标题： 求解具有所需容差的标量初值问题的方法 显示英文标题

标题： The method of solving a scalar initial value problem with a required tolerance

摘要： 一种新的数值方法用于求解具有给定初始条件的标量常微分方程。该方法使用等效积分方程的数值积分过程，并在本文中称为积分方法。在特定约束条件下，该方法可确保在用户提供的容差范围内返回近似解。这使其不同于大量单步和多步方法，这些方法提供结果的误差形式为O(h^k)，其中h为步长，k与方法的精度有关。该方法的优点和缺点得到了阐述。为了避免后者的一些问题也进行了改进。数值实验支持这些理论结果。 显示英文摘要

摘要： A new numerical method for solving a scalar ordinary differential equation with a given initial condition is introduced. The method is using a numerical integration procedure for an equivalent integral equation and is called in this paper an integrating method. Bound to specific constraints, the method returns an approximate solution assuredly within a given tolerance provided by a user. This makes it different from a large variety of single- and multi-step methods for solving initial value problems that provide results up to some undefined error in the form O(h^k), where h is a step size and k is concerned with the method's accuracy. Advantages and disadvantages of the method are presented. Some improvements in order to avoid the latter are also made. Numerical experiments support these theoretical results.