Multistep differential transform method for numerical solution of classical neutron point kinetic equation (Q487205)

The authors apply the multistep differential transform method applied to the neutron point kinetic equations \[ \frac{dn(t)}{dt} = \left(\frac{\rho(t)-\beta}{l}\right) n(t) + \sum_{i=1}^m \lambda_i c_i + S(t), \] and \[ \frac{dc_i(t)}{dt} = \frac{\beta_i}{l} n(t) - \lambda_i c_i(t),\quad i=1,\dots, m. \] This is written in matrix form as \[ \frac{d \vec{x}(t)}{dt} = A x(t) + B(t) \vec{x}(t) + S(t), \qquad \vec{x}(0)= \vec{x}_0. \] For a function \(f(t)\) the differential transform method (DTM) gives the function \(F\) defined by \[ F(k) = \frac{1}{k!} \left[ \frac{d^k f(t)}{dt^k}\right]_{t=t_0}. \] If \(f(t) = \frac{d \vec{x}(t)}{dt}\), then \(F(k) = (k+1) X(k+1)\). This allows to transform the neutron point kinetic equation, which gives an explicit equation for \(X(k+1)\). In order to obtain an approximate solution of the kinetic equations, the DTM uses the values \(X(k)\) for \(k=0,\dots,p\), where \(X(0)=\vec{x}(0)=\vec{x}_0\) and \(X(k)\) for \(k\geq 1\) is determined from the transformed differential equation, namely with \(p=m+1\) \[ \vec{x}(t) = \sum_{k=0}^p t^k X(k). \] In the classical DTM, \(\vec{x}(t)\) is defined on \([0,T]\). The multistep DTM employs a partition \(0=t_0<t_1<\dots<t_n=T\) of the interval \([0,T]\), and sets \(\vec{x}_0(t) = \vec{x}(t)\) for \(0\leq t \leq t_1\) as obtained above. Then \(\vec{x}_i(t)\) is obtained as above, but with initial data \(\vec{x}_i(t_i) = \vec{x}_{i-1}(t_i)\) for \(i=1,\dots, n-1\). The approximate solution is then \(\vec{x}(t) = \vec{x}_i(t)\) for \(t\in [t_{i},t_{i+1}]\).