19th Ave New York, NY 95822, USA

# 数学代写|数值分析代写Numerical analysis代考|MATH/CS514 Finite Difference Methods for Partial Differential Equations

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|数值分析代写Numerical analysis代考|Finite Differences

In general, to approximate the derivative of a function at a point, say $f^{\prime}(x)$ or $f^{\prime \prime}(x)$, one constructs a suitable combination of sampled function values at nearby points. The underlying formalism used to construct these approximation formulae is known as the calculus of finite differences. Its development has a long and influential history, dating back to Newton. The resulting finite difference numerical methods for solving differential equations have extremely broad applicability, and can, with proper care, be adapted to most problems that arise in mathematics and its many applications.
The simplest finite difference approximation is the ordinary difference quotient
$$\frac{u(x+h)-u(x)}{h} \approx u^{\prime}(x),$$
used to approximate the first derivative of the function $u(x)$. Indeed, if $u$ is differentiable at $x$, then $u^{\prime}(x)$ is, by definition, the limit, as $h \rightarrow 0$ of the finite difference quotients. Throughout our discussion, $h$, the step size, which may be either positive or negative, is assumed to be small: $|h| \ll 1$. When $h>0$, (11.1) is referred to as a forward difference, while $h<0$ gives a backward difference. Geometrically, the difference quotient equals the slope of the secant line through the two points $(x, u(x))$ and $(x+h, u(x+h))$ on the graph of the function. For small $h$, this should be a reasonably good approximation to the slope of the tangent line, $u^{\prime}(x)$, as illustrated in the first picture in Figure 11.1.

