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Computational and numerical developments
Published in Seán M. Stewart, R. Barry Johnson, Blackbody Radiation, 2016
Seán M. Stewart, R. Barry Johnson
Another useful tool in the computational analysis of the general fractional function Φls(z) is its asymptotic expansion about a certain point. An asymptotic expansion usually takes the form of an infinite series, known as an asymptotic series, which may diverge, yet when the first few terms of the series are considered still manages to produce accurate numerical values for the value of the function in the vicinity of the point of expansion.
Analysis of the long-term thermal response of geothermal heat exchangers by means of asymptotic expansion techniques
Published in Science and Technology for the Built Environment, 2020
Santiago Ibáñez, Miguel Hermanns
The asymptotic expansion technique (Lagerstrom 1988) is a mathematical method used in numerous fields of science to obtain approximate solutions to a given problem when a small parameter is present. First, an asymptotic expansion is generated by expanding each unknown variable as a sum of terms that are proportional to increasingly higher powers of that small parameter. The number of terms used in the asymptotic expansion defines the so-called order of the solution. Introducing the asymptotic expansions into the equations and boundary conditions of the problem and grouping terms of similar order, a set of simplified problems is generated that can be solved sequentially for each power of the small parameter.
Gaps in the spectrum of two-dimensional square packing of stiff disks
Published in Applicable Analysis, 2023
In view of the particular position of the zeros of the Bessel function and consequently of sequence (19), we do not provide a complete result of the existence of spectral gaps (see Figure 2). More specifically, we only prove the existence of the spectral gaps between bands generated by eigenvalues whose leading term is the simple eigenvalue and the double one , the double eigenvalue and the simple one , the simple eigenvalue and the double one and finally, the double eigenvalues and (see Corollary 3.3). We highlight that, up to now, we cannot detect a gap between the spectral bands generated by eigenvalues whose leading terms are double eigenvalues, i.e. , for . Moreover, since the first-order correction terms associated with and vanish, we are not able to give an estimate of the length of the spectral gaps. Hence, an analysis of higher order terms of asymptotic expansions combined with a numerical computation is needed. This is not the aim of our paper and is left as an open problem.
Viscosity approximation of the solution to Burgers' equations with shock layers
Published in Applicable Analysis, 2023
Junho Choi, Youngjoon Hong, Chang-Yeol Jung, Hoyeon Lee
Using more terms from the outer expansions and more correctors, we can achieve more accurate error estimates between the solution and the corresponding asymptotic expansion. We now address this question.