Modelling and the scientific method: Objectives of mathematical modelling; the iterative nature of modelling; falsifiability and predictive accuracy; Occam’s razor, paradigms and model components

这是一份Bath巴斯大学MA20221作业代写的成功案

Proof. Using the above estimate for $f$ we have ( $t_{0}=0$ without loss of generality)
$$
|\phi(t)| \leq\left|x_{0}\right|+\int_{0}^{t}(M+L|\phi(s)|) d s, \quad t \in[0, T] \cap I .
$$
Setting $\psi(t)=\frac{M}{L}+|\phi(t)|$ and applying Gronwall’s inequality (Lemma $2.7$ ) shows
$$
|\phi(t)| \leq\left|x_{0}\right| \mathrm{e}^{L T}+\frac{M}{L}\left(\mathrm{e}^{L T}-1\right)
$$
Thus $\phi$ lies in a compact ball and the result follows by the previous lemma.
Again, let me remark that it suffices to assume
$$
|f(t, x)| \leq M(t)+L(t)|x|, \quad x \in \mathbb{R}^{n}
$$
where $M(t), L(t)$ are locally integrable.

MA20221 COURSE NOTES ：

and hence by induction
$$
x_{m}(t)=\sum_{j=0}^{m} \frac{t^{j}}{j !} A^{j} x_{0}
$$
The limit as $m \rightarrow \infty$ is given by
$$
x(t)=\lim {m \rightarrow \infty} x{m}(t)=\sum_{j=0}^{\infty} \frac{t^{j}}{j !} A^{j} x_{0} .
$$
In the one dimensional case $(n=1)$ this series is just the usual exponential and hence we will write
$$
x(t)=\exp (t A) x_{0}
$$