An algorithm of approximate integration of second order linear differential equations (Q2730541)

This article deals with the approximate integration of the linear differential equation \(y''+p(x)y=q(x)\), \(0<x\leq l\) with the initial conditions \(y(0)=y_0, y'(0)=y_0'\). Let the function \(p(x)\) be continuously differentiable and let \(q(x)\) be continuous on \([0,l ]\). On every interval \([x_{k-1},x_{k} ]\) \((x_{k}=kh\), \(h=l/N)\) the approximate solution of the given problem is sought in the form \(z_{k}''(x)+p_{\overline k}z_{k}(x)= q(x)\), \(x_{k-1}<x\leq x_{k}\), \(z_{k}(x_{k-1})=z_{k-1}(x_{k-1})\), \(z_{k}'(x_{k-1})= z_{k-1}'(x_{k-1})\), \(k=1,\ldots,N\), where \(p_{\overline k}=(1/h) \int_{x_{k-1}}^{x_{k}}p(x) dx\), \(\overline k=1,\ldots,N\), \(x_0=0\), \(z_0(0)= y_0\), \(z_0'(0)=y'_0\). The author finds the approximate solution in the form of splines composed of elementary functions. Let \(y_{k}, y_{k}'\) and \(z_{k},z_{k}'\) denote respectively the values of explicit solution, its derivative, approximate solution and its derivative of the given problem at the mesh points \(x_{k}\), \(k=0,\ldots,N\). If the functions \(p(x), q(x)\) satisfy the above mentioned conditions and \(h<1/\sqrt{\|p\|}\), then \(|y_{k}-z_{k}|\) and \(|y_{k}'-z_{k}'|\) converges uniformly to zero.