## 数学代写|数值分析代写numerical analysis代考|CIVL5458

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Solving linear systems of equations

The need to solve systems of linear equations $\mathbf{A x}=\mathbf{b}$ arises across nearly all of engineering and science, business, statistics, economics, and many other fields. In a standard undergraduate linear algebra course, we have learned how to solve this problem using Gaussian Elimination (GE). We will show here how such a procedure is equivalent to an LU factorization of the coefficient matrix A, followed by a forward and a back substitution. To achieve stability of the factorization in computer arithmetic, a strategy called pivoting is necessary, which leads to the LU factorization with partial pivoting. This is the standard direct method for solving linear systems where $\mathbf{A}$ is a dense matrix.

Linear systems with large and sparse (most entries are zero) coefficient matrices arise often in numerical solution methods of differential equations, for example, by the finite element and finite difference discretizations. State-of-the-art direct methods can nowadays efficiently solve such linear systems up to an order of a few million, using advanced strategies to keep the LU factors as sparse as possible and the factorization stable. However, problems of ever-increasing dimension need be tackled, and sparse linear systems of order tens of millions to billions have become more routine. To efficiently solve these large systems approximately, iterative methods such as the Conjugate Gradient (CG) method are typically used, and on sufficiently large problems, can be advantageous over direct methods.

This chapter will mainly focus on direct methods but will also discuss the CG method. “Linear solvers” has become a vast field and is a very active research area. We aim here to provide a fundamental understanding of the basic types of solvers, but note that we are just scratching the surface, in particular for iterative methods.

## 数学代写|数值分析代写numerical analysis代考|Solving triangular linear systems

Consider a system of linear equations $\mathbf{A x}=\mathbf{b}$, where the coefficient matrix $\mathbf{A}$ is square and nonsingular. Recall that the GE procedure gradually eliminates all entries in the coefficient matrix below the main diagonal by elementary row operations, until the modified coefficient matrix becomes an upper triangular matrix U. The solution remains unchanged during the entire procedure. In this section, we consider how to solve a linear system where the coefficient matrix is upper or lower triangular. The procedure of elimination will be reviewed and explored in the new perspective of matrix factorization in the next section.

Example 3 (Back substitution for an upper triangular system). Consider the linear system $x_1+2 x_2+3 x_3=2,4 x_2+5 x_3=3$ and $6 x_3=-6$. It can be written in matrix form as $$\left(\begin{array}{lll} 1 & 2 & 3 \ & 4 & 5 \ & & 6 \end{array}\right)\left(\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right)=\left(\begin{array}{c} 2 \ 3 \ -6 \end{array}\right)$$
where the coefficient matrix is upper triangular. To solve this linear system, we start from the last equation $6 x_3=-6$, which immediately gives $x_3=\frac{-6}{6}=-1$.

Then, from the second equation $4 x_2+5 x_3=3$, we get $x_2=\frac{3-5 x_3}{4}=2$. Finally, the first equation $x_1+2 x_2+3 x_3=2$ leads to $x_1=2-2 x_2-3 x_3=1$.

This procedure illustrates the general procedure of back substitution. Given an upper triangular linear system with nonzero diagonal entries
$$\mathbf{U x}=\left(\begin{array}{cccc} u_{11} & u_{12} & \ldots & u_{1 n} \ & \ddots & \ddots & \vdots \ & & u_{(n-1)(n-1)} & u_{(n-1) n} \ & & & u_{n n} \end{array}\right)\left(\begin{array}{c} x_1 \ \vdots \ x_{n-1} \ x_n \end{array}\right)=\left(\begin{array}{c} b_1 \ \vdots \ b_{n-1} \ b_n \end{array}\right),$$
we start with the last equation and evaluate $x_n=\frac{b_n}{u_{n n}}$ directly, then substitute it into the previous equation and compute $x_{n-1}=\frac{b_{n-1}-u_{(n-1)} x_{n n}}{u_{(n-1)}(n-1)}$. Assume in general that we have already solved for $x_{i+1}, \ldots, x_n$, then $x_i=\frac{\left.b_i-\sum_{i-i+1}^n u_{i j} x_i-1\right)}{u_{i i}}$ can be evaluated. We continue until the value of $x_1$ is found.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Solving triangular linear systems

$$\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \end{array}\right)\left(\begin{array}{lll} x_1 & x_2 & x_3 \end{array}\right)=\left(\begin{array}{lll} 2 & 3 & -6 \end{array}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|数值分析代写numerical analysis代考|MATH3003

