zbMATH — the first resource for mathematics

Sinc and the numerical solution of fifth-order boundary value problems. (English) Zbl 1121.65087
The present paper deals with numerical methods for fifth-order two-point boundary value problems on the unit interval. Problems of that type arise from modelling viscoelastic flows. The author introduces the new class of sinc-Galerkin methods, a special variant of spectral Galerkin schemes, to solve the boundary value problem. The suggested methods admit an elegant treatment of the boundary conditions. As usual, the corresponding integral relations are approximated by quadrature sums. Finally, some numerical computations display the power of the suggested method in comparison to standard schemes.

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
Full Text: DOI
[1] Agarwal, R.P., Boundary value problems for high ordinary differential equations, (1986), World Scientific Singapore · Zbl 0598.65062
[2] Bialecki, B., Sinc-collocation methods for two-point boundary value problems, IMA J. numer. anal., 11, 357-375, (1991) · Zbl 0735.65052
[3] Broyden, C.G., The convergence of an algorithm for solving sparse nonlinear systems, Math. comp., 25, 285-294, (1971) · Zbl 0227.65038
[4] Caglar, 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, 25-30, (1999) · Zbl 0941.65073
[5] Davis, A.R.; Karageorghis, A.; Phillips, T.N., Spectral Galerkin methods for the primary two-point boundary-value problem in modelling viscoelastic flows, Int. J. numer. methods eng., 26, 647-662, (1988) · Zbl 0635.73091
[6] Davis, A.R.; Karageorghis, A.; Phillips, T.N., Spectral collection methods for the primary two-point boundary-value problem in modelling viscoelastic flows, Int. J. numer. methods eng., 26, 805-813, (1988) · Zbl 0637.76008
[7] Dockery, J.D., Numerical solution of travelling waves for reaction – diffusion equations via the sinc-Galerkin method, (), 95-113 · Zbl 0829.65116
[8] M. El-Gamel, A. Zayed, A comparison between the Wavelet-Galerkin and the Sinc-Galerkin methods in solving nonhomogeneous heat equations, in: Zuhair Nashed, Otmar Scherzer (Eds.), Contemporary Mathematics of the American Mathematical Society, Series, Inverse Problem, Image Analysis, and Medical Imaging, vol. 313, AMS, Providence, 2002. · Zbl 1028.65107
[9] El-Gamel, M.; Zayed, A., Sinc-Galerkin method for solving nonlinear boundary-value problems, Comput. math. appl., 48, 1285-1298, (2004) · Zbl 1072.65111
[10] El-Gamel, M.; Cannon, J.R.; Zayed, A., Sinc-Galerkin method for solving linear sixth order boundary-value problems, Math. comput., 73, 1325-1343, (2004) · Zbl 1054.65085
[11] El-Gamel, M.; Cannon, J.R., On the solution of second order singularly-perturbed boundary value problem by the sinc-Galerkin method, Z. angew. math. phys., 56, 45-58, (2005) · Zbl 1058.65082
[12] El-Gamel, M., A numerical scheme for solving nonhomogeneous time-dependent problems, Z. angew. math. phys., 57, 369-383, (2006) · Zbl 1093.65092
[13] El-Gamel, M., The sinc-Galerkin method for solving singularly-perturbed reaction – diffusion problem, Etna, 23, 129-140, (2006) · Zbl 1112.65094
[14] Grenander, V.; Szego, G., Toeplitz forms and their applications, (1985), Chelsea Publishing Co. Orlando
[15] Lund, J.; Bowers, K., Sinc methods for quadrature and differential equations, (1992), SIAM Philadelphia, PA · Zbl 0753.65081
[16] M.S. Khan, Finite-difference solutions of fifth-order boundary value problems, Ph.D. thesis, Brunel University, England, 1994.
[17] A. Mohsen, M. El-Gamel, A Sinc-collocation method for the linear Fredholm integro-differential equations, Z. Angew. Math. Phys., in press.
[18] Smith, R.; Bogar, G.; Bowers, K.; Lund, J., The sinc-Galerkin method for fourth-order differential equations, SIAM J. numer. anal., 28, 760-788, (1991) · Zbl 0735.65058
[19] Stenger, F., Numerical methods based on sinc and analytic functions, (1993), Springer New York · Zbl 0803.65141
[20] Wazwaz, A.M., The numerical solution of fifth-order boundary value problems by the decomposition method, J. comp. appl. math., 136, 259-270, (2001) · Zbl 0986.65072
[21] Yin, G., Sinc-collocation method with orthogonalization for singular problem-like Poisson, Math. comput., 62, 21-40, (1994) · Zbl 0796.65121
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.