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# 数学代写|数值分析代写Numerical analysis代考|MATH345 Initial-Value Problems

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## 数学代写|数值分析代写Numerical analysis代考|One-Step Methods: Basic Concepts

As one can already surmise from Section 7.1, the methods and results for initial-value problems for systems of ordinary differential equations of first order are essentially independent of the number $n$ of unknown functions. In the following we therefore limit ourselves to the case of only one ordinary differential equation of first order for only one unknown function (i.e., $n=$ 1). The results, however, are valid, as a rule, also for systems (i.e., $n>1$ ), provided quantities such as $y$ and $f(x, y)$ are interpreted as vectors, and $|\cdot|$ as norm $|\cdot|$. For the following, we assume that the initial-value problem under consideration is always uniquely solvable.

A first numerical method for the solution of the initial-value problem $y^{\prime}=f(x, y), \quad y\left(x_0\right)=y_0$
is suggested by the following simple observation: Since $f(x, y(x))$ is just the slope $y^{\prime}(x)$ of the desired exact solution $y(x)$ of $(7.2 .1 .1)$, one has for $h \neq 0$ approximately

$$\frac{y(x+h)-y(x)}{h} \approx f(x, y(x)) \text {, }$$
or
$(7.2 .1 .2) \quad y(x+h) \approx y(x)+h f(x, y(x))$.
Once a steplength $h \neq 0$ is chosen, starting with the given initial values $x_0$, $y_0=y\left(x_0\right)$, one thus obtains at equidistant points $x_i=x_0+i h, i=1,2$, $\ldots$, approximations $\eta_i$ to the values $y_i=y\left(x_i\right)$ of the exact solution $y(x)$ as follows:
$(7.2 .1 .3)$
$$\begin{gathered} \eta_0:=y_0 \ \text { for } i=0,1,2, \ldots: \ \eta_{i+1}:=\eta_i+h f\left(x_i, \eta_i\right) \ x_{i+1}:=x_i+h \end{gathered}$$
One arrives at the polygon method of Euler shown in Figure 14.

## 数学代写|数值分析代写Numerical analysis代考|Convergence of One-Step Methods

In this section we wish to examine the convergence behavior as $h \rightarrow 0$ of an approximate solution $\eta(x ; h)$ furnished by a one-step method. We assume that $f \in F_1(a, b)$ and denote by $y(x)$ the exact solution of the initial-value problem $(7.2 .1 .1)$
$$y^{\prime}=f(x, y), \quad y\left(x_0\right)=y_0 .$$
Let $\Phi(x, y ; h)$ define a one-step method,
\begin{aligned} \eta_0 &:=y_0, \ \text { for } i=0, & 1, \ldots: \ \eta_{i+1} &:=\eta_i+h \Phi\left(x_i, \eta_i ; h\right), \ x_{i+1} &:=x_i+h, \end{aligned}
which for $x \in R_h:=\left{x_0+i h \mid i=0,1,2, \ldots\right}$ produces the approximate solution $\eta(x ; h)$ :
$$\eta(x ; h):=\eta_i, \quad \text { if } x=x_0+i h .$$
We are interested in the behavior of the global discretization error
$$e(x ; h):=\eta(x ; h)-y(x)$$
for fixed $x$ and $h \rightarrow 0, h \in H_x:=\left{\left(x-x_0\right) / n \mid n=1,2, \ldots\right}$. Since $e(x ; h)$, like $\eta(x ; h)$, is only defined for $h \in H_x$, this means a study of the convergence of
$$e\left(x ; h_n\right), \quad h_n:=\frac{x-x_0}{n}, \quad \text { as } n \rightarrow \infty$$

## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|ONESTEP METHODS: BASIC CONCEPTS

\frac{y(x+h)-y(x)}{h} \approx f(x, y(x)),
$$或者 (7.2 .1 .2) \quad y(x+h) \approx y(x)+h f(x, y(x)). 一旦一个步长 h \neq 0 被选中，从给定的初始值开始 x_0, y_0=y\left(x_0\right), 因此在等距点获得 x_i=x_0+i h, i=1,2, \ldots, 近似值 \eta_i 价值观 y_i=y\left(x_i\right) 的精确解 y(x) 如下: (7.2 .1 .3)$$
\eta_0:=y_0 \text { for } i=0,1,2, \ldots: \eta_{i+1}:=\eta_i+h f\left(x_i, \eta_i\right) x_{i+1}:=x_i+h
$$得出图 14 所示的欧拉多边形方法。 ## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|CONVERGENCE OF ONE-STEP METHODS 在本节中，我们希望检育收敛行为 h \rightarrow 0 的近似解 \eta(x ; h) 通过一步法提供。我们假设 f \in F_1(a, b) 并表示为 y(x) 初值问题的精确解( (7.2 .1 .1)$$
y^{\prime}=f(x, y), \quad y\left(x_0\right)=y_0 .
$$让 \Phi(x, y ; h) 定义一个一步法，$$
\eta_0:=y_0, \text { for } i=0, \quad 1, \ldots: \eta_{i+1}:=\eta_i+h \Phi\left(x_i, \eta_i ; h\right), x_{i+1} \quad:=x_i+h,

\eta(x ; h):=\eta_i, \quad \text { if } x=x_0+i h .
$$我们对全局离散化误差的行为感兴趣$$
e(x ; h):=\eta(x ; h)-y(x)

e\left(x ; h_n\right), \quad h_n:=\frac{x-x_0}{n}, \quad \text { as } n \rightarrow \infty


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