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|bit floating-point numbers

By far the most common computer number representation system is the 64-bit “double” floating-point number system. This is the default used by all major mathematical and computational software. In some cases, it makes sense to use 32 or 128 bit number systems, but that is a discussion for later (later, as in “not in this book”), as first we must learn the basics. Each “bit” on a computer is a 0 or a 1, and each number on a computer is represented by 640 ‘s and 1’s. If we assume each number is in standard binary form, then the important information for each number is (i) sign of the number, (ii) exponent, and (iii) the digits after the decimal point. Note that the number 0 is an exception and is treated as a special case for the number system.
The IEEE standard divides up the 64 bits as follows:

• 1 bit sign: 0 for positive, 1 for negative;
• 11 bit exponent: the base 2 representation of (standard binary form exponent + 1023);
• 52 bit mantissa: the first 52 digits after decimal point from standard binary form.
The reason for the “shift” (sometimes also called bias) of 1023 in the exponent is so that the computer does not have to store a sign for the exponent (more numbers can be stored this way). The computer knows internally that the number is shifted, and knows how to handle it.

With the bits from above denoted as sign $s$, exponent $E$, and mantissa $b_1, \ldots, b_{52}$ the corresponding number is standard binary form is $(-1)^s \cdot 1 . b_1 \ldots b_{52} \times 2^{E-1023}$.
Example 2. Convert the base 10 number $d=11.5625$ to 64 bit double floating-point representation.

From a previous example, we know that $11.5625=(1011.1001)_{\text {base2 }}$, and so has standard binary representation of $1.0111001 \times 2^3$. Hence, we immediately know that
\begin{aligned} & \text { sign bit }=0 \ & \text { mantissa }=0111001000000000000000000000000000000000000000000000 \end{aligned}

As we saw in Example 1 in this chapter, if we add 1 to $10^{-16}$, it does not change the 1 at all. Additionally, the next computer representable number after 1 is $1+2^{-52}=$ $1+2.22 \times 10^{-16}$. Since $1+10^{-16}$ is closer to 1 than it is to $1+2.22 \times 10^{-16}$, it gets rounded to 1 , leaving the $10^{-16}$ to be lost forever.

We have seen this effect in the example at the beginning of this chapter when repeatedly adding $10^{-16}$ to 1 . Theoretically speaking, addition in floating-point computation is not associative, meaning $(A+B)+C=A+(B+C)$ may not hold, due to rounding.

One way to minimize this type of error when adding several numbers is to add from smallest to largest (if they all have the same sign) and to use factorizations that lessen the problem. There are other more complicated ways to deal with this kind of error that is out of the scope of this book, for example, the “Kahan Summation Formula.”

The issue here is that insignificant digits can become significant digits, and the problem is illustrated in Example 2, earlier in this chapter. Consider the following MATLAB command and output:
\begin{aligned} & \gg 1+1 e-15-1 \ & \text { ans }= \ & \text { 1. } 110223024625157 \mathrm{e}-15 \ & \end{aligned}
Clearly, the answer should be $10^{-15}$, but we do not get that, as we observe error in the second significant digit. It is true that the digits of accuracy in the subtraction operation is 16 , but there is a potential problem with the “garbage” digits 110223024625157 (these digits arise from rounding error). If we are calculating a limit, for example, they could play a role.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|bit floating-point numbers

• 1位符号: 0为正, 1 为负；
• 11位指数： (标准二进制形式指数 $+1023$ ) 的2进制表示;
• 52 位尾数：标准二进制形式的小数点后的前 52 位。
指数中 1023 的”移位” (有时也称为偏差) 的原因是计算机不必为指数存储符号 (可以通过这种方式 存储更多数字) 。计算机内部知道数字被移动了，并且知道如何处理它。
上面的位表示为符号 $s$ ，指数 $E$ ，和尾数 $b_1, \ldots, b_{52}$ 相应的数字是标准的二进制形式是 $(-1)^s \cdot 1 . b_1 \ldots b_{52} \times 2^{E-1023}$.
示例 2. 转换以 10 为底的数字 $d=11.5625$ 到 64 位双浮点表示。
从前面的例子我们知道 $11.5625=(1011.1001)_{\text {base22 }}$ ，所以有标准的二进制表示 $1.0111001 \times 2^3$. 因此，我们立即知道
$$\text { sign bit }=0 \quad \text { mantissa }=011100100000000000000000000000000000000000000000$$

$$\gg 1+1 e-15-1 \quad \text { ans }=1.110223024625157 \mathrm{e}-15$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|数值分析代写numerical analysis代考|MATHS7104

