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http://dx.doi.org/10.4134/CKMS.c150245

THE COMPUTATION OF POSITIVE SOLUTIONS FOR A BOUNDARY VALUE PROBLEM OF THE LINEAR BEAM EQUATION  

Ji, Jun (Department of Mathematics Kennesaw State University)
Yang, Bo (Department of Mathematics Kennesaw State University)
Publication Information
Communications of the Korean Mathematical Society / v.32, no.1, 2017 , pp. 215-224 More about this Journal
Abstract
In this paper, we propose a method of order two for the computation of positive solutions to a boundary value problem of the linear beam equation. The method is based on the Power method for the eigenvector associated with the dominant eigenvalue and the Crout-like factorization algorithm for the banded system of linear equations. It is extremely fast due to the linear complexity of the linear system solver. Numerical result of a test problem is included.
Keywords
boundary value problem; Crout-like factorization algorithm; linear beam equation; positive eigenvector;
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1 R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference, and Integral Equations, Kluwer Academic, Dordrecht, 1998.
2 R. L. Burden and J. D. Faires, Numerical Analysis, Thomson Brooks/Cole, Belmont, CA, 2005.
3 J. M. Davis, L. H. Erbe, and J. Henderson, Multiplicity of positive solutions for higher order Sturm-Liouville problems, Rocky Mountain J. Math. 31 (2001), no. 1, 169-184.   DOI
4 J. R. Graef and J. Henderson, Double solutions of boundary value problems for 2mth-order differential equations and difference equations, Comput. Math. Appl. 45 (2003), no. 6-9, 873-885.   DOI
5 J. R. Graef and B. Yang, Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems, Appl. Anal. 74 (2000), no. 1-2, 201-214.   DOI
6 C. P. Gupta, Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. Anal. 26 (1988), no. 4, 289-304.   DOI
7 J. Ji and B. Yang, Eigenvalue comparisons for boundary value problems of the discrete beam equation, Adv. Difference Equ. 2006 (2006), Article ID 81025, 1-9.
8 J. Ji and B. Yang, Positive solutions for boundary value problems of second order difference equations and their computation, J. Math. Anal. Appl. 367 (2010), no. 2, 409-415.   DOI
9 J. Ji and B. Yang, Eigenvalue Comparisons for a class of boundary value problems of discrete beam equation, Appl. Math. Comput. 218 (2012), no. 9, 5402-5408.   DOI
10 P. Pietramala, A note on a beam equation with nonlinear boundary conditions, Bound. Value Probl. 2011 (2011), Article ID 376782, 14 pages.   DOI
11 Q. Yao, An existence theorem for a nonlinear elastic beam equations with all order derivatives, J. Math. Study 38 (2005), no. 1, 24-28.
12 R. A. Usmani, Discrete variable methods for a boundary value problem with engineering applications, Math. Compu. 32 (1978), no. 144, 1087-1096.   DOI
13 R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.