Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series

On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable function

“…Downloaded by [University of Glasgow] at 10:38 19 September 2013 (ii) to show how to use these expressions for solving ordinary differential equations with polynomial coefficients by reducing them to a recurrence relations of lowest order in the expansion coefficients of their solutions. (iii) to compare our results with those obtained by applying the optimal algorithm claimed by Lewanowicz (1976).…”

“…, p, # 0 are polynomials in x, and the coefficients of the Legendre series of the function p(x) are known. Lewanowicz (1976) gives a method of constructing in view of equation (32) the linear recurrrence relation of the lowest order, where a,, a,, . .…”

“…For the sake of comparisons, we mention very briefly the optimum algorithm of Lewanowicz (1976). Let f ( x ) be a function defined on [-l , 1 ] , and has the Legendre expansion (I), and assume that it satisfies the linear nonhomogeneous differential equation of order n where po, p,, .…”

“…Two numerical examples of how to apply these theorems for solving ordinary differential equations with varying coefficients are given and discussed in Section 5. Comparisons with the results obtained by the optimum algorithm of Lewanowicz (1976) are noted. Some concluding remarks are made in Section 6.…”

“…Orthogonal polynomials are extensively used for the numerical solution of differential equations in spectral and pseudospectral methods, see for instance, Canuto et al (1988), Doha (1990Doha ( ,1991, Gottlieb and Orszag (1977), Karageorghis (1988a), Phillips (1988,1989) and Lewanowicz (1976Lewanowicz ( ,1986. If these polynomials are used as basis functions, then the rate of decay of the expansion coefficients is determined by the smoothness properties of the function being expanded.…”