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Examples of the effects of roundoff error

To motivate the need to study computer representation of numbers, let us consider first some examples taken from MATLAB-but we note that the same thing happens in C, Java, etc.:

1. The order in which you add numbers on a computer makes a difference!
\begin{aligned} & >1+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16 \ & \text { ans }= \ & \quad 1 \ & >1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \ & \text { ans }= \ & 1.000000000000001 \end{aligned}
Note: AAAeBBB is a common notation for a floating-point number with the value $A A A \times$ $10^{B B B}$. So $1 \mathrm{e}-16=10^{-16}$.

As we will see later in this chapter, the computer stores about 16 base 10 digits for each number; this means we get 15 digits after the first nonzero digit of a number. Hence, if you try to add 1e-16 to 1, there is nowhere for the computer to store the 1e-16 since it is the 17 th digit of a number starting with 1 . It does not matter how many times you add 1e-16; it just gets lost in each intermediate step, since operations are always done from left to right. So even if we add $1 \mathrm{e}-16$ to 1,10 times in a row, we get back exactly 1 . However, if we first add the $1 \mathrm{e}-16$ ‘s together, then add the 1 , these small numbers get a chance to combine to become big enough not to be lost when added to 1 .

$$f(x)=\frac{e^x-e^{-x}}{x}$$
Suppose we wish to calculate
$$\lim _{x \rightarrow 0} f(x) .$$
By L’Hopital’s theorem, we can easily determine the answer to be 2. However, how might one do this on a computer? A limit is an infinite process, and moreover, it requires some analysis to get an answer. Hence on a computer one is seemingly left with the option of choosing small $x$ ‘s and plugging them into $f$. Table $1.1$ shows what we get back from MATLAB by doing so.

## 数学代写|数值分析代写numerical analysis代考|Binary numbers

Computers and software allow us to work in base 10 , but behind the scenes everything is done in base 2. This is because numbers are stored in computer memory (essentially) as “voltage on” (1) or “voltage off” (0). Hence, it is natural to represent numbers in their base 2, or binary, representation. To explain this, let us start with base 10 , or decimal, number system. In base 10 , the number $12.625$ can be expanded into powers of 10 , each multiplied by a coefficient:
$$12.625=1 \times 10^1+2 \times 10^0+6 \times 10^{-1}+2 \times 10^{-2}+5 \times 10^{-3} .$$
It should be intuitive that the coefficients of the powers of 10 must be digits between 0 and 9. Also, the decimal point goes between the coefficients of $10^{\circ}$ and $10^{-1}$.

Base 2 numbers work in an analogous fashion. First, note that it only makes sense to have digits of 0 and 1 , for the same reason that digits in base 10 must be 0 through 9 . Also, the decimal point goes between the coefficients of $2^0$ and $2^{-1}$. Hence in base 2 we have, for example, that
$$(11.001)_{\text {base } 2}=1 \times 2^1+1 \times 2^0+0 \times 2^{-1}+0 \times 2^{-2}+1 \times 2^{-3}=2+1+\frac{1}{8}=3.125 .$$
Converting a base 2 number to a base 10 number is nothing more than expanding it into powers of 2. To get an intuition for this, consider Table $1.3$ that converts the base 10 numbers 1 through 10.

The following algorithm will convert a base 10 number to a base 2 number. Note this is not the most efficient computational algorithm, but perhaps it is the easiest to understand for beginners.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Examples of the effects of roundoff error

1. 您在计算机上添加数字的顺序会有所不同!
$$1+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16 \quad \text { ans }$$

$$f(x)=\frac{e^x-e^{-x}}{x}$$

$$\lim _{x \rightarrow 0} f(x) .$$

## 数学代写|数值分析代写numerical analysis代考|Binary numbers

$$12.625=1 \times 10^1+2 \times 10^0+6 \times 10^{-1}+2 \times 10^{-2}+5 \times 10^{-3} .$$

$(11.001)_{\text {base } 2}=1 \times 2^1+1 \times 2^0+0 \times 2^{-1}+0 \times 2^{-2}+1 \times 2^{-3}=2+1+\frac{1}{8}=3.125$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|数值分析代写numerical analysis代考|MATH2722

