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ISSN Online: 2377-424X

ISBN Print: 978-1-56700-474-8

ISBN Online: 978-1-56700-473-1

International Heat Transfer Conference 16
August, 10-15, 2018, Beijing, China

PROPERTY & FIELD DARBOUX TRANSFORMATIONS (PROFIDT METHOD): A VERSATILE TOOLBOX FOR GENERATING EXACT ANALYTICAL SOLUTIONS TO DIFFUSION-LIKE AND WAVE-LIKE EQUATIONS IN GRADED MEDIA

Get access (open in a dialog) DOI: 10.1615/IHTC16.kn.000015
pages 293-333

Abstract

Transfer processes, whether by diffusion or wave propagation, often take place in media that show continuous variations in their properties. The modeling of these processes should then take into account the local variations of the influential properties, in particular (but not only) those defining the diffusion coefficient, respectively the wave celerity. Due to the lack of a general analytic solution for arbitrary distributions of these parameters, even in the one-dimensional case, it is often expedient to replace the medium with a piecewise-constant approximate model. Several methodologies have been developed to provide solutions of higher order, namely with a higher fidelity to the actual property profiles, as it is outlined in a review focused on the heat diffusion problem. However, their weaknesses are limited applicability, limited scalability or numerical burden due to the use of special mathematical functions. A method with quite remarkable properties has recently been described to solve dynamic heat-transfer problems in 1D graded media, namely media with heterogeneous heat capacity and conductivity. Its particularity is not to propose a solution to a particular problem, but a way to build cascades of solutions for problems of increasing complexity. Among these solutions is a class of profiles flexible enough to play the role of smart bricks in so-called "solvable splines". Arbitrary complex profiles can then be synthetized by joining those bricks and applying the classical quadrupole methodology (with however ad-hoc quadrupole matrices). The procedure was then extended to electromagnetic waves for media with graded permittivity and permeability. This paper intends to review the major aspects of this general analytical approach, in particular the Liouville transformation, which merges the influential properties into a single leading property, and then the Darboux transformation, which expands the number of solutions and their sophistication. Examples related to photothermal modeling will be presented and some reference will be given to other applications such as advection-diffusion or light-wave propagation in graded-index dielectrics.