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# 数学代写|数值分析代写Numerical analysis代考|MATH/CS514 Linear multistep methods – LMMs

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## 数学代写|数值分析代写Numerical analysis代考|Linear multistep methods – LMMs

In the previous chapters, the effectiveness of $\operatorname{TS}(p)$ methods was shown. For order $p>1$, these methods have a disadvantage in that they require the right-hand side of the dynamic equation to be differentiable a number of times. This often rules out their use in the real-world applications. The families of linear multistep methods (I.MMs) achieve higher order by exploiting $x$ and $x^{\Delta}$ that were computed at the previous $k$-steps and combining them to generate an approximation of the next step. We begin with two-step methods to describe the strategy.

Suppose that $\mathbb{T}$ is a time scale with forward jump operator $\sigma$ and delta differentiation operator $\Delta$. Let also, $a, b \in \mathbb{T}, a<b$, and $t_j \in[a, b] \subset \mathbb{T}, j \in{0,1, \ldots, m}$, so that
$$a=t_0<t_1<\cdots<t_m=b .$$
Consider the IVP
$$\left{\begin{array}{l} x^{\Delta}(t)=f(t, x(t)), \quad t \in[a, b], \ x(a)=x_0, \end{array}\right.$$
where $x_0 \in \mathbb{R}$. Set $r_0=r_{m+1}=0$ and
$$t_j=t_{j-1}+r_j, \quad j \in{1, \ldots, m}, \quad r=\max _{j \in{1, \ldots, m}} r_j .$$

## 数学代写|数值分析代写Numerical analysis代考|Two-step methods

For a delta differentiable function $z$, we need to find constants
$$\alpha_{0 j}, \quad \alpha_{1 j}, \quad \alpha_{2 j}, \quad \beta_{0 j}, \quad \beta_{1 j}, \quad \beta_{2 j}, \quad \gamma_{0 j}, \quad \gamma_{1 j}, \quad \gamma_{2 j},$$
so that
\begin{aligned} z\left(t_j\right.&\left.+r_{j+1}+r_{j+2}\right)+\alpha_{1 j} z\left(t_j+r_{j+1}\right)+\alpha_{0 j} z\left(t_j\right) \ =& r_{j+2}\left(\beta_{2 j} z^{\Delta}\left(t_j+r_{j+1}+r_{j+2}\right)+\beta_{1 j} z^{\wedge}\left(t_j+r_{j+1}\right)+\beta_{0 j} z^{\Delta}\left(t_j\right)\right) \ &+r_{j+1}\left(y_{2 j} z^{\Delta}\left(t_j+r_{j+1}+r_{j+2}\right)+y_{1 j} z^{\Delta}\left(t_j+r_{j+1}\right)+y_{0 j} z^{\Delta}\left(t_j\right)\right) \ &+O\left(r^{p+1}\right), \quad j \in{0, \ldots, m-2}, \end{aligned}
where $p$ might be specified in some cases, or one might try to make $p$ as large as possible in others. We choose $z=x$, where $x$ is a solution of the IVP (7.1) and, dropping the $O\left(r^{p+1}\right)$, we find
\begin{aligned} &x\left(t_j+r_{j+1}+r_{j+2}\right)+\alpha_{1 j} \chi\left(t_j+r_{j+1}\right)+\alpha_{0 j} x\left(t_j\right) \ &\quad=r_{j+2}\left(\beta_{2 j} f\left(t_j+r_{j+1}+r_{j+2}, x\left(t_j+r_{j+1}+r_{j+2}\right)\right)\right. \end{aligned}

$\left.+\beta_{1 j} f\left(t_j+r_{j+1}, x\left(t_j+r_{j+1}\right)\right)+\beta_{0 j} f\left(t_j, x\left(t_j\right)\right)\right)$
$+r_{j+1}\left(y_{2 j} f\left(t_j+r_{j+1}+r_{j+2}, x\left(t_j+r_{j+1}+r_{j+2}\right)\right)\right.$
$\left.+y_{1 j} f\left(t_j+r_{j+1}, x\left(t_j+r_{j+1}\right)\right)+y_{0 j} f\left(t_j, x\left(t_j\right)\right)\right), \quad j \in{0, \ldots, m-2}$.

## 数学代写数值分析代写Numerical analysis代考|Linear multistep methods -LMMs

$$a=t_0<t_1<\cdots<t_m=b .$$

$\$ \$$\eft }$$
x^{\Delta}(t)=f(t, x(t)), \quad t \in[a, b], x(a)=x_0,
$$【正确的。 where \ x_0 \in \mathbb{R} \$$.Set $\$ r_0=r_{m+1}=0 \$$and \ \$$

## 数学代写|数值分析代写Numerical analysis代考|Two-step methods

$$\alpha_{0 j}, \quad \alpha_{1 j}, \quad \alpha_{2 j}, \quad \beta_{0 j}, \quad \beta_{1 j}, \quad \beta_{2 j}, \quad \gamma_{0 j}, \quad \gamma_{1 j}, \quad \gamma_{2 j},$$

$$z\left(t_j+r_{j+1}+r_{j+2}\right)+\alpha_{1 j} z\left(t_j+r_{j+1}\right)+\alpha_{0 j} z\left(t_j\right)=\quad r_{j+2}\left(\beta_{2 j} z^{\Delta}\left(t_j+r_{j+1}+r_{j+2}\right)+\beta_{1 j} z^{\wedge}\left(t_j+r_{j+1}\right)+\beta_{0 j} z^{\Delta}\left(t_j\right)\right)+r_{j+1}\left(y _ { 2 j } z ^ { \Delta } \left(t_j+r_{j+1}+r_{j+}\right.\right.$$

\begin{aligned} &\quad x\left(t_j+r_{j+1}+r_{j+2}\right)+\alpha_{1 j} \chi\left(t_j+r_{j+1}\right)+\alpha_{0 j} x\left(t_j\right) \quad=r_{j+2}\left(\beta_{2 j} f\left(t_j+r_{j+1}+r_{j+2}, x\left(t_j+r_{j+1}+r_{j+2}\right)\right)\right. \ &\left.+\beta_{1 j} f\left(t_j+r_{j+1}, x\left(t_j+r_{j+1}\right)\right)+\beta_{0 j} f\left(t_j, x\left(t_j\right)\right)\right) \ &+r_{j+1}\left(y_{2 j} f\left(t_j+r_{j+1}+r_{j+2}, x\left(t_j+r_{j+1}+r_{j+2}\right)\right)\right. \ &\left.+y_{1 j} f\left(t_j+r_{j+1}, x\left(t_j+r_{j+1}\right)\right)+y_{0 j} f\left(t_j, x\left(t_j\right)\right)\right), \quad j \in 0, \ldots, m-2 . \end{aligned}

## MATLAB代写

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