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Solving Fredholm Integral Equation via Tight Framelets

Many methods have been presented to find exact and approximate solutions of different integral equations. In this work, we introduce a new method for solving the above-mentioned class of equations. We use quasi-affine tight framelets systems generated by the UEP and OEP for solving some types of integral equations. Consider the second-kind linear Fredholm integral equation of the form:
$$u(x)=f(x)+\lambda \int_a^b \mathcal{K}(x, t) u(t) d t,-\infty<a \leq x \leq b<\infty,$$
where $\lambda$ is a real number, $f$ and $\mathcal{K}$ are given functions and $u$ is an unknown function to be determined. $\mathcal{K}$ is called the kernel of the integral Equation (10). A function $u(x)$ defined over $[a, b]$ can be expressed by quasi-affine tight framelets as Equation (5). To find an approximate solution $u_n$ of (10), we will truncate the quasi-affine framelet representation of $u$ as in Equation (6). Then,
$$u(x) \approx u_n(x)=\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x),$$
where
$$c_{j, k}^{\ell}=\int_{\mathbb{R}} u_n(x) \psi_{j, k}^{\ell}(x) d x .$$
Substituting (11) into (10) yields
$$\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x)=f(x)+\lambda \sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) d t$$
Multiply Equation (12) by $\sum_{s=1}^r \psi_{p, q}^s(x)$ and integrate both sides from $a$ to $b$. This can be a generalization of Galerkin method used in Reference [29,30]. Then, with a few algebra, Equation (12) can be simplified to a system of linear equations with the unknown coefficients $c_{j, k}^{\ell}$ (to be determined) given by
$$\sum_{s, \ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} m_{j, k, p, q}^{\ell, s}=g_{p, q \prime} \quad p, q \in \mathbb{Z},$$
where
$$m_{j, k, p, q}^{\ell, s}=\int_a^b \psi_{j, k}^{\ell}(x) \psi_{p, q}^s(x) d x-\lambda \int_a^b \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) \psi_{p, q}^s(x) d x d t, \quad p, q \in \mathbb{Z}$$
and
$$g_{p, q}=\sum_{s=1}^r \int_a^b f(x) \psi_{p, q}^s(x) d x, \quad p, q \in \mathbb{Z} .$$

## 数学代写|数值分析代写numerical analysis代考|Error Analysis

In this section, we get an upper bound for the error of our method. Let $\phi$ be as in Equation (1) and $W_2^m(\mathbb{R})$ is the Sobolev space consists of all square integrable functions $f$ such that $\left{f^{(k)}\right}_{k=0}^m \in L^2(\mathbb{R})$. Then, $X^0(\Psi)$ provides approximation order $m$, if
$$\left|f-S_n f\right|_2 \leq C 2^{-n m} \mid f^{(m)} |_2, \quad \forall f \in W_2^m(\mathbb{R}), n \in \mathbb{N} .$$
The approximation order of the truncated function $S_n$ was studied in References [20,31]. It is well known in the literature that the vanishing moments of the framelets can be determined by its low and high pass filters $\hat{h}_{\ell} \ell=0, \ldots, r$. Also, if the quasi-affine framelet system has vanishing moments of order say $m_1$ and the low pass filter of the system satisfy the following,
$$1-\left|\hat{h}_0(\xi)\right|^2=\mathcal{O}\left(|\cdot|^{2 m}\right),$$
at the origin, then the approximation order of $X^0(\Psi)$ is equal to $\min \left{m_1, m\right}$. Therefore, as the OEP increases the vanishing moments of the quasi-affine framelet system, the accuracy order of the truncated framelet representation, will increase as well.

As mentioned earlier, integral equations describe many different events in applications such as image processing and data reconstructions, for which the regularity of the function $f$ is low and does not meet the required order of smoothness. This makes the determination of the approximation order difficult from the functional analysis side. Instead, it is assumed that the solution function to satisfy a decay condition with a wavelet characterization of Besov space $B_{2,2}^s$. We refer the reader to Reference [32] for more details. Hence, we impose the following decay condition such that
$$N_f=\sum_{\ell=1}^r \sum_{j \geq 0} \sum_{k \in \mathbb{Z}} 2^{s j}\left|\left\langle f, \psi_{j, k}^{\ell}\right\rangle\right|<\infty,$$
where $s \geq-1$.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Solving Fredholm Integral Equation via Tight Framelets

