“Picard’s theorem”的概念、定义、翻译、参考文献-科学参考

Picard’s theorem

Mathematics

单词	Picard’s theorem
释义	Picard’s theorem Mathematics An iterative method for solving the initial-value problem: $\frac{d y}{d x} = f (x, y (x)), y (x_{0}) = y_{0} .$ Assume that f is a continuous function in the first variable and Lipschitz in the second variable. Then there exists an open interval about x₀ on which there is a unique solution y(x). The solution can be constructively found as the limit of functions y_n(x), where $y_{n + 1} (x) = y_{0} + \int_{x_{0}}^{x} f (t, y_{n} (t)) d t, y_{0} (x) = y_{0} .$ As an example, suppose dy/dx = y and y(0) = 1. Then $\begin{matrix} y_{1} (x) = 1 + \int_{0}^{x} 1 d t = 1 + x, \\ y_{2} (x) = 1 + \int_{0}^{x} (1 + t) d t = 1 + x + \frac{x^{2}}{2} . \end{matrix}$ So y_n(x) is the first n + 1 terms of the exponential series and converges to the solution y(x) = e^x.
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