zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
B-spline collocation for solution of two-point boundary value problems. (English) Zbl 1222.65076
The numerical solution based on B-spline collocation for boundary value problems for nonlinear differential equations up to sixth order is studied. By using the sextic B-spline collocation at the midpoints of a uniform mesh, a numerical method of order 6 is proposed. Then, an error analysis and the convergence of the method are investigated in detail. Several numerical examples, including numerical solutions for some stiff problems, are given for illustrating the efficiency of the B-spline collocation method.
MSC:
65L10Boundary value problems for ODE (numerical methods)
65L20Stability and convergence of numerical methods for ODE
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
65L70Error bounds (numerical methods for ODE)
65L04Stiff equations (numerical methods)
34B15Nonlinear boundary value problems for ODE
References:
[1]Timoshenko, S.; Woinowsky-Krieger, S.: Theory of plates and shells, (1959)
[2]Chawla, M. M.; Katti, C. P.: Finite difference methods for two-point boundary value problems involving high order differential equations, BIT numer. Math. 19, 27-33 (1979) · Zbl 0401.65053 · doi:10.1007/BF01931218
[3]Chawla, M. M.; Katti, C. P.: A finite difference method for a class of singular two-point boundary value problems, IMA J. Numer. anal. 4, 457-466 (1984) · Zbl 0571.65076 · doi:10.1093/imanum/4.4.457
[4]Chawla, M. M.; Subramanian, R.; Sathi, L. H.: A fourth order method for a singular two-point boundary value problem, BIT numerical mathematics 28, 88-97 (1988) · Zbl 0636.65079 · doi:10.1007/BF01934697
[5]Mohanty, R. K.: A fourth-order finite difference method for the general one dimensional nonlinear biharmonic problems of first kind, J. comput. Appl. math. 114, 275-290 (2000) · Zbl 0963.65083 · doi:10.1016/S0377-0427(99)00202-2
[6]Ahlberg, J. H.; Nilson, E. N.; Walsh, J. L.: The theory of splines and their applications, (1967) · Zbl 0158.15901
[7]C.D. Boor, The method of projection as applied to the numerical solution of two point boundary value problems using cubic splines. Dissertation, University of Michigan, Ann Arbor, Michigan, 1966.
[8]Russell, R. D.; Shampine, L. F.: A collocation method for boundary value problems, Numer. math. 19, 1-28 (1972) · Zbl 0221.65129 · doi:10.1007/BF01395926
[9]Bickley, W. G.: Piecewise cubic interpolation and two-point boundary value problems, Comput. J. 11, 206-208 (1968) · Zbl 0155.48004
[10]Ablasiny, E. L.; Hoskins, W. D.: Cubic spline solutions to two-point boundary value problems, Comput. J. 12, 151-153 (1969) · Zbl 0185.41403 · doi:10.1093/comjnl/12.2.151
[11]Fyfe, D. J.: The use of cubic splines in the solution of certain fourth order boundary value problems, Comput. J. 13, 204-205 (1970) · Zbl 0191.16701
[12]Fyfe, D. J.: The use of cubic splines in the solution of two-point boundary value problems, Comput. J. 12, 188-192 (1969) · Zbl 0185.41404 · doi:10.1093/comjnl/12.2.188
[13]Sakai, M.: Piecewise cubic interpolation and two-point boundary value problems, Publ. res. Inst. math. Sci. 7, 345-362 (1971) · Zbl 0236.65054 · doi:10.2977/prims/1195193546
[14]Papamichael, N.; Worsey, A. J.: A cubic spline method for the solutions of a linear fourth order two-point boundary value problem, Brunel university /101 (1981)
[15]Gladwell, I.; Mullings, D. T.: On the effect of boundary conditions in collocation by polynomial splines for the solution of boundary value problems in ordinary differential equations, J. lnst. Maths. appl. 16, 93-107 (1975) · Zbl 0341.65061 · doi:10.1093/imamat/16.1.93
[16]Daniel, J. W.; Swartz, B. K.: Extrapolated collocation for two-point boundary value problems using cubic splines, J. inst. Maths appl. 16, 161-174 (1975) · Zbl 0402.65051 · doi:10.1093/imamat/16.2.161
[17]Irodotou-Ellina, M.; Houstis, E. N.: An o(h6) quintic spline collocation method for fourth order two-point boundary value problems, BIT numerical mathematics 28, 288-301 (1988) · Zbl 0651.65063 · doi:10.1007/BF01934092
[18]Hoskins, W. D.; Meek, D. S.: Linear dependence relations for polynomial splines at mid knots, BIT numerical mathematics 15, 272-276 (1975) · Zbl 0311.65002 · doi:10.1007/BF01933659
[19]Lucas, T. R.: Error bounds for interpolating cubic spline under various end conditions, SIAM J. Numer. anal. 11, No. 3, 569-584 (1974) · Zbl 0286.65004 · doi:10.1137/0711049
[20]Boor, C. D.; Swartz, B.: Collocation on Gaussian points, SIAM J. Numer. anal. 10, No. 4, 582-606 (1973) · Zbl 0232.65065 · doi:10.1137/0710052
[21]Agarwall, R. P.; Akrivis, G.: Boundary value problems occurring in plate deflection theory, J. comput. Appl. math. 8, 145-154 (1982) · Zbl 0503.73061 · doi:10.1016/0771-050X(82)90035-3
[22]Ccglar, H. N.; Caglar, S. H.; Twizell, E. H.: The numerical solution of fifth-order boundary value problems with sixth-degree b-spline functions, Appl. math. Lett. 12, 20-30 (1999) · Zbl 0941.65073 · doi:10.1016/S0893-9659(99)00052-X
[23]Noor, M. A.; Mohyud-Din, S. T.: Homotopy perturbation method for solving sixth-order boundary value problems, Comput. math. Appl. 55, 2953-2972 (2008) · Zbl 1142.65386 · doi:10.1016/j.camwa.2007.11.026
[24]Noor, M. A.; Mohyud-Din, S. T.: Variational iteration method for fifth-order boundary value problems using hes polynomials, Math. prob. Eng. (2008)
[25]Noor, M. A.; Noor, K. I.; Mohyud-Din, S. T.: Variational iteration method for solving sixth-order boundary value problems, Commun. non. Sci. numer. Simulat. 14, 2571-2580 (2009) · Zbl 1221.65176 · doi:10.1016/j.cnsns.2008.10.013
[26]Twizell, E. H.: A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems, Comp. meth. Appl. mech. Eng. 62, 293-303 (1987) · Zbl 0602.73089 · doi:10.1016/0045-7825(87)90064-8
[27]Twizell, E. H.; Tirmizi, S. I. A.: Multiderivative methods for nonlinear beam problems, Commun. appl. Numer. meth. 4, 43-50 (1988) · Zbl 0627.73082 · doi:10.1002/cnm.1630040107
[28]Wang, C. C.; Lee, Z. Y.; Kuo, Y.: Application of residual correction method in calculating upper and lower approximate solutions of fifth-order boundary-value problems, Appl. math. Comput. 199, 677-690 (2008) · Zbl 1143.65065 · doi:10.1016/j.amc.2007.10.030
[29]Wazwaz, A.: The numerical solution of fifth-order boundary-value problems by domain decomposition method, J. comput. Appl. math. 136, 259-270 (2001) · Zbl 0986.65072 · doi:10.1016/S0377-0427(00)00618-X
[30]Wazwaz, A. M.: The numerical solution of sixth-order boundary value problems by the modified decomposition method, Appl. math. Comput. 118, 311-325 (2001) · Zbl 1023.65074 · doi:10.1016/S0096-3003(99)00224-6
[31]Zhang, J.: The numerical solution of fifth-order boundary value problems by the variational iteration method, Computers and mathematics with applications 58, 2347-2350 (2009) · Zbl 1189.65183 · doi:10.1016/j.camwa.2009.03.073
[32]Conte, S. D.: The numerical solution of linear boundary value problems, SIAM rev. 8, 309-321 (1966) · Zbl 0168.14101 · doi:10.1137/1008063