A collocation/quadrature-based Sturm–Liouville problem solver

Abstract

ISR develops, applies and teaches advanced methodologies of design and analysis toKeywords: Sturm-Liouville problems; collocation; quadrature; eigenfunction expansions; computational methods. AbstractWe present a computational method for solving a class of boundary-value problems in Sturm-Liouville form. The algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions. Error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid; eigenfunctions that produce residuals exceeding an infinity-norm bound are discarded. Because the computational approach involves the generation of quadrature weights and discrete differentiation operations, our computational methods provide a convenient framework for solving boundary-value problems by eigenfunction expansion and other projection methods.

Collocation on uniform grids

“…where ψ(0) = ψ(1) and ψ (0) = ψ (1). In table 7 we compare the eigenvalues obtained using our LSF 1 , for N = 200, with the results of [27]. Our results agree with those of [27] to the accuracy reported in that paper.…”

“…We have collocated this problem on the uniform grid of the LSF 4 , which fulfils the boundary conditions requested in this case. Table 6 contains the results for the first three eigenvalues obtained with meshes with N ranging from N = 10 to N = 100, which agree with the results obtained in [27] following a different approach.…”

“…Here x ∈ (0, 1) and ψ (0) = ψ (1) = 0. Following [27] we use the constant parameters q = 10 and g = 20. Note that the eigenfunction corresponding to the lowest eigenvalue is constant and therefore λ 1 = −g.…”

A Computational Framework for Boundary-Value Problem Based Simulations

“…sl Trial functions used in MWR commonly are eigenfunctions found as solutions to a Sturm-Liouville problem related to the BVP to be solved. Therefore, we developed a Sturm-Liouville problem solver [9] that generates a vector of eigenvalues A, discretized eigenfunction array T, discretized adjoint eigenfunction array (D, and weight functions, wef and wad, that define orthogonality for non-self adjoint problems. The class of regular Sturm-Liouville problems solved is described by:…”

Modelling of Displacement Washing of Pulp Fibers Using the Hermite Collocation Method

“…The diffusion dispersion problems arising from packed bed reactors have always motivated the scientists for the development of new models (Pellet, 1966;Grähs, 1974;Neretnieks, 1974Neretnieks, , 1976Perron & Lebeau, 1977;Al-Jabari et al, 1994;Kukreja et al, 1995;Eriksson et al, 1996;Potůček, 1997;Potůček & Pulcer, 2004;Arora et al, 2006Arora et al, , 2008. It has given rise to the use of a variety of analytical and numerical techniques for the solution of these models such as Laplace transforms (Brenner, 1962;Pellett, 1966;Rasmuson & Neretnieks, 1980;Liao & Shiau, 2000;Aminikhah, 2012), orthogonal collocation method (Villadsen & Stewart, 1967;Michelsen & Villadsen, 1971;Raghvan & Ruthven, 1983;Adomaitis & Lin, 2000;Solsvik & Jakobsen, 2012), orthogonal collocation on finite elements (Carey & Finlayson, 1975;Ma & Guiochon, 1991;Arora et al, 2005Arora et al, , 2006Arora et al, , 2008, Galerkin method (Liu & Bhatia, 2001;Onah, 2002;Bhrawy & El-Soubhy, 2010;Nadukandi et al, 2010;Shen et al, 2011;Zhu et al, 2011;Solsvik & Jakobsen, 2012), Tau method (ElDaou & Al-Matar, 2010;Vanani & Aminataei, 2011;…”