Integro-differential, Integral, and Algebraic Equations

Abstract

For the kinetic theory equation (constant velocity, isotropic scattering)

$$\frac{{\partial \varphi }}{{\partial t}} + \,\sum\limits_{k - 1}^{m - 1} {{u_k}} \frac{{\partial \varphi }}{{\partial {x_k}}} + \sigma \varphi = \frac{{{\sigma _s}}}{{4\pi }}\int {\varphi \left( {x,u,t} \right)\,du + S\left( {x,u,t} \right)} $$

(7.1.1)

the following scheme was mentioned in the work of G. I. Marchuk and the author [69] (incomplete splitting)

$$\frac{{{\varphi ^{n + 1/2}} - {\varphi ^n}}}{{r!\left( {n - r} \right)!}} = {\Lambda _1}\left( {\alpha {\varphi ^{n + 1/2}} + \beta {\varphi ^n}} \right) + \bar S,$$

(7.1.2)

$$\frac{{{\varphi ^{n + 1/2}} - {\varphi ^n}}}{{r!\left( {n - r} \right)!}} = {\Lambda _2}\left( {\alpha {\varphi ^{n + 1}} + \beta {\varphi ^{n + 1/2}}} \right).$$

(7.1.3)