An application of the decomposition method to the matrix Riccati equation in a neutron transport process
The differential matrix Riccati equation \[ R'(x)=B(x)+D(x)R(x)+R(x)D(x)+R(x)B(x)R(x) \] subject to the initial condition \(R(0)=0\), where B(x), D(x) are continuous non-negative \(n\times n\) matrix valued functions, is considered. The Adomian decomposition method is applied to give an analytical approximation of the solution which is searched in the form \(\Phi =\sum^{m}_{k=0}R_ k\) with \[ R_ 0=L^{-1}B,\quad R_ k=L^{-1}\{DR_{k-1}+R_{k-1}D\}+L^{- 1}\{\sum^{k-1}_{i=0}R_ iBR_{\quad k-i}\},\quad k\geq 1, \] where \(L^{-1}=\int^{x}_{0}[.]ds.\) To show the accuracy of the method, a model of a neutron steady-state transport process is proposed, which allows to evaluate the matrix coefficient functions in a rather simple way. Different order approximations of the solution are obtained and compared with the direct numerical integration of the equation.
scientific article; zbMATH DE number 4101611