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# 数学代写|差分方程作业代写difference equation代考|NONLINEAR INITIAL VALUE PROBLEMS

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## 数学代写|差分方程作业代写difference equation代考|Predictor–corrector methods

The idea behind predictor-corrector methods is easy. In marching from time level $n$ to time level $n+1$, we first ‘predict’ an intermediate and ‘rough’ solution using some explicit finite difference scheme and we then ‘correct’ it at time level $n+1$. The advantage of this approach is that we can approximate a nonlinear IVP by a sequence of simpler (and linear!) finite difference schemes.

In order to motivate the current scheme, let us first discretise (6.37) by the trapezoidal rule
$$y_{n+1}-y_{n}=\frac{1}{2} h\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n+1}\right)\right], \quad n=0,1,2, \ldots$$
This is an example of an implicit method because the unknown value of $y$ at time level $n+1$ appears implicitly on the right-hand side of equation (6.38). Thus we cannot directly solve this problem at time level $n+1$. If $f$ is a nonlinear function we then have to solve a nonlinear system at each time level because the unknown function lives on both sides of equation (6.38) as it were. This complicates matters somewhat but not all is lost because we modify (6.38) so that the unknown value is removed from the right-hand side. To this end, we propose the following (iterative) algorithm:

• Step 1: Calculate an ‘intermediate’ value (called the predictor) as follows:
$$y_{n+1}^{(0)}=y_{n}+h f\left(t_{n}, y_{n}\right)$$
Please note that we calculate the predictor by using the explicit Euler method. We now adapt equation (6.38), by using the predicted value on the right-hand side instead of the unknown function to get the approximation
$$y_{n+1}^{(1)}=y_{n}+\frac{h}{2}\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n+1}^{(0)}\right)\right]$$
• Step 2: The general iteration is given by
$$y_{n+1}^{(k)}=y_{n}+\frac{h}{2}\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n+1}^{(k-1)}\right)\right], \quad k=1,2, \ldots$$
• Step 3: We compute the left-hand side of (6.41) until
$$\frac{\left|y_{n+1}^{(k)}-y_{n+1}^{(k-1)}\right|}{\left|y_{n+1}^{(k)}\right|} \leq \epsilon \text { for prescribed tolerance } \epsilon$$

## 数学代写|差分方程作业代写difference equation代考|Runge-Kutta methods

There is a vast literature on Runge-Kutta $(\mathrm{RK})$ methods and their applications to initial value problems (Stoer and Bulirsch, 1980; Conte and de Boor, 1980, Crouzeix, 1975). We give the essentials of these methods in this section. Basically, Runge-Kutta methods are based on the idea of comparing the value of $f(t, y)$ to several strategically chosen points near the solution curve in the interval $\left(t_{n}, t_{n+1}\right)$ and then to combine these values in such a way as to get good accuracy in the computed increment $y_{n+1}-y_{n}$.
The simplest RK method is called Heun’s method:
\begin{aligned} &k_{1}=h f\left(t_{n}, y_{n}\right) \ &k_{2}=h f\left(t_{n}+h, y_{n}+k_{1}\right) \ &y_{n+1}=y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right) \end{aligned}
This is a second-order scheme, as can be seen from the series
$$y(t, h)=y(t)+c_{2}(t) h^{2}+\sum_{j=3}^{\infty} c_{j}(t) h^{j}$$
where $y(t, h)$ is the solution of $(6.44)$ at the value $t$. Notice that we are using $h$ as the time step value. Thus, we can apply Richardson extrapolation to improve the accuracy.
A well-known RK method is the fourth-order method defined as follows:
\begin{aligned} &k_{1}=h f\left(t_{n}, y_{n}\right) \ &k_{2}=h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{k_{1}}{2}\right) \ &k_{3}=h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{k_{2}}{2}\right) \ &k_{4}=h f\left(t_{n}+h, y_{n}+k_{3}\right) \ &y_{n+1}=y_{n}+\frac{1}{6}\left(k_{1}+2 k_{2}+2 k_{3}+k_{4}\right) \end{aligned}

## 数学代写|差分方程作业代写DIFFERENCE EQUATION代考|PREDICTOR–CORRECTOR METHODS

• 第 1 步：计算“中间”值C一种ll和d吨H和pr和d一世C吨这r如下：
是n+1(0)=是n+HF(吨n,是n)
请注意，我们使用显式欧拉方法计算预测变量。我们现在调整方程6.38，通过使用右侧的预测值而不是未知函数来获得近似值
是n+1(1)=是n+H2[F(吨n,是n)+F(吨n+1,是n+1(0))]
• 第 2 步：一般迭代由下式给出
是n+1(ķ)=是n+H2[F(吨n,是n)+F(吨n+1,是n+1(ķ−1))],ķ=1,2,…
• 第 3 步：我们计算左手边6.41直到
|是n+1(ķ)−是n+1(ķ−1)||是n+1(ķ)|≤ε 对于规定的公差 ε

## 数学代写|差分方程作业代写DIFFERENCE EQUATION代考|RUNGE-KUTTA METHODS

\begin{aligned} &k_{1}=h f\left(t_{n}, y_{n}\right) \ &k_{2}=h f\left(t_{n}+h, y_{n}+k_{1}\right) \ &y_{n+1}=y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right) \end{aligned}
This is a second-order scheme, as can be seen from the series
$$y(t, h)=y(t)+c_{2}(t) h^{2}+\sum_{j=3}^{\infty} c_{j}(t) h^{j}$$

\begin{aligned} &k_{1}=h f\left(t_{n}, y_{n}\right) \ &k_{2}=h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{k_{1}}{2}\right) \ &k_{3}=h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{k_{2}}{2}\right) \ &k_{4}=h f\left(t_{n}+h, y_{n}+k_{3}\right) \ &y_{n+1}=y_{n}+\frac{1}{6}\left(k_{1}+2 k_{2}+2 k_{3}+k_{4}\right) \end{aligned}

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。