19th Ave New York, NY 95822, USA

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

my-assignmentexpert™ 差分方程difference equation作业代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。my-assignmentexpert™， 最高质量的差分方程difference equation作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此差分方程difference equation作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

my-assignmentexpert™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在差分方程difference equation代写方面经验极为丰富，各种差分方程difference equation相关的作业也就用不着 说。

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

A special case of an initial value problem is when the number of dimension $n$ in the initial value problem (6.11)-(6.12) is equal to 1 . In this case we speak of a scalar problem and it is useful to study these problems if one wishes to get some insights into how finite difference methods work. A numerical and computational discussion of scalar IVP is given in Duffy (2004). In this section we discuss some numerical properties of one-step finite difference schemes for the linear scalar problem:
\begin{aligned} &L u \equiv \frac{\mathrm{d} u}{\mathrm{~d} t}+a(t) u=f(t), \quad 00, \quad \forall t \in[0, T] The reader can check that the one-step methods (equations (6.17), (6.18) and (6.19)) can be cast as the general form recurrence relation
U^{n+1}=A^{n} U^{n}+B^{n}, \quad n \geq 0
$$Then, using this formula and mathematical induction we can give an explicit solution at any time level as follows:$$
U^{n}=\left(\prod_{j=0}^{n-1} A^{j}\right) U^{0}+\sum_{v=0}^{n-1} B^{v} \prod_{j=v+1}^{n-1} A^{j}, \quad n \geq 1 \quad \text { with } \prod_{j=I}^{j=J} g^{j} \equiv 1 \text { if } I>J
$$A special case is when the coefficients A and B are constant, that is:$$
U^{n+1}=A U^{n}+B, \quad n \geq 0
$$Then the general solution is given by$$
U^{n}=A^{n} U^{0}+B \frac{1-A^{n}}{1-A}, \quad n \geq 0
$$where in equation (6.52) we note that A^{n} \equiv n^{t h} power of constant A and A \neq 1. The proof of this requires the formula for the sum of a series$$
$$For a readable introduction to difference schemes we refer the reader to Goldberg (1986). Learning finite difference theory for the Black-Scholes equation involves not only understanding the main concepts but also developing skills in basic arithmetic. This is absolutely vital if you wish to become proficient in this area of numerical analysis. ## 数学代写|差分方程作业代写difference equation代考|Exponentially fitted schemes We now introduce a special class of schemes that prove to be very useful in approximating the solution of the Black-Scholes PDE. In particular, these so-called exponentially fitted schemes are able to handle discontinuities (near a strike price and at barriers, for example). In general, a fitted scheme is a modification of the Crank-Nicolson scheme (see equation (6.19)) except that we introduce a new coefficient into the difference equation. In order to find this coefficient we argue as follows: Consider the trivial IVP \frac{\mathrm{d} u}{\mathrm{~d} t}+a u=0, \quad a>0 constant u(0)=A with solution u(t)=A \mathrm{e}^{-a t} \sigma \frac{U^{n+1}-U^{n}}{k}+a \frac{U^{n+1}+U^{n}}{2}=0 U^{0}=A We now demand that the solution of (6.55) should equal the solution of (6.54) at the mesh points. This will determine the value of \sigma, and some arithmetic shows that$$
\sigma=\frac{a k}{2} \operatorname{coth} \frac{a k}{2}
$$where \quad \operatorname{coth} x=\left(\mathrm{e}^{2 x}+1\right) /\left(\mathrm{e}^{2 x}-1\right). This is the famous fitting factor and it has been known since the 1950 s (de Allen and Southwell, 1955), elaborated upon by Soviet scientists (Il’in, 1969) and generalised to convection-diffusion equations in Duffy (1980). Based on the fitting factor defined in equation (6.56), we propose the generalised finite difference scheme when the coefficient a in equation (6.54) is variable a=a(t) and non-zero right-hand side f=f(t) : \sigma^{n} \frac{U^{n+1}-U^{n}}{k}+a^{n+\frac{1}{2}} \frac{U^{n+1}+U^{n}}{2}=f^{n+\frac{1}{2}}, \quad n \geq 0 u^{0}=A \sigma^{n} \equiv \frac{a^{n+\frac{1}{2}} k}{2} \operatorname{coth} \frac{a^{n+\frac{1}{2}} k}{2} A full discussion of this scheme, its applicability to the Black-Scholes equations and its implementation in \mathrm{C}++ is given in Duffy (2004). We shall also reuse this fitting factor finite difference schemes for the Black-Scholes equation in later chapters in this book. ## 差分方程代写 ## 数学代写|差分方程作业代写DIFFERENCE EQUATION代考|SCALAR INITIAL VALUE PROBLEMS 初始值问题的一个特例是当维数n在初值问题6.11-6.12等于 1 。在这种情况下，我们谈到了一个标量问题，如果希望了解有限差分方法的工作原理，研究这些问题是有用的。Duffy 中给出了标量 IVP 的数值和计算讨论2004. 在本节中，我们讨论线性标量问题的一步有限差分格式的一些数值性质：$$
\begin{aligned}
&L u \equiv \frac{\mathrm{d} u}{\mathrm{~d} t}+a(t) u=f(t), \quad 00, \quad \forall t \in[0, T]$The reader can check that the one-step methods (equations (6.17), (6.18) and (6.19)) can be cast as the general form recurrence relation $$U^{n+1}=A^{n} U^{n}+B^{n}, \quad n \geq 0$$ Then, using this formula and mathematical induction we can give an explicit solution at any time level as follows: $$U^{n}=\left(\prod_{j=0}^{n-1} A^{j}\right) U^{0}+\sum_{v=0}^{n-1} B^{v} \prod_{j=v+1}^{n-1} A^{j}, \quad n \geq 1 \quad \text { with } \prod_{j=I}^{j=J} g^{j} \equiv 1 \text { if } I>J$$ A special case is when the coefficients$A$and$B$are constant, that is: $$U^{n+1}=A U^{n}+B, \quad n \geq 0$$ Then the general solution is given by $$U^{n}=A^{n} U^{0}+B \frac{1-A^{n}}{1-A}, \quad n \geq 0$$ where in equation (6.52) we note that$A^{n} \equiv n^{t h}$power of constant$A$and$A \neq 1\$.
The proof of this requires the formula for the sum of a series
$$1+A+\cdots+A^{n}=\frac{1-A^{n+1}}{1-A}, \quad A \neq 1$$

## 数学代写|差分方程作业代写DIFFERENCE EQUATION代考|EXPONENTIALLY FITTED SCHEMES

d在 d吨+一种在=0,一种>0持续的在(0)=一种有溶液在(吨)=一种和−一种吨 σ在n+1−在nķ+一种在n+1+在n2=0 在0=一种

σ=一种ķ2考特⁡一种ķ2

σn在n+1−在nķ+一种n+12在n+1+在n2=Fn+12,n≥0

σn≡一种n+12ķ2考特⁡一种n+12ķ2

## 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。