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# 数学代写|数值分析代写Numerical analysis代考|MTH405 The Euler method

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## 数学代写|数值分析代写Numerical analysis代考|The Euler method

Many problems in science and engineering involve a change of one quantity with respect to another, usually with respect to time. Therefore, they are modeled by differential or difference equations, more specifically, initial or boundary value problems. The differential and difference equations are unified as dynamic equations on time scales. Especially for the nonlinear equations, an exact analytical solution cannot be obtained in many cases. The development of efficient numerical methods having high accuracy has been an extremely important problem in numerical analysis $[4,6,15]$.
The basic and most simple method to find an approximate solution of an initial value problem for a first order differential equation is the so-called Euler method. It is not very accurate, but the simplicity of its derivation is used in the construction of more advanced and complicated numerical methods. It is also used to provide the necessary extra information for the application of such more advanced and complicated methods.

In this chapter, we generalize the Euler method for initial value problems associated with first order dynamic equations on time scales. It can also be adapted to higher order dynamic equations by transforming the $n$th order dynamic equation into a system of $n$ first order dynamic equations.

Let $\mathbb{T}$ be a time scale with forward jump operator $\sigma$, backward jump operator $\rho$, and delta differentiation operator $\Delta$.

## 数学代写|数值分析代写Numerical analysis代考|Analyzing the method

Consider the Cauchy problem
$$\left{\begin{array}{l} x^{\Delta}(t)=f(t, \chi(t)) \ \chi\left(t_0\right)=x_0 \end{array}\right.$$
where $f: \mathbb{T} \times \mathbb{R} \rightarrow \mathbb{R}, x_0 \in \mathbb{R}, t_0 \in \mathbb{T}$ are given, and $x: \mathbb{T} \rightarrow \mathbb{R}$ is unknown. Suppose that $t \in\left[t_0, \bar{t}\right]{\mathbb{T}}$ and take the points $$t_0{i-1}^{l_{i-1}}\left(t_{i-1}\right), l_{i-1} \in \mathbb{N} & \text { if } t_{i-1} \text { is right-scattered } \ t_{i-1}+q_{i-1}, q_{i-1} \in \mathbb{R} & \text { if } t_{i-1} \text { is right-dense }\end{cases}$$
$i=1, \ldots, N$. Define $\left{x_i\right}_{i=1}^N$ by

\begin{aligned} x_{i+1} &=x_i+\left(t_{i+1}-t_i\right) f\left(t_i, x_i\right) \ &=x_i+ \begin{cases}\left(\sigma^{l_i}\left(t_i\right)-t_i\right) f\left(t_i, x_i\right) & \text { if } t_i \text { is right-scattered, } \ q_i f\left(t_i, x_i\right) & \text { if } t_i \text { is right-dense. }\end{cases} \end{aligned}
Formula (4.3) is known as the Euler method for the numerical solution of the initial value problem (4.1) [3].

## 数学代写|数值分析代写Numerical analysis代考|Analyzing the method

$\$ \$$\backslash left {$$
x^{\Delta}(t)=f(t, \chi(t)) \chi\left(t_0\right)=x_0
$$【正确的。 where \ f: \mathbb{T} \times \mathbb{R} \rightarrow \mathbb{R}, x_0 \in \mathbb{R}, t_0 \in \mathbb{T} \aregiven, and \ x: \mathbb{T} \rightarrow \mathbb{R} \$$ isunknown. Supposethat $\$ t \in\left[t_0, \bar{t}\right] \mathbb{T}$\$andtakethepoints
$t_{-} 0{i-1} \wedge\left{1_{-}{i-1}\right} \backslash$ left $\left(t_{-}{i-1} \backslash\right.$ right $), I_{-}{i-1} \backslash$ in $\backslash \operatorname{mathbb}{N} \& \backslash$ text ${$ if $} t_{-}{i-1} \backslash$ text ${$ 是右散布 $} \backslash t_{-}{i-$
$1}+q_{-}{i-1}, q_{-}{i-1} \backslash$ in $\backslash \operatorname{mathbb}{R} \& \backslash$ text ${$ if $} t_{-}{i-1} \backslash$ text ${$ 是右密集 $} \backslash$ lend ${$ cases $}$
$\$ \
$i=1, \ldots, N$. 定义 lleft 的分隔符缺失或无法识别 经过
$x_{i+1}=x_i+\left(t_{i+1}-t_i\right) f\left(t_i, x_i\right) \quad=x_i+\left{\quad\left(\sigma^{l_i}\left(t_i\right)-t_i\right) f\left(t_i, x_i\right) \quad\right.$ if $t_i$ is right-scattered, $q_i f\left(t_i, x_i\right) \quad$ if $t_i$ is right-dense.

## MATLAB代写

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