## 数学代写|数值分析代写Numerical analysis代考|Numerical Algorithms for the Heat Equation

Consider the heat equation
$$\frac{\partial u}{\partial t}=\gamma \frac{\partial^2 u}{\partial x^2}, \quad 00. To be concrete, we impose time-dependent Dirichlet boundary conditions$$
$$specifying the temperature at the ends of the bar, along with the initial conditions$$
u(0, x)=f(x), \quad 0 \leq x \leq \ell,
$$specifying the bar’s initial temperature distribution. In order to effect a numerical approximation to the solution to this initial-boundary value problem, we begin by introducing a rectangular mesh consisting of points \left(t_i, x_j\right) with$$
$$For simplicity, we maintain a uniform mesh spacing in both directions, with$$
$$representing, respectively, the time step size and the spatial mesh size. It will be essential that we do not a priori require the two to be the same. We shall use the notation$$
u_{i, j} \approx u\left(t_i, x_j\right) \quad \text { where } \quad t_i=i \Delta t, \quad x_j=j \Delta x
$$to denote the numerical approximation to the solution value at the indicated mesh point. As a first attempt at designing a numerical method, we shall employ the simplest finite difference approximations to the derivatives. The second order space derivative is approximated by (11.6), and hence$$
\begin{aligned}
\frac{\partial^2 u}{\partial x^2}\left(t_i, x_j\right) & \approx \frac{u\left(t_i, x_{j+1}\right)-2 u\left(t_i, x_j\right)+u\left(t_i, x_{j-1}\right)}{(\Delta x)^2}+\mathrm{O}\left((\Delta x)^2\right) \
& \approx \frac{u_{i, j+1}-2 u_{i, j}+u_{i, j-1}}{(\Delta x)^2}+\mathrm{O}\left((\Delta x)^2\right),
\end{aligned}
$$where the error in the approximation is proportional to (\Delta x)^2. Similarly, the one-sided finite difference approximation (11.4) is used to approximate the time derivative, and so$$
\frac{\partial u}{\partial t}\left(t_i, x_j\right) \approx \frac{u\left(t_{i+1}, x_j\right)-u\left(t_i, x_j\right)}{\Delta t}+\mathrm{O}(\Delta t) \approx \frac{u_{i+1, j}-u_{i, j}}{\Delta t}+\mathrm{O}(\Delta t),
$$where the error is proportion to \Delta t. In practice, one should try to ensure that the approximations have similar orders of accuracy, which leads us to choose$$
\Delta t \approx(\Delta x)^2
$$## 数值分析代写 ## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|FINITE DIFFERENCES 一般来说，为了通近函数在某一点的导数，比如说 f^{\prime}(x) 或者 f^{\prime \prime}(x) ，一个人在附近的点构造一个合适的采样函数值组合。用于构造这些近似公式的基础形式被称 为有限差分微积分。它的发展有着悠久而有影响的历史，可以追湖到牛顿。由此产生的用于求解微分方程的有限差分数值方法具有极其广泛的适用性，并且在适当 注意的情况下可以适用于数学及其许多应用中出现的大多数问题。 最简单的有限差分近似是常差商$$
\frac{u(x+h)-u(x)}{h} \approx u^{\prime}(x),
$$用于近似函数的一阶导数 u(x). 的确，如果 u 可微于 x ，然后 u^{\prime}(x) 根据定义，是极限，因为 h \rightarrow 0 的有限差商。在我们的整个讨论中， h ，假设步长很小，可以是 正的也可以是负的: |h| \ll 1. 什么时候 h>0, 11.1被称为前向差异，而 h<0 给出了落后的差异。在几何上，差商等于通过两点的割线的斜率 (x, u(x)) 和 (x+h, u(x+h)) 在函数图上。对于小 h ，这应该是对切线斜率的合理近似， u^{\prime}(x) ，如图11.1第一张图所示。 ## 数学代写|数值分析代写NUMERICAL ANALYSIS代考INUMERICAL ALGORITHMS FOR THE HEAT EQUATION 考虑热方程$$
\frac{\partial u}{\partial t}=\gamma \frac{\partial^2 u}{\partial x^2}, \quad 00 \$. \text { Tobeconcrete, weimposetime – dependentDirichletboundaryconditions } $$在 t, 0=\backslash 阿尔法 t, 四你 t, \ell=\backslash 测试版 t, \backslash ¡quad \mathrm{t} \backslash geq 0 , specifyingthetemperatureattheendsofthebar, alongwiththeinitialconditions 在 0, x=\mathrm{f} x, \quad 0 \eq \mathrm{x} \backslash leq \ell, specifyingthebar’sinitialtemperaturedistribution. Inordertoef fectanumericalapproximationtothesolutiontothisinitial – boundaryvalueproblem, Forsimplicity, wemaintainauniformmeshspacinginbothdirections, with \Delta t=t_{i+1}-t_i, \quad \backslash Delta x=x_{-}{j+1}-x_{-} j=\backslash frac \left.\left.{\backslash e l}\right} n\right}, representing, respectively, thetimestepsizeandthespatialmeshsize. Itwillbeessentialthatwedonotapriorirequirethetwotobethesame. Weshallusethe u_{-}{i, j} \backslash a p p r o x u \backslash left t_{-} i, x_{-} j \backslash 对 quad \backslash text { 其中 } \backslash quad t_{-} i=i \backslash Delta t, \backslash q u a d x_{-} j=j \backslash Delta x \ \$$ 表示指定网格点处解值的数值近似值。 作为设计数值方法的第一次尝试，我们将对导数采用最简单的有限差分近似。二阶空间导数近似为$11.6$，因此 $$\frac{\partial^2 u}{\partial x^2}\left(t_i, x_j\right) \approx \frac{u\left(t_i, x_{j+1}\right)-2 u\left(t_i, x_j\right)+u\left(t_i, x_{j-1}\right)}{(\Delta x)^2}+\mathrm{O}\left((\Delta x)^2\right) \quad \approx \frac{u_{i, j+1}-2 u_{i, j}+u_{i, j-1}}{(\Delta x)^2}+\mathrm{O}\left((\Delta x)^2\right),$$ 其中近似误差与$(\Delta x)^2$. 类似地，单边有限差分近似 11.4用于近似时间导数，因此 $$\frac{\partial u}{\partial t}\left(t_i, x_j\right) \approx \frac{u\left(t_{i+1}, x_j\right)-u\left(t_i, x_j\right)}{\Delta t}+\mathrm{O}(\Delta t) \approx \frac{u_{i+1, j}-u_{i, j}}{\Delta t}+\mathrm{O}(\Delta t),$$ 误差与$\Delta t\$. 在实践中，应该层量确保近似值具有相似的精度等级，这导致我们选择
$$\Delta t \approx(\Delta x)^2$$

## 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。