The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations

[摘要] This second paper adopts a more rigorous, in-depth approach to modelling the resulting dynamic-pressures in the human brain, following a transitory improvised explosive device (IED) shock-wave entering the head. Determining more complicated boundary conditions, a set of particular-solutions for both Burgers’ and the Transport equations has been obtained to describe the highly damped neurological pressures, complete with respective graphical plots. Many of these two-dimensional solution-curves closely resemble the Friedlander curve, [6; 7; 8; 9], not only illustrating enormous over-pressures that result almost immediately after the initial impact, but under-pressures experimentally depicted in all cases, due to oscillatory motion. It appears, given experimental evidence, that most – if not all – of these models can be aptly described by damped sinusoidal functions, these facts being further corroborated by existing literature, referencing models expounded by Friedlander’s seminal work, [6; 7; 8; 9]. Using other advanced mathematical techniques, such as the Hopf-Cole Transformation, application of the Dirac-delta function and the Heat-Diffusion equation, expressions have been determined to model and predict the associated energies and temperatures both within this paper and the next.

[学科分类] 数学（综合）

[关键词] MTBI;Traumatic Brain Injury;Brain;Skull;Head;shock-wave;Partial differential equation;PDEs;Burgers’ equation;transport equation;blast;Pressure;Under-pressure;mathematical model;Friedlander curve;Hopf-cole transformation;Dirac-delta function;Energy;temperature;Heat-diffusion equation;IED