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radiative transfer ; diffuse reflection problem ; Ambartsumian’s nonlinear functional equation ; eigenfunctions and eigenvalues problem
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A new approach to solving of the Ambartsumian’s functional equation is presented. Its application is illustrated on the two classical cases: a) in the case of a plane-parallel semi-infinite medium with monochromatic anisotropic scattering and b) the simple one-dimensional problem of diffuse reflection of radiation from the scattering-absorbing semi-infinite medium in the case of the general law of redistribution of radiation by frequencies. The desired: a) azimuthal harmonics of reflection function dependent on two angular variables are explicitly expressed through the according eigenfunctions of one angle variable and b) diffuse reflection function of two frequency variables also expressed through a system of according eigenfunctions which have only one frequency variable. This does not require the use of any simplifying assumptions or special decompositions of the characteristics of elementary act of scattering: a) of scattering indicatrix and b) of the redistribution function of radiation by frequencies.