In this paper, we propose two new methods to solve large-scale systems of differential equations, which are based on the Krylov method. In the first one, the exact solution with the exponential projection technique of the matrix. In the second, we get a new problem of small size, by dropping the initial problem, and then we solve it in ways, such as the Rosenbrock and the BDF. Some theoretical results are presented such as an accurate expression of the remaining criteria. We give an expression of error report and numerical values to compare the two methods in terms of how long each method takes, and we also compare the approaches.

中文翻译：

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求解大型微分方程组的数值方法

在本文中，我们基于Krylov方法，提出了两种求解大型微分方程组的新方法。在第一个中，使用矩阵的指数投影技术来精确求解。在第二个问题中，通过删除初始问题，我们得到了一个尺寸较小的新问题，然后以诸如Rosenbrock和BDF之类的方式解决了该问题。提出了一些理论结果，例如准确表达了其余标准。我们给出了一个错误报告和数值表达式，以比较每种方法花费的时间，并比较两种方法。