The radiation diffusion equation: Explicit analytical solutions by improved integral-balance method

Abstract

Университет химической технологии и металлургии; 1756 София, 8 Kl, Охридский бульвар, Болгария; jordan.hristov@mail.bg Разработаны приближенные явные аналитические решения уравнения диффузии излуче-ния с применением двойного интегрирования в методе интегрального баланса. Этот метод поз-воляет разрабатывать приближенные решения закрытой формы. Решена проблема со ступенча-тым изменением температуры поверхности и две задачи с зависящими от времени граничными условиями. Минимизация погрешности приближенных решений была проведена непосред-ственно путем минимизации остаточной функции определяющего уравнения.Ключевые слова: диффузия, метод интегрального баланса, приближенные решения, температура поверхности. IntroductionHeat wave behaviour of thermal diffusion due to radiation is a reasonable physical and mathematic interpretation of thermal energy transfer in a variety of applied problem related to astrophysical phenomena [1], plasma physics [2,3], building insulation [4], etc. In general, this physical model assumes a diffusion approximation relating the local thermal flux at any point of the medium by the local gradient of the radiation energy density that is an approach known from the classical Fourier law. Following Smith The presentation of the thermal flux density  as a gradient of the 4 th power of the local temperature is an accordance with the Rosseland approximation [1,5,6] which is valid for thick, non-opaque media in absence of fluid motion [5][6][7][8][9].The diffusion equation (1) was solved for the first time by Barenblatt [10] by a selfsimilar solution and then refined by Zeldovich and Raizer [11]. As commented by Smith [5] these early attempts, especially the report of Hammer and Rosen [14] repeated the idea of the Barenblatt but met problems in defining the shape of the spatial temperature distribution profile. At the same time as the Barenblatt solution appeared the study of Marshak [9] was carried out (performed in 1944-1945 in Los Alamos Laboratory but published only in 1958 as it is especially mentioned in the publication) . The Marshak solution also tried to express the spatial temperature distribution as a series of parabolic profiles. We will especially consider this model in the article since the new method used applies a generalized parabolic profile.1 Статья подготовлена по материалам доклада, представленного на XII Международной научно-практической конференции «Фундаментальные и прикладные проблемы математики и инфор-матики», которая прошла 19-22 сентября 2017 года в Дагестанском государственном универси-тете (г. Махачкала, РФ). For further deep reading on the solutions related to radiation-diffusion equation we refer to [5][6][7][8] and the references therein.The aim of the present work is to present an approximate closed form analytical solution allowing to estimate the penetration depth of the heat wave and the spatial temperature distribution applying an improved integral-balance approach [12,13] already successfully applied to nonlinear heat conduction problems modelled by de...