Product Integration Quadratures for the Radiative Transfer Problem with Hopf’s Kernel

Abstract

Either $X: = {C^0}\left( {\left[ {0,1} \right]} \right)orX: = {L^1}\left( {\left[ {0,1} \right]} \right)$ can be used as theoretical framework for the integral operator $T:X \to X$ defined by

$$x \mapsto Tx:s \in [0,1] \mapsto (Tx)(s): = \frac{{{{\tau }_{0}}\varpi }}{2}\int_{0}^{1} {{{E}_{1}}({{\tau }_{0}}|s - t|)x(t)dt,}$$

where E₁ denotes the first exponential-integral function, that is, the function E₁ of the sequence $\left( {{E_v}} \right)_v^{\infty } = 1$ defined by

$$\begin{array}{*{20}{c}} {{{E}_{\nu }}(\tau ): = \int_{1}^{\infty } {\frac{{\exp ( - \tau \mu )}}{{{{\mu }^{\nu }}}}d\mu ,} } & {\tau > 0,} & {\nu \in [\kern-0.15em[ 1,\infty [\kern-0.15em[ .} \\ \end{array}$$

and where ${\tau_0} > 0$ and $\varpi \in \left[ {0,1} \right].$