$$u(x)=f(x)+\lambda \int_a^b \mathcal{K}(x, t) u(t) d t,-\infty<a \leq x \leq b<\infty,$$

$$u(x) \approx u_n(x)=\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x),$$

$$c_{j, k}^{\ell}=\int_{\mathbb{R}} u_n(x) \psi_{j, k}^{\ell}(x) d x .$$

$$\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x)=f(x)+\lambda \sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) d t$$

$$\sum_{s, \ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} m_{j, k, p, q}^{\ell, s}=g_{p, q^{\prime}} \quad p, q \in \mathbb{Z},$$

$$m_{j, k, p, q}^{\ell, s}=\int_a^b \psi_{j, k}^{\ell}(x) \psi_{p, q}^s(x) d x-\lambda \int_a^b \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) \psi_{p, q}^s(x) d x d t, \quad p, q \in \mathbb{Z}$$
$$g_{p, q}=\sum_{s=1}^r \int_a^b f(x) \psi_{p, q}^s(x) d x, \quad p, q \in \mathbb{Z}$$

## 数学代写|数值分析代写numerical analysis代考|Error Analysis

$$\left|f-S_n f\right|2 \leq C 2^{-n m}\left|f^{(m)}\right|_2, \quad \forall f \in W_2^m(\mathbb{R}), n \in \mathbb{N} .$$ 截断函数的逼近阶数 $S_n$ 在参考文献 [20,31] 中进行了研究。在文献中众所周知，小框架的消失时刻可以通 过其低通和高通滤波器来确定 $\hat{h}{\ell} \ell=0, \ldots, r$. 此外，如果准仿射框架系统具有消失的秩序时刻说 $m_1$ 并 且系统的低通滤波器满足以下条件，
$$1-\left|\hat{h}0(\xi)\right|^2=\mathcal{O}\left(|\cdot|^{2 m}\right),$$ 在原点，那么近似阶 $X^0(\Psi)$ 等于 \min \eft{m_1, m\right } } \text { . 因此，随着 OEP 增加准仿射小框架系统的消失 } 矩，截断小框架表示的精度阶数也将增加。 如前所述，积分方程描述了图像处理和数据重建等应用中的许多不同事件，其中函数的正则性 $f$ 低且不满 足所需的平滑度顺序。这使得从泛函分析方面难以确定近似阶数。相反，假设解函数满足具有 Besov 空 间小波特征的衰减条件 $B{2,2}^s$. 我们建议读者参阅参考文献 [32] 了解更多详细信息。因此，我们施加以下衰 减条件，使得
$$N_f=\sum_{\ell=1}^r \sum_{j \geq 0} \sum_{k \in \mathbb{Z}} 2^{s j}\left|\left\langle f, \psi_{j, k}^{\ell}\right\rangle\right|<\infty$$

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Preliminary Results

Frame theory is a relatively emerging area in pure as well as applied mathematics research and approximation. It has been applied in a wide range of applications in signal processing [13], image denoising [14], and computational physics and biology [15]. Interested readers should consult the references therein to get a complete picture.

The expansion of a function in general is not unique. So, we can have a redundancy for a given representation. This happens, for instance, in the expansion using tight frames. Frames were introduced in 1952 by Duffin and Schaeffer [8]. They used frames as a tool in their paper to study a certain class of non-harmonic Fourier series. Thirty years later, Young introduced a beautiful development for abstract frames and presented their applications to non-harmonic Fourier series [16]. Daubechies et al. constructed frames for $L^2(\mathbb{R})$ based on dilations and translation of functions [17]. These papers and others spurred a dramatic development of wavelet and framelet theory in the following years.
The space $L^2(\mathbb{R})$ is the set of all functions $f(x)$ such that
$$|f|_{L^2(\mathbb{R})}=\left(\int_{\mathbb{R}}|f(x)|^2\right)^{1 / 2}<\infty .$$ Definition 1. A sequence $\left{f_k\right}_{k=1}^{\infty}$ of elements in $L^2(\mathbb{R})$ is a frame for $L^2(\mathbb{R})$ if there exist constants $A, B>0$ such that
$$A|f|^2 \leq \sum_{k=1}^{\infty}\left|\left\langle f, f_k\right\rangle\right|^2 \leq B|f|^2, \forall f \in L^2(\mathbb{R}) .$$
A frame is called tight if $A=B$.
Let $\ell_2(\mathbb{Z})$ be the set of all sequences of the form $h[k]$ defined on $\mathbb{Z}$, satisfying
$$\left(\sum_{k=-\infty}^{\infty}|h[k]|^2\right)^{1 / 2}<\infty .$$
The Fourier transform of a function $f \in L^2(\mathbb{R})$ is defined by
$$\widehat{f}(\xi)=\int_{\mathbb{R}} f(t) \mathbf{e}^{-i \zeta \bar{t}} d t, \xi \in \mathbb{R},$$
and its inverse is
$$f(x)=\frac{1}{2 \pi} \int_{\mathbb{R}} \widehat{f}(\xi) \mathbf{e}^{i \xi \bar{\zeta} x} d \xi, x \in \mathbb{R} .$$
Similarly, we can define the Fourier series for a sequence $h \in \ell_2(\mathbb{Z})$ by
$$\widehat{h}(\xi)=\sum_{k \in \mathbb{Z}} h[k] \mathbf{e}^{-i \xi k}$$

