xn−1 < xn = b is established by constructing it. The construction of s(x) is done with the help of the quintic B-splines. Introduce ten additional knots x−5, x−4, x−3, ...
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.10(2010) No.2,pp.222-230
Numerical Solutions of Fourth Order Boundary Value Problems by Galerkin Method with Quintic B-splines 1
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K.N.S. Kasi Viswanadham1 ∗ , P. Murali Krishna1 , Rao S. Koneru2
Department of Mathematics, National Institute of Technology Warangal -506 004 INDIA. Retired Professor of Mathematics, Indian Institute of Technology Bombay, Mumbai - 400 076 INDIA. (Received 11 June 2010, accepted 5 July 2010)
Abstract: In this paper, Galerkin method with quintic B-splines as basis functions is presented to solve a fourth order boundary value problem with two different cases of boundary conditions. In the method, the basis functions are redefined into a new set of basis functions which vanish at the boundary where the Dirichlet type of boundary conditions are prescribed. The proposed method is tested on several numerical examples of fourth order linear and nonlinear boundary value problems. Numerical results obtained by the proposed method are in good agreement with the exact solutions available in the literature. Keywords: Galerkin method; quintic B-splines; basis functions; fourth order boundary value problems; band matrix; absolute error
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Introduction
Generally, fourth order boundary value problems arises in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plates deflection theory, beam element theory and many more applications of engineering and applied mathematics. Solving such type of boundary value problems analytically is possible only in very rare cases. Many researchers worked for the numerical solutions of fourth order boundary value problems [1–5]. In this paper, we try to present a simple finite element method which involves Galerkin approach with quintic B-splines as basis functions to solve the fourth order boundary value problems with two different cases of boundary conditions. In this paper, we consider a general fourth order linear boundary value problem given by 𝑎0 (𝑥)𝑦 (4) + 𝑎1 (𝑥)𝑦 ′′′ + 𝑎2 (𝑥)𝑦 ′′ + 𝑎3 (𝑥)𝑦 ′ + 𝑎4 (𝑥)𝑦 = 𝑏(𝑥),
