Analysing the deflection of plates or beams generally requires the solution of a two-point boundary value probzem associated with an ordinary differential equation. Such problems can be solved by finite difference methods, by shooting techniques, or by spline approximation. We develop and analyse these three types of methods.
We also cite sufficient conditions under which unique solutions of these boundary value problems exist and the requirements for obtaining solutions of various orders. Finite difference methods are easy to im plement, cheaper, and more accurate than the shooting methods of the same order. Spline methods are com parable to finite difference methods.
Albasiny, E.L.Hoskins, W.D.Cubic Spline Solutions to Two-Point Boundary Value ProblemsComputer Journal vol. 121969 pp. 151-153
3.
AZIZ, A.K., editor Numerical Solution of Boundary Value Problems for Ordinary Differential EquationsAcademic PressNew York1975
4.
Aziz, A.K.Hubbard, B.E.Bounds for the Solution of the Sturm-Liouville Problem with Applications to Finite Difference MethodsJournal of the Society for Industrial and Applied Mathematics vol. 121964 pp. 163-178
5.
6.
Collatz, L.The Numerical Treatment of Differential EquationsSpringerBerlin1966 3rd edition
7.
Fischer, C.F.Usmani, R.A.Properties of Some Tridiagonal Matrices and Their Applications to Boundary Value ProblemsJournal of the Society for Industrial and Applied Mathmatics: Numerical Analysis volume 61969 pp. 127-142
8.
9.
10.
Froberg, C.-E.Introduction to Numerical AnalysisAddison-WesleyReading, Massachusetts19692nd edition
11.
12.
Jain, M.K.Iyenger, S.R.K.Saldanha, J.S.V.Numerical Integration of a Fourth Order Ordinary Differential EquationJournal of Engineering Mathematics vol. 111977 pp. 373-380
13.
Keller, H.B.Numerical Solution to Boundary Value Problems for Ordinary Differential Equations; Survey and Recent Results In Aziz, A.K., editor, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Academic Press, New York, 1975), pp. 27-88
14.
Kreiss, H.O.Difference Approximations for Boundary and Eigenvalue Problems for Ordinary Differential EquationsMathematics of Computation vol. 261972 pp. 605-624
15.
Lees, M.Discrete Methods for Nonlinear Two-Point Boundary Value Problems In James Hubbard, editor, Numerical Solutions of Partial Differential Equations (Academic Press, New York , 1966), pp. 59-72
16.
17.
18.
Richardson, L.F.The Deferred Approach to the Limit. I: Single LatticeTransactions of the Royal Society London 1927 pp. 299-349
19.
20.
Sakai, M.Cubic Spline Function and Difference Methods Memoirs of the Faculty of Science, Kyushu University . Series A: Mathematics vol. 28 1974 pp. 43-58
21.
Sakai, M.Spline Interpolation and Two-Sided Approximate Methods for Two-Point Boundary Value Problems Reports of the Faculty of Science Kagoshima University vol. 11 1978 pp. 1-19
22.
Salvadori, M.G.Baron, M.L.Numerical Methods in EngineeringPrentice-Hall Englewood Cliffs, New Jersey1961
23.
24.
Steinberg, J.Numerical Solution of a Boundary Value Problem Arising in the Deflection of Beams and ShellsJournal of the Institute of Mathematics and Its Applications vol. 101972 pp. 10-14
25.
26.
Usmani, R.A.On the Numerical Integration of a Boundary Value Problem Involving a Fourth Order Linear Differential EquationTidskrift Informationsbehandling (BIT) vol. 171977 pp. 227-234
27.
Usmani, R.A.An O(h6) Finite Difference Analogue for the Solution of Some Differential Equations Occurring in Plate Deflection TheoryJournal of the Institute of Mathematics and Its Applications vol. 201977 pp. 331-333
28.
Usmani, R.A.Discrete Variable Methods for a Boundary Value Problem with Engineering ApplicationsMathematics of Computation vol. 321978 pp. 1087-1096
29.
Usmani, R.A.A Uniqueness Theorem for a Boundary Value ProblemProceedings of the American Mathematical Society vol. 771979 pp. 329-335
30.
Usmani, R.A.Discrete Methods for Boundary Value Problems with Applications in Plate Deflection TheoryJournal of Applied Mathematics and Physics (ZAMP) vol. 301979 pp. 87-99
31.
Usmani, R.A.Smooth Spline Approximations for the Solution of a Boundary Value Problem with Engineering ApplicationsJournal of Computational and Applied Mathematics vol. 61980 pp. 93-98
32.
Usmani, R.A.Andres, T.H.Walton, D.J.Error Estimation in the Integration of Ordinary Differential EquationsInternational Journal of Computer Mathematics Section B vol. 51976 pp. 241-256
33.
Usmani, R.A.Marsden, M.J.Numerical Solution of Some Ordinary Differential Equations Occurring in Plate Deflection TheoryJournal of Engineering Mathematics vol. 91975 pp. 1-10
34.
Usmani, R.A.Marsden, M.J.Convergence of a Numerical Procedure for the Solution of a Fourth Order Boundary Value ProblemProceedings of the Indian Academy of Science vol. 881979 pp. 21-30
35.
Usmani, R.A.Meek, D.S.On the Application of a Five-Band Matrix in the Numerical Solution of a Boundary Value ProblemUtilitas Mathematica vol. 141978 pp. 21-29
36.
Usmani, R.A.Warsi, S.A.Quintic Spline Solutions of Boundary Value ProblemsJournal of Computer and Mathematics with Applications vol. 61980 pp. 197-203
37.
Usmani, R.A.Warsi, S.A.Smooth Spline Solutions for Boundary Value Problems in Plate Deflection TheoryJournal of Computer and Mathematics with Applications vol. 61980 pp. 205-211
38.
Varga, R.S.Matrix Iterative AnalysisPrentice-Hall Englewood Cliffs, New Jersey1962
