We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane-Emden equations and the Thomas-Fermi equation.

翻译：针对（方程组形式的）拟线性微分方程的奇异初值问题，本文提出一种几何方法进行数值积分。该方法将原始问题转化为计算关联向量场驻点处不稳定流形的问题，从而以高效且稳健的方式求解。通过结合打靶法，该方案同样适用于边值问题。作为示例，我们处理了若干（广义）Lane-Emden方程及Thomas-Fermi方程。