Integral transform solution of eigenvalue problems

Abstract

SUMMARYThe generalized integral transform technique is used to reduce eigenvalue problems described by partial differential equations to algebraic ones, that can be solved by existing codes for matrix eigensystem analysis. The method is illustrated for the operators that correspond to heat and mass diffusion, but can be employed in different fields. Three examples demonstrate the potential of the method.

Hybrid solution for the laminar flow of power-law fluids inside rectangular ducts

“…On the other hand, along the last two decades, a hybrid numerical-analytical approach has been developed for partial differential equations, the well-established generalized integral transform technique (GITT) as reviewed by Cotta (1993Cotta ( , 1994. This spectral-type approach is based on eigenfunction expansions yielding to solutions where the most features are: (i) an automatic and straightforward global error control and, (ii) an only mild cost increase in overall computational effort for multidimensional situations.…”

“…This spectral-type approach is based on eigenfunction expansions yielding to solutions where the most features are: (i) an automatic and straightforward global error control and, (ii) an only mild cost increase in overall computational effort for multidimensional situations. Due to its hybrid nature, this scheme has been well indicated for benchmarking purposes and for the validation of different numerical methods in many classes of problems such as, non-linear heat and¯uid¯ow problems, including the Navier±Stokes equations (Pe Ârez Guerrero and Lima et al, 1997;Quaresma and Cotta, 1997), the laminar and turbulent boundary layer equations in duct¯ows (Cotta and Carvalho, 1991;Carvalho et al, 1993;Machado and Cotta, 1995;Figueira da Silva and Cotta, 1996;Cotta and Pimentel, 1998) and, convection-diffusion and eigenvalue problems (Campos Silva et al, 1992;Mikhailov and Cotta, 1994), as well as, laminar boundary layer equations in non-Newtonian¯ows in channels (Magno et al, 1999).…”

Analytical and hybrid solutions of diffusion problems within arbitrarily shaped regions via integral transforms

“…Under this representation, the integral transformation of problem (5a, b) into an algebraic eigensystem is as simple as in the case of regular domains [24]. According to the domain description introduced by Eqs.…”

Inverse analysis of forced convection in micro-channels with slip flow via integral transforms and Bayesian inference

“…Thus, the Generalized Integral Transform Technique (GITT) is here employed in the solution of the Sturm-Liouville problem (3) via the proposition of a simpler auxiliary eigenvalue problem, and expansion of the unknown eigenfunctions in terms of the chosen basis [19]. Also, the variable equation coefficients may themselves be expanded in terms of known eigenfunctions [13], so as to allow for a fully analytical implementation of the coefficients matrices in the transformed system.…”