## 数学代写|数值分析代写numerical analysis代考|Quasi-Affine B-Spline Tight Framelet Systems

There is an interesting family of refinable functions known as B-splines. It has an important role in applied mathematics, geometric modeling and many other areas [23,24]. An investigation of the frame set using a class of functions that called generalized $B$-spline and which includes the $B$-spline has been studied extensively in Reference [25].

In applications, the $B$-splines of order 2 and 4 are more popular than those of other orders. Also, it is preferred to have the $B$-splines to be centered at $x=0$. Therefore, we define the centered $B$-splines as follows:
Definition 3 ([26]). The B-spline $B_{m+1}$ is defined as follows by using the convolution
$$B_{m+1}(x):=\left(B_m * B_1\right)(x), x \in \mathbb{R},$$
where $B_1(x)$ is defined to be $\chi_{\left[-\frac{1}{2}, \frac{1}{2}\right)}(x)$, the characteristic function for the interval $\left[-\frac{1}{2}, \frac{1}{2}\right)$.
Figure 1 shows the graphs of the first few $B$-splines.

One can easily show that the Fourier transform of the $B$-spline, $B_m$, of order $m$ is given by
$$\widehat{B}m(\xi)=\mathrm{e}^{-i \xi \bar{\zeta} d}\left(\frac{\sin (\xi / 2)}{\xi / 2}\right)^m \text { and } \widehat{h}_0^m(\xi)=\mathrm{e}^{-i \xi d / 2} \cos ^m(\xi / 2) \text {, }$$ where $d=0$ if $m$ is even, and $d=1$ if $m$ is odd. We refer to [27] for more details. 3.1. Framelets by the UEP and Its Generalization The UEP is a method to construct tight framelets from a given refinable function. For a given refinable function and to construct tight framelets system, the function $\Theta$, which is non-negative, essentially bounded and continuous at the origin with $\Theta(0)=1$, should satisfy the following conditions $$\left{\begin{array}{l} \Theta(2 \xi)\left|\widehat{h}_0(\xi)\right|^2+\sum{\ell=1}^r\left|\widehat{h}{\ell}(\xi)\right|^2=\Theta(\xi) ; \ \Theta(2 \xi) \widehat{h}_0(\xi) \widehat{h}_0(\xi+\pi)+\sum{\ell=1}^r \widehat{h}{\ell}(\xi) \widehat{h}{\ell}(\xi+\pi)=0 . \end{array}\right.$$
In applications, it is recommended to use tight framelet systems that are shift-invariant. The set of functions is said to be $\rho$-shift-invariant if for any $k \in \mathbb{Z}$ and $\psi \in \mathcal{S}$, we have $\psi(\cdot-\rho k) \in \mathcal{S}$. Hence, the quasi-affine system was introduced to convert the system $X(\Psi)$ (not shift-invariant) to a shift-invariant system. Next, we present a quasi-affine system that allows us to construct a quasi-affine tight framelet. This system is not an orthonormal basis [28].

