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# 数学代写|数值分析代写Numerical analysis代考|MAT12004 One-step methods

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

A one-step method expresses $y_{n+1}$ in terms of the previous value $y_n$; later on we shall consider $k$-step methods, where $y_{n+1}$ is expressed in terms of the $k$ previous values $y_{n-k+1}, \ldots, y_n$, where $k \geq 2$. The simplest example of a one-step method for the numerical solution of the initial value problem (12.1), (12.2) is Euler’s method.

Euler’s method. Given that $y\left(x_0\right)=y_0$, let us suppose that we have already calculated $y_n$, up to some $n, 0 \leq n \leq N-1, N \geq 1$; we define
$$y_{n+1}=y_n+h f\left(x_n, y_n\right) .$$
Thus, taking in succession $n=0,1, \ldots, N-1$, one step at a time, the approximate values $y_n$ at the mesh points $x_n$ can be easily obtained. This numerical method is known as Euler’s method.

In order to motivate the definition of Euler’s method, let us observe that on expanding $y\left(x_{n+1}\right)=y\left(x_n+h\right)$ into a Taylor series about $x_n$, retaining only the first two terms, and writing $y^{\prime}\left(x_n\right)=f\left(x_n, y\left(x_n\right)\right)$, we have that
$$y\left(x_n+h\right)=y\left(x_n\right)+h f\left(x_n, y\left(x_n\right)\right)+\mathcal{O}\left(h^2\right) .$$

## 数学代写|数值分析代写Numerical analysis代考|Consistency and convergence

Returning to the general one-step method (12.13), we consider the choice of the function $\Phi$. Theorem $12.2$ suggests that if the truncation error ‘approaches zero’ as $h \rightarrow 0$, then the global error ‘converges to zero’ also. This observation motivates the following definition.

Definition 12.1 The numerical method (12.13) is consistent with the differential equation (12.1) if the truncation error, defined by (12.14), is such that for any $\varepsilon>0$ there exists a positive $h(\varepsilon)$ for which $\left|T_n\right|<\varepsilon$ for $0<h<h(\varepsilon)$ and any pair of points $\left(x_n, y\left(x_n\right)\right),\left(x_{n+1}, y\left(x_{n+1}\right)\right)$ on any solution curve in $D$.

For the general one-step method $(12.13)$ we have assumed that the function $\Phi(\cdot, \cdot ; \cdot)$ is continuous; since $y^{\prime}$ is also a continuous function on $\left[x_0, X_M\right]$ it follows from (12.14) that, in the limit of
$h \rightarrow 0$ and $n \rightarrow \infty$, with $\lim {n \rightarrow \infty} x_n=x \in\left[x_0, X_M\right]$, we have $$\lim {n \rightarrow \infty} T_n=y^{\prime}(x)-\Phi(x, y(x) ; 0)$$

## 数学代写|数值分析代写NUMERICAL ANALYSIS代 考|ONE-STEP METHODS

$$y_{n+1}=y_n+h f\left(x_n, y_n\right) .$$

$$y\left(x_n+h\right)=y\left(x_n\right)+h f\left(x_n, y\left(x_n\right)\right)+\mathcal{O}\left(h^2\right) .$$

## 数学代写|数值分析代奇UUERICAL ANALYSIS代 䒴|CONSISTENCY AND CONVERGENCE

$h \rightarrow 0$ 和 $n \rightarrow \infty$ ， 和 $\lim n \rightarrow \infty x_n=x \in\left[x_0, X_M\right]$ ，我们有
$$\lim n \rightarrow \infty T_n=y^{\prime}(x)-\Phi(x, y(x) ; 0)$$

## Matlab代写

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