Fixed mesh approximation of ordinary differential equations with impulses (Q1377021)

An effective algorithm is presented for approximation to the solution of an ordinary differential equation with impulsive forcing function. The system has the form \[ \dot x(t)= f(x(t),t)+ \sum^\infty_{j= 0}\alpha_i \delta(t- t_j),\quad 0\leq t\leq T;\quad x(0)= x_0,\tag{i} \] where \(\sum^\infty_{j= 0}|\alpha_j|< \infty\), and \(f\) is integrable satisfying a Lipschitz condition. Let \(P_N(h)\) be a partition of \([0,T]\), \(0= t_0< t_1\cdots< t_N= T\), with \(0< ch\leq| t_n- t_{n- 1}|\leq h\), \(n= 1,2,\dots, N\). The authors introduce a \(P_N(h)\) partition dependent variational problem whose solution \(x(t)\) satisfies (i) in a weak sense. The solution of the variational problem is approximated by a Galerkin method using piecewise polynomials of degree \(k\) which may be solved numerically. It is proved that the order of error of the approximate solution at mesh nodes is \(O(h)\) and that the \(L^2\) error has order \(O(h^{1/2})\). In the linear case with \(f= ax+b\), \(b(t)\in H^{k+1}\), columns of \(a(t)\) in \(H^{k+ 1}\), the order of error at nodal points is proved to be \(O(h^{k+2})\) while the \(L^2\) error is again \(O(h^{1/2})\). The results are confirmed numerically by application of the method to a linear and a nonlinear test problem.