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Preliminary Results

$$|f|{L^2(\mathbb{R})}=\left(\int{\mathbb{R}}|f(x)|^2\right)^{1 / 2}<\infty .$$
$$A|f|^2 \leq \sum_{k=1}^{\infty}\left|\left\langle f, f_k\right\rangle\right|^2 \leq B|f|^2, \forall f \in L^2(\mathbb{R})$$

$$\left(\sum_{k=-\infty}^{\infty}|h[k]|^2\right)^{1 / 2}<\infty$$

$$\widehat{f}(\xi)=\int_{\mathbb{R}} f(t) \mathbf{e}^{-i \zeta \bar{t}} d t, \xi \in \mathbb{R}$$

$$f(x)=\frac{1}{2 \pi} \int_{\mathbb{R}} \widehat{f}(\xi) \mathbf{e}^{i \xi \bar{\zeta} x} d \xi, x \in \mathbb{R}$$

$$\widehat{h}(\xi)=\sum_{k \in \mathbb{Z}} h[k] \mathbf{e}^{-i \xi k}$$

## 数学代写|数值分析代写numerical analysis代考|Quasi-Affine B-Spline Tight Framelet Systems

$$B_{m+1}(x):=\left(B_m * B_1\right)(x), x \in \mathbb{R},$$

$$\widehat{B} m(\xi)=\mathrm{e}^{-i \xi \bar{\zeta} d}\left(\frac{\sin (\xi / 2)}{\xi / 2}\right)^m \text { and } \widehat{h}_0^m(\xi)=\mathrm{e}^{-i \xi d / 2} \cos ^m(\xi / 2),$$

$$\Theta(2 \xi)\left|\widehat{h}_0(\xi)\right|^2+\sum \ell=1^r|\widehat{h} \ell(\xi)|^2=\Theta(\xi) ; \Theta(2 \xi) \widehat{h}_0(\xi) \widehat{h}_0(\xi+\pi)+\sum \ell=1^r \widehat{h} \ell(\xi) \widehat{h} \ell(\xi+\pi)$$
、正确的。
$\$ \$$在应用程序中，建议使用移位不变的紧框架系统。据说这组函数是 \rho-shift-invariant 如果对于任何 k \in \mathbb{Z} 和 \psi \in \mathcal{S} ，我们有 \psi(\cdot-\rho k) \in \mathcal{S}. 因此，引入准仿射系统来转换系统 X(\Psi) (不是移位不变的) 到移位 不变的系统。接下来，我们提出了一个准仿射系统，它允许我们构建一个准仿射紧框架。该系统不是正交 基础。 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## 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。 ## 数学代写|数值分析代写numerical analysis代考|MATHS7104 如果你也在 怎样代写数值分析numerical analysis这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。 数值分析是数学的一个分支，使用数字近似法解决连续问题。它涉及到设计能给出近似但精确的数字解决方案的方法，这在精确解决方案不可能或计算成本过高的情况下很有用。 statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。 我们提供的数值分析numerical analysis及其相关学科的代写，服务范围广, 其中包括但不限于: • Statistical Inference 统计推断 • Statistical Computing 统计计算 • Advanced Probability Theory 高等概率论 • Advanced Mathematical Statistics 高等数理统计学 • (Generalized) Linear Models 广义线性模型 • Statistical Machine Learning 统计机器学习 • Longitudinal Data Analysis 纵向数据分析 • Foundations of Data Science 数据科学基础 ## 数学代写|数值分析代写numerical analysis代考|Swift–Hohenberg Type of Equation on a Narrow Band Domain The SH type of equation on a surface \mathcal{S} is given by$$
\frac{\partial \phi(\mathbf{x}, t)}{\partial t}=-\left(\phi^3(\mathbf{x}, t)-g \phi^2(\mathbf{x}, t)+\left(-\epsilon+\left(1+\Delta_{\mathcal{S}}\right)^2\right) \phi(\mathbf{x}, t)\right), \quad \mathbf{x} \in \mathcal{S}, 0<t \leq T,
$$where \Delta_{\mathcal{S}} is the Laplace-Beltrami operator [19,20]. Next, let \Omega_\delta={\mathbf{y} \mid \mathbf{x} \in \mathcal{S}, \mathbf{y}=\mathbf{x}+ \eta \mathbf{n}(\mathbf{x}) for |\eta|<\delta} be a \delta-neighborhood of \mathcal{S}, where \mathbf{n}(\mathbf{x}) is a unit normal vector at \mathbf{x}. Then, we extend the Equation (1) to the narrow band domain \Omega_\delta :$$
\frac{\partial \phi(\mathbf{x}, t)}{\partial t}=-\left(\phi^3(\mathbf{x}, t)-g \phi^2(\mathbf{x}, t)+\left(-\epsilon+\left(1+\Delta_{\mathcal{S}}\right)^2\right) \phi(\mathbf{x}, t)