Volume 9, Issue 2
Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations
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J. Comp. Math., 9 (1991), pp. 171-183

Published online: 1991-09

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• Abstract

A method is presented for obtaining recurrence relations for the coefficients in ultrapherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zz), -1\leq x\leq 1,z\in C,|z|\leq 1$,where y is a solution of a linear differential equation. In particular, rational approximations of y(z) result if x is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants.

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@Article{JCM-9-171, author = {}, title = {Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations}, journal = {Journal of Computational Mathematics}, year = {1991}, volume = {9}, number = {2}, pages = {171--183}, abstract = { A method is presented for obtaining recurrence relations for the coefficients in ultrapherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zz), -1\leq x\leq 1,z\in C,|z|\leq 1$,where y is a solution of a linear differential equation. In particular, rational approximations of y(z) result if x is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9390.html} }
TY - JOUR T1 - Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations JO - Journal of Computational Mathematics VL - 2 SP - 171 EP - 183 PY - 1991 DA - 1991/09 SN - 9 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9390.html KW - AB - A method is presented for obtaining recurrence relations for the coefficients in ultrapherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zz), -1\leq x\leq 1,z\in C,|z|\leq 1$,where y is a solution of a linear differential equation. In particular, rational approximations of y(z) result if x is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants.
E. H. Doha. (1970). Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations. Journal of Computational Mathematics. 9 (2). 171-183. doi:
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