Finite section method for linear ordinary differential equations
The aim of this paper is to study the solutions to linear ordinary differential equations \[ \dot x(t)- A(t)x(t)= f(t),\quad 0\leq t<\infty,\quad Lx(0)= 0,\tag{1} \] as limits of the corresponding equations on a finite interval of the form \[ \dot x(t)- A(t) x(t)= f(t),\quad 0\leq t\leq \tau,\quad Lx(0)+ N(\tau) x(\tau)= 0,\tag{2} \] where \(A(t)\) is a locally integrable \(n\times n\)-matrix function on \([0,\infty)\), \(L\) and \(N(\tau)\) are \(n\times n\)-matrices such that \(\text{rank }L+ \text{rank }N(\tau)= n\), and \(L\) is an exponential dichotomy of \[ \dot x(t)- A(t)x(t)= 0,\quad 0\leq t<\infty,\tag{3} \] i.e. \(L\) is a projection and there exist positive real constants \(M\) and \(\alpha\) such that \[ \|U(t)LU(s)^{-1}\|\leq Me^{-\alpha(t- s)},\quad 0\leq s\leq t<\infty, \] \[ \|U(t)(I- L)U(s)^{-1}\|\leq Me^{- \alpha(s-t)},\quad 0\leq t\leq s<\infty, \] \(U(t)\) being the fundamental matrix of (3). The problem which is going to be solved is that the finite section method with respect to the boundary value matrices \(\{N(\tau)\}\) converges in \(L_p\), i.e. given \(f\in L^n_p[0,\infty)\), the unique solution \(x\) to (1) can be approximated by the solution \(x_\tau\) to (2) for \(\tau\to\infty\). The authors solve this problem even in the more general case in which the right side of (2) is replaced by a function \(f_\tau\) such that for \(\tau\to\infty\) the function \(f_\tau\) converges in \(L_p\) to \(f\). If \(L+ N(\tau)U(\tau)\) is invertible for each \(\tau\geq \tau_0\) and \[ \sup_{\tau\geq\tau_0}\|U(\tau)(L+ N(\tau) U(\tau))^{- 1}N(\tau)\|< \infty, \] then it is shown that the finite section method relative to the boundary value matrices \(\{N(\tau)\}\) converges in \(L_p\). In the time-invariant case it is considered the pair of equations \[ \dot x(t)- A(t) x(t)= f(t),\quad 0\leq t<\infty,\quad Qx(0)= 0,\tag{4} \] and \[ \dot x(t)- A(t)x(t)= f(t),\quad 0\leq t<\tau,\quad Qx(0)+ (I- Q) x(\tau)= 0,\tag{5} \] where \(Q\) is a projection on \(\mathbb{C}^n\) such that \(\text{ker }Q\oplus \text{Im }L_A= \mathbb{C}^n\) and \(L_A\) is the spectral projection of \(A\) corresponding to the eigenvalues of \(A\) in the left half-plane. It is shown that the finite section method relative to the boundary value matrix \(Q\) converges if and only if \((I-Q)(I- L_A)+ QL_A\) is invertible. The authors apply these results to a differential equation that appears in linear transport theory, and to integral equations with semi-separable kernels.
integral equations with semi-separable kernels
scientific article; zbMATH DE number 1474625