Computational procedure for Sturm-Liouville problems

Milkhailov, Mikhail D; Vulchanov, N.L.

doi:10.1016/0021-9991(83)90101-8

Cited by 76 publications

(27 citation statements)

References 16 publications

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“…For general functional forms of u(z) and K(z), the solution of the eigenvalue problem could be obtained by using the semi-analytic sign-count method [22] or from algorithms for automatic computation of eigenvalue and eigenfunctions of Sturm-Liouville problems [23,24]. The procedure using the GITT given by Cotta [20], and Mikhailov & Cotta [16], is another alternative to solve the eigenvalue problem.…”

Section: Problem Formulationmentioning

confidence: 99%

Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions

et al. 2014

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In this work, the performance of a unified formal analytical solution for the simulation of atmospheric diffusion problems under stable conditions is evaluated. The eigenquantities required by the formal analytical solution are obtained by solving numerically the associated eigenvalue problem based on a newly developed algorithm capable of being used in high orders and without missing eigenvalues. The performance of the formal analytical solution is evaluated by comparing the converged predicted results against the observed values in the stable runs of the Prairie Grass experiment as well as the simulated results available in the literature. It was found that the developed algorithm was efficient and that the convergence rate depends on the stability condition and the considered parametrizations for wind speed and turbulence. The comparisons among predicted and observed concentrations showed a good agreement and indicate that the considered dispersion formulations are appropriate to simulate dispersion under slightly to moderate atmospheric stable conditions.

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Section: Problem Formulationmentioning

confidence: 99%

Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions

et al. 2014

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“…For the case λ 2 [19]. From the general Sturm-Liouville theory we know that the smallest eigenvalue gives rise to an eigenfunction with no zeros in the interval (0, L N ), see [20,Chap.…”

Section: Generalized Fourier Transformmentioning

confidence: 99%

Strategy for solving semi-analytically three-dimensional transient flow in a coupled N-layer aquifer system

Veling

Maas

2008

J Eng Math

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Efficient strategies for solving semi-analytically the transient groundwater head in a coupled N -layer aquifer system φ i (r, z, t), i = 1, . . . , N , with radial symmetry, with full z-dependency, and partially penetrating wells are presented. Aquitards are treated as aquifers with their own horizontal and vertical permeabilities. Since the vertical direction is fully taken into account, there is no need to pose the Dupuit assumption, i.e., that the flow is mainly horizontal. To solve this problem, integral transforms will be employed: the Laplace transform for the t-variable (with transform parameter p), the Hankel transform for the r -variable (with transform parameter α) and a particular form of a generalized Fourier transform for the vertical direction z with an infinite set of eigenvalues λ 2 m (with the discrete index m). It is possible to solve this problem in the form of a semi-analytical solution in the sense that an analytical expression in terms of the variables r and z, transform parameter p, and eigenvalues λ 2 m ( p) of the generalized Fourier transform can be given or in terms of the variables z and t, transform parameter α, and eigenvalues λ 2 m (α). The calculation of the eigenvalues λ 2 m and the inversion of these transformed solutions can only be done numerically. In this context the application of the generalized Fourier transform is novel. By means of this generalized Fourier transform, transient problems with horizontal symmetries other than radial can be treated as well. The notion of analytical solution versus numerical solution is discussed and a classification of analytical solutions is proposed in seven classes. The expressions found in this paper belong to Class 6, meaning that the transformed solutions are written in terms of eigenvalues which depend on one transform parameter (here p or α). Earlier solutions to the transient problem belong to Class 7, where the eigenvalues depend on two transform parameters. The theory is applied to three examples.

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“…A1-A4 in the Appendix and parameters k 1 and k 2 are the related thermal conductivities of the two layers (k = ρca). Positive numbers β n represent eigenvalues of the solution, and they can be found by the efficient and reliable "sign-count" method proposed by Mikhailov and Vulchanov [15]. According to that procedure, the number of eigenvalues N (β 0 , S) below an arbitrary value β 0 is equal to…”

Section: Mathematical Modelsmentioning

confidence: 99%

Determination of Transient Thermal Interface Resistance Between Two Bonded Metal Bodies using the Laser-Flash Method

Milošević

2008

Int J Thermophys

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The paper presents the data reduction analysis for measurements of the transient thermal interface resistance between two bonded metal bodies using the laser-flash method. By using two different mathematical models, i.e., a two-layered and a three-layered model, whose complete analytical solutions for realistic conditions are provided, different results for final values and their uncertainties can be obtained. The analysis has been applied to experimental data measured from samples prepared with three different bonding materials, cyanoacrylate, metal epoxy resin, and silicone rubber.

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Computational procedure for Sturm-Liouville problems

Cited by 76 publications

References 16 publications

Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions

Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions

Strategy for solving semi-analytically three-dimensional transient flow in a coupled N-layer aquifer system

Determination of Transient Thermal Interface Resistance Between Two Bonded Metal Bodies using the Laser-Flash Method

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