zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Troesch’s problem: a B-spline collocation approach. (English) Zbl 1235.65086
Summary: A finite-element approach, based on cubic B-spline collocation, is presented for the numerical solution of Troesch’s problem. The method is used on both a uniform mesh and a piecewise-uniform Shishkin mesh, depending on the magnitude of the eigenvalues. This is due to the existence of a boundary layer at the right endpoint of the domain for relatively large eigenvalues. The problem is also solved using an adaptive spline collocation approach over a non-uniform mesh via exploiting an iterative scheme arising from Newton’s method.

65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
Full Text: DOI
[1] Weibel, E. S.: R.k.m.landshoffthe plasma in magnetic field, The plasma in magnetic field (1958)
[2] Roberts, S. M.; Shipman, J. S.: On the closed form solution of troesch’s problem, Journal of computational physics 21, 291 (1976) · Zbl 0334.65062 · doi:10.1016/0021-9991(76)90026-7
[3] Feng, X.; Mei, L.; He, G.: An efficient algorithm for solving troesch’s problem, Applied mathematics and computation 189, 500-507 (2007) · Zbl 1122.65373 · doi:10.1016/j.amc.2006.11.161
[4] Khuri, S. A.: A numerical algorithm for solving the troesch’s problem, International journal of computer mathematics 80, No. 4, 493-498 (2003) · Zbl 1022.65084 · doi:10.1080/0020716022000009228
[5] Scott, M. R.: On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms, Numerical solutions of boundary-value problems for ordinary differential equations, 89-146 (1975) · Zbl 0335.65032
[6] Chang, S. H.; Chang, I. L.: A new algorithm for calculating the one-dimensional differential transform of nonlinear functions, Applied mathematics and computation 195, 799-808 (2008) · Zbl 1132.65062 · doi:10.1016/j.amc.2007.05.026
[7] Chang, Shih-Hsiang: A variational iteration method for solving troesch’s problem, Journal of computational and applied mathematics 234, 3043-3047 (2010) · Zbl 1191.65101 · doi:10.1016/j.cam.2010.04.018
[8] Bahadir, A. R.: Application of cubic B-spline finite element technique to the thermistor problem, Applied mathematics and computation 149, 379-387 (2004) · Zbl 1038.65057 · doi:10.1016/S0096-3003(03)00146-2
[9] Caglar, H. N.; Caglar, S. H.; Twizell, E. H.: The numerical solution of third-order boundary value problems with fourth-degree B-spline functions, International journal of computer mathematics 71, 373-381 (1999) · Zbl 0929.65048 · doi:10.1080/00207169908804816
[10] Caglar, H. N.; Caglar, S. H.; Twizell, E. H.: The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions, Applied mathematics letters 12, 25-30 (1999) · Zbl 0941.65073 · doi:10.1016/S0893-9659(99)00052-X
[11] Deeba, E.; Khuri, S. A.: Nonlinear equations, Wiley encyclopedia of electrical and electronics engineering 14, 562-570 (1999)
[12] Luigi Brugnano, Francesca Mazzia, Donato Trigiante, Fifty years of stiffness. arXiv:0910.3780v1 [math.NA] (20.10.09). · Zbl 1216.65083
[13] Ahlberg, J. H.; Ito, T.: A collocation method for two-point boundary value problems, Mathematics of computation 29, No. 131, 761-776 (1975) · Zbl 0312.65056 · doi:10.2307/2005287
[14] Farrell, P. A.; Hegarty, A. F.; Miller, J. J. H.; O’riordan, E.; Shishkin, G. I.: Robust computational techniques for boundary layers, (2000) · Zbl 0964.65083
[15] Miller, J. J. H.; O’riordan, E.; Shishkin, G. I.: Fitted numerical methods for singular perturbation problems, (1996)
[16] Carey, G. F.; Dinh, H. T.: Grading functions and mesh redistribution, SIAM journal on numerical analysis 22, No. 5, 1028-1040 (1985) · Zbl 0577.65076 · doi:10.1137/0722061
[17] Christara, C. C.; Ng, Kit Sun: Adaptive techniques for spline collocation, Computing 76, 259-277 (2006) · Zbl 1086.65077 · doi:10.1007/s00607-005-0141-3