Abstract. The purpose in this paper is to compute the eigenvalues of SturmLiouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the regularized sampling method, an improvement on the method based on Shannon sampling theory, which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost. A few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions.

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Purpose This paper aims to study analytically and numerically the problem of transient natural convection heat transfer in a trapezoidal cavity with spatial side-wall temperature variation. Design/methodology/approach The governing equations subject to the initial and boundary conditions are solved numerically by the finite difference scheme consisting of the alternating direction implicit method and the tri-diagonal matrix algorithm. The left sloping wall of the cavity is heated to non-uniform temperature, and the right sloping wall is maintained at a constant cold temperature, while the horizontal walls are kept adiabatic. Findings It is shown that the heat transfer rate increases in non-uniform heating increments, whereby low wave number values are more affected by the convection. The best heat transfer enhancement results from larger side wall inclination angle; however, trapezoidal cavities require longer time compared to that of square to reach steady state. Originality/value The study of natural convection heat transfer in a trapezoidal cavity filled with nanofluid and heated by spatial side-wall temperature has not yet been undertaken. Thus, the authors of the present study believe that this work is valuable.

This paper deals with the computation of the eigenvalues of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method.A few numerical examples among which singular ones will be presented to illustrate the merit of the method and comparison made with the exact eigenvalues when they are available.

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