On Bürmann’s Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion

Abstract

This article presents a compact analytic approximation to the solution of a nonlinear partial differential equation of the diffusion type by using Bürmannʼs theorem. Expanding an analytic function in powers of its derivative is shown to be a useful approach for solutions satisfying an integral relation, such as the error function and the heat integral for nonlinear heat transfer. Based on this approach, series expansions for solutions of nonlinear equations are constructed. The convergence of a Bürmann series can be enhanced by introducing basis functions depending on an additional parameter, which is determined by the boundary conditions. A nonlinear example, illustrating this enhancement, is embedded into a comprehensive presentation of Bürmannʼs theorem. Besides a recursive scheme for elementary cases, a fast algorithm for multivalued Bürmann expansions and inverse functions is developed using integer partitions. The present approach facilitates the search for expansions of analytic functions superior to commonly used Taylor series and shows how to apply these expansions to nonlinear PDEs of the diffusion type.

Wireless Channel Modeling Perspectives for Ultra-Reliable Communications

“…with x = (ln(r) − µ l )/(σ l √ 2) and erf being the error function. Using Bürmann-type asymptotic approximation [35] leads to F LN (x) ≈ 1 2 1 + sgn(x) √ 1 − e −x 2 ≈ 1 4 e −x 2 , when omitting higher order terms and approximating the square root for |x| 0. A tighter approximation can be obtained if we use F LN (x) ≈ 1 4 e −f (x) with a polynomial fitting function f (x) [36].…”

Unravelling the relative contributions of climate change and ground disturbance to subsurface temperature perturbations: Case studies from Tyneside, UK

“…We note in passing that the practical difficulty regarding accurate evaluation of erf(x) for the general case of solving equation 6, which arises because the Taylor series for this function typically converges slowly, requiring the summation of many terms, led Westaway and Younger (2013) to adopt a computation scheme that solved for T/z then numerically integrated the result to obtain T. However, as discussed by Schöpf and Supancic (2014), the Bürmann series for erf(x) converges much more rapidly than its Taylor series, making efficient accurate computation of this function much more straightforward; this approach can therefore be used with programming languages (such as PHP) that currently lack inbuilt facilities for calculating this function. Nonetheless, for determining conductive temperature profiles affected by GST perturbations we have reused the code from Westaway and Younger (2013), notwithstanding the more efficient alternative computational procedure now feasible.…”

Linear Response of General Observables in Spiking Neuronal Network Models

“…However, when this quantity becomes large in absolute value, one has to consider more and more terms in the expansion to approach sufficiently well the function . This is a well-known property of the function (which can be written in terms of the error function): the Taylor expansion converges very slowly near infinity and other expansions are more efficient (e.g., Bürmann series [ 60 ]). Here, we consider the effect of a perturbation in a range where the function does not saturate.…”