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Collocation method via Jacobi polynomials for solving nonlinear ordinary differential equations. (English) Zbl 1221.65173
Summary: We extend a collocation method for solving a nonlinear ordinary differential equation (ODE) via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth. Choosing the optimal polynomial for solving every ODEs problem depends on many factors, for example, smoothing continuously and other properties of the solutions. In this paper, we show intuitionally that in some problems choosing other members of Jacobi polynomials gives better result compared to Chebyshev or Legendre polynomials.

65L10Boundary value problems for ODE (numerical methods)
34B15Nonlinear boundary value problems for ODE
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
Full Text: DOI
[1] G. Szeg, Orthogonal Polynomials, vol. 23, Amer. Math. Soc., 1985. · Zbl 65.0278.03
[2] Y. Luke, The Special Functions and Their Approximations, Academic Press, New York, NY, USA, 1969. · Zbl 0193.01701
[3] E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” Journal of Physics. A, vol. 35, no. 15, pp. 3467-3478, 2002. · Zbl 0997.33004 · doi:10.1088/0305-4470/35/15/308
[4] E. H. Doha, “On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials,” Journal of Physics. A, vol. 37, no. 3, pp. 657-675, 2004. · Zbl 1055.33007 · doi:10.1088/0305-4470/37/3/010
[5] E. H. Doha, “The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function,” Journal of Computational and Applied Mathematics, vol. 89, no. 1, pp. 53-72, 1998. · Zbl 0909.33007
[6] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 2, SIAM, Philadelphia, Pa, USA, 1977. · Zbl 0412.65058
[7] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer, New York, NY, USA, 1988. · Zbl 0658.76001
[8] R. G. Voigt, D. Gottlieb, and M. Y. Hussaini, Spectral Methods for Partial Differential Equations, SIAM, Philadelphia, Pa, USA, 1984. · Zbl 0534.00017
[9] C. de Boor and B. Swartz, “Collocation at Gaussian points,” SIAM Journal on Numerical Analysis, vol. 10, pp. 582-606, 1973. · Zbl 0232.65065 · doi:10.1137/0710052
[10] F. A. Oliveira, “Numerical solution of two-point boundary value problems and spline functions,” in Numerical methods (Third Colloq., Keszthely, 1977), vol. 22 of Colloq. Math. Soc. János Bolyai, pp. 471-490, North-Holland, Amsterdam, The Netherlands, 1980. · Zbl 0442.65072
[11] R. D. Russell and L. F. Shampine, “A collocation method for boundary value problems,” Numerische Mathematik, vol. 19, pp. 1-28, 1972. · Zbl 0221.65129 · doi:10.1007/BF01395926 · eudml:132126
[12] R. Weiss, “The application of implicit Runge-Kutta and collection methods to boundary-value problems,” Mathematics of Computation, vol. 28, pp. 449-464, 1974. · Zbl 0284.65067 · doi:10.2307/2005918
[13] R. Frank and C. W. Ueberhuber, “Collocation and iterated defect correction,” in Numerical Treatment of Differential Equations, Lecture Notes in Math., Vol. 631, pp. 19-34, Springer, Berlin, Germany, 1978. · Zbl 0392.65032
[14] \DI. \cCelik, “Approximate calculation of eigenvalues with the method of weighted residuals-collocation method,” Applied Mathematics and Computation, vol. 160, no. 2, pp. 401-410, 2005. · Zbl 1064.65073 · doi:10.1016/j.amc.2003.11.011
[15] \DI. \cCelik and G. Gokmen, “Approximate solution of periodic Sturm-Liouville problems with Chebyshev collocation method,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 285-295, 2005. · Zbl 1082.65556 · doi:10.1016/j.amc.2004.11.038
[16] I. Babu\vska and B. Guo, “Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces,” SIAM Journal on Numerical Analysis, vol. 39, no. 5, pp. 1512-1538, 2002. · Zbl 1008.65078 · doi:10.1137/S0036142901356551
[17] I. Babu\vska and B. Guo, “Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions,” Numerische Mathematik, vol. 85, no. 2, pp. 219-255, 2000. · Zbl 0970.65117 · doi:10.1007/s002110000130
[18] B.-Y. Guo and L. I. Wang, “Jacobi interpolation approximations and their applications to singular differential equations,” Advances in Computational Mathematics, vol. 14, no. 3, pp. 227-276, 2001. · Zbl 0984.41004 · doi:10.1023/A:1016681018268
[19] P. Junghanns, “Uniform convergence of approximate methods for Cauchy-type singular integral equations over (-1, 1),” Wissenschaftliche Zeitschrift der Technischen Hochschule Karl-Marx-Stadt, vol. 26, no. 2, pp. 251-256, 1984. · Zbl 0575.65136
[20] E. P. Stephan and M. Suri, “On the convergence of the p-version of the boundary element Galerkin method,” Mathematics of Computation, vol. 52, no. 185, pp. 31-48, 1989. · Zbl 0661.65118 · doi:10.2307/2008651
[21] Z.-Q. Wang and B.-Y. Guo, “A rational approximation and its applications to nonlinear partial differential equations on the whole line,” Journal of Mathematical Analysis and Applications, vol. 274, no. 1, pp. 374-403, 2002. · Zbl 1121.41303 · doi:10.1016/S0022-247X(02)00334-7
[22] \DI. \cCelik, “Collocation method and residual correction using Chebyshev series,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 910-920, 2006. · Zbl 1090.65096 · doi:10.1016/j.amc.2005.05.019