On the stability of the transport equation
The author considers the differential equation of the type \[ T(\partial \psi /\partial x)(x)=-A\psi(x),\quad x\in J=(0,a),\quad a>0\text{ or } a=\infty \] in a Hilbert space \(H\), where \(T\) and \(A\) are bounded, self-adjoint, injective, and \(A\) has a bounded inverse. The stability of the solution \(\psi\) when \(A\) is perturbed by a self-adjoint operator is discussed. The time independent neutron transport equation is a special case of the above equation with the Hilbert space \(L_ 2[(-1,+1);d\mu]\), \(\psi: J\to L^ 2[(-1,+1);d\mu]\), \(Tv(\mu):=\mu v(\mu)\), \[ Av(\mu):=v(\mu)- \frac{c}{4\pi}\int^{+1}_{-1}v(\mu_ 1)\int^{2\pi}_{0}f(\mu \mu_ 1+\sqrt{1-\mu^ 2}\sqrt{1-\mu^ 2\cos \theta})d\theta d\mu_ 1 \] for \(\mu\in [-1,+1]\), \(v\in L_ 2[(-1,+1);d\mu]\). This equation is rewritten equivalently in the form \[ (*)\quad S\frac{d\psi}{dx}(x)=-\psi(x),\quad x\in J \] where \(S=A^{-1}T\) is self-adjoint relative to \((.,.)_ A\). It is proved that if \(A\) is a self-adjoint operator in \(H:=L_ 2[(-1,+1);d\mu]\) with positive bounds relative to the usual innerproduct in \(H\) and \((A_ n)\) is a sequence of self-adjoint operators in \(H\) such that \(A_ n\to A\) then for each \(x\in J\) the solutions \(\psi_ n(x)\) of (*) in \(\hat H=L_ 2[(-1,1);| \mu | d\mu]\) corresponding to \(A_ n\) given in two different forms of boundary conditions for \(J=(0,\infty)\) and \(J=(0,a)\) \((0<a<\infty)\) converge strongly in \(\hat H\) to the solutions \(\psi(x)\) which corresponds to \(A\). For the linear transport equations the integral operator \(C\) in \(A=I-C\) is approximated by finite rank operators \(C_ n\). It is also shown that the spectral measure \(\sigma_ n\) corresponding to \(A_ n=I-C_ n\) converge to \(\sigma\), the spectral measure corresponding to A in the weak* sense on \(C(I)\). Also it turns out that the spectrum \(\Sigma(S)\) is continuous for \(C_ n\to C\).
time independent neutron transport equation
scientific article; zbMATH DE number 4000822