than evaluate them eactly using the complicated antiderivatives of the integrands. Construction of quadrature rules for numerical integration based on interpolating polynomials is done by many authors that these polynomials are used to find weights corresponding to nodes. Gauss Legendre quadrature rules are etremely efficient when the functions to be integrated are well approimated by polynomials. However, the integrands in practical situations are not always polynomial but rational or irrational epressions of trigonometric, eponential, logarithmic etc. For which there is no order of Gaussian quadrature rules cannot evaluate these integrals eactly. From the literate review we may observe that various numerical integration schemes, the use of Gaussian quadrature is attractive and it can evaluate eactly the (2n -)th order polynomials with n Gauss integration points in []. Rational epressions for which there is no order of Gauss quadrature that will evaluate these integrals eactly in [- ] The integration rule proposed in this paper require zeros of P 2n () and computations of the associated weights. The integration points have to be increased in order to improve the integration accuracy. This method is applicable to a wide class of functions, functions with end point singularities, such as those encountered in the solution of integral equations, comple analysis, potential theory, and several other areas. The remainder of this paper is organized as follows. Section 2. presents the mathematical preliminaries required for understanding the derivation and also calculated the Gaussian Legendre quadrature nodes and weights of order N=,,,2,,. In Section. we provide the numerical results, with comparisons, error estimates and some illustrative eamples. II. Formulation of integrals with weight function k 2. If k = The Numerical integration of an arbitrary function f is given by I = f d = t f(t m )d = W k t k f(t m k= k ) = W k k f( k ) 2.2 If k = I 2 = f() The Numerical integration of an arbitrary function f is given by d = tf(t )d m = W k t k f(t m k= k ) = W k k f( k ) 2. If k = The Numerical integration of an arbitrary function f is given by I = f d = t f(t m )d = W k t k f(t m k= k ) = W k k f( k ) 2. If k = The Numerical integration of an arbitrary function f is given by k= () k= (2) k= () DOI:.979/ Page

2 Numerical Evaluation of Integrals with weight function k Using Gauss I = f() d = t 2 f(t )d m 2 = W k t k f(t m 2 k= k ) = W k k f( k ) k= () 2. If k = 2 The Numerical integration of an arbitrary function f is given by I = 2 f()d = t f(t m )d = W k t k f(t m k= k ) = W k k f( k ) 2.6 If k = 2 The Numerical integration of an arbitrary function f is given by I 6 = f() d = f(t m )d = W 2 k f(t m k= k ) = k= W k f( k ) (6) k= () We find out new Gaussian points k and weights coefficients W k of various order N =,,,, by using equation () - (6)and tabulated in Table. 6 III. Numerical Results We have compared the Numerical results obtained using the proposed method with that of the results given in J.Ma et al. [8] and also estimated error of various order of N are tabulated in Table Table : Integration by the Gauss Legendre quadrature for the weight function w() = 2 2 d = sin () d = d = Table 2: Integration by the Gauss Legendre quadrature for the weight function w() = 2 d = sin () d = d = DOI:.979/ Page

3 Numerical Evaluation of Integrals with weight function k Using Gauss Table : Integration by the Gauss Legendre quadrature for the weight function w() = 2 2 d = sin () d = d = Table : Integration by the Gauss Legendre quadrature for the weight function w() = 2 d = sin () d = d = DOI:.979/ Page

4 Numerical Evaluation of Integrals with weight function k Using Gauss Table : Integration by the Gauss Legendre quadrature for the weight function w() = 2 d = sin () d = d = IV. Conclusions In this paper, numerical integration of the form k f d where k = ±, ±, ± 2 New Gaussian points and weight coefficients are calculated of order N =,,,2,,. Most of the results obtained here are eact up to decimal places and the proposed method can be used to integrate a wide class of functions including functions with end point singularities Table:6 Weight function = Weight function = Order k W k k W k N= N= N= DOI:.979/ Page

5 Numerical Evaluation of Integrals with weight function k Using Gauss DOI:.979/ Page N= N=

6 Numerical Evaluation of Integrals with weight function k Using Gauss References []. O.C. Zienkiewicz, The Finite Element Method, rd. Edition, Mcgraw - Hill Inc., New York (977). [2]. W. B. Bickford. (99). A First Course In The Finite Element Method, Irwin, Homewood, (99). []. G. Yogawa, G. W. Ye And S. Yoshimura. A Numerical Integration Scheme For Finite Element Method Based On Symbolic Manipulation, Int. J. Numer. Methods. Eng 29, Pp. 9-9, (99). []. M. Abramowicz And I.A. Stegun. Handbook Of Mathematical Functions. Dover Publications, Inc. New York (96). []. J. N. Reddy. An Introduction To The Finite Element Method. Tata Mcgraw- Hill, Third Edition (2). [6]. H. T. Rathod. Gauss Legendre Over A Triangle. J. Indian Institute Of Science., Vol.8, Pp.8-88, (2). [7]. Farzana Hussain, M.S. Karim, Accurate Evaluation Schemes For Triangular Domain Integrals. Journal Of Mechanical And Civil Engineering, Vol.2, Pp. 8-, (22), [8]. J. Ma, V. Rokhlin, S. Wandzura, Generalized Gaussian Quadrature Rules For Systems Of Arbitrary Functions, Siam, J. Numer. Anal. Vol., No.,Pp , (996) [9]. K. T. Shivaram. Generalised Gaussian Quadrature Over Triangle, American Journals Of Engineering Research..Vol.2,Pp (2), []. Bharath Rathod, K.T. Shivaram And H.T. Rathod, Some Generalized Gaussian Quadrature Rule For Integrands With Singularities. []. International E-Journal Of Numerical Analysis And Related Topic, Vol.6,Pp.2-6,.(2), [2]. K.T.Shivaram, Numerical Integration Over An Arbitrary Rectangle And Square Region Using Generalised Gaussian Quadrature Rules, []. International Journal Of Mathematical Archive, Vol.(), May 2. DOI:.979/ Page