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# 数学代写|数值分析代写Numerical analysis代考|STAT721 Initial value problems

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## 数学代写数值分析代写Numerical analysis代考|Initial value problems

In this chapter, we consider numerical methods for solving first-order initial value problems (IVPs) of the form
\begin{aligned} & y^{\prime}(t)=f(t, y), \quad t_0 \leq t \leq T, \ & y\left(t_0\right)=y_0, \end{aligned}
where $t_0$ is the start time, $T$ is the end time, and $y_0$ is the initial condition.
While students generally learn about certain types of IVPs in a sophomore differential equations course, they typically only learn about IVPs of very specific types, where exact solutions can be derived as closed-form expressions. While theoretically this may be possible for many types of IVPs, it is unknown and maybe even not possible for most. Moreover, if one cannot find an exact solution in a book, deriving the exact solution one’s self can be very difficult and time consuming, and may not be possible in a reasonable amount of time. The methods we discuss in this chapter approximate the solution at some finite number of $t$-points, and from this we can interpolate the values at every $t$ in the interval $\left[t_0, T\right]$. That is, the “solution” of a numerical ODE solver is a set of points
$$\left(t_0, y_0\right),\left(t_1, y_1\right), \ldots,\left(T, y_n\right)$$

## 数学代写|数值分析代写Numerical analysis代考|Reduction of higher order IVPs to first order

This section reviews that many higher order ODEs can be written as vector systems of first-order ODEs. Recall that the order of an ODE is the highest number of derivatives in any of its terms. For example, the ODE
$$y^{\prime \prime \prime}(t)+y(t) y^{\prime \prime}(t)-t^2=0$$
is a third-order ODE. Provided the ODE can be written in the form
$$y^{(n)}(t)=F\left(t, y, y^{\prime}, \ldots, y^{(n-1)}\right)$$
then it can be written as a first-order vector ODE by the following process:

• An $n$th order ODE will be turned into a first-order ODE with $n$ equations.
• Define functions $u_1, u_2, \ldots, u_n$ by $u_1(t)=y(t)$ and $u_i(t)=y^{(i-1)}(t)$ for $i=2,3, \ldots, n$.
• The equations (identities) $u_i^{\prime}=u_{i+1}$ for $i=1,2, \ldots, n-1$ form the first $n-1$ equations.
• For the last equation, use that $u_n^{\prime}=y^{(n)}(t)=F\left(t, y, y^{\prime}, y^{\prime \prime}, \ldots, y^{(n-1)}\right)=F\left(t, u_1, u_2, u_3\right.$, $\left.\ldots, u_n\right)$
Consider the following example.

## 数学代写数值分析代写Numerical analysis代考|Initial value problems

$$y^{\prime}(t)=f(t, y), \quad t_0 \leq t \leq T, \quad y\left(t_0\right)=y_0$$

$$\left(t_0, y_0\right),\left(t_1, y_1\right), \ldots,\left(T, y_n\right)$$

## 数学代写|数值分析代写Numerical analysis代考|Reduction of higher order NPs to first order

$$y^{\prime \prime \prime}(t)+y(t) y^{\prime \prime}(t)-t^2=0$$

$$y^{(n)}(t)=F\left(t, y, y^{\prime}, \ldots, y^{(n-1)}\right)$$

• 一个 $n$th order ODE 将变成一阶 ODE $n$ 方程式。
• 定义函数 $u_1, u_2, \ldots, u_n$ 经过 $u_1(t)=y(t)$ 和 $u_i(t)=y^{(i-1)}(t)$ 为了 $i=2,3, \ldots, n$.
• 方程式 (恒等式) $u_i^{\prime}=u_{i+1}$ 为了 $i=1,2, \ldots, n-1$ 形成第一个 $n-1$ 方程式。
• 对于最后一个等式，使用那个 $u_n^{\prime}=y^{(n)}(t)=F\left(t, y, y^{\prime}, y^{\prime \prime}, \ldots, y^{(n-1)}\right)=F\left(t, u_1, u_2, u_3\right.$ ， $\ldots, u_n$ )
考虑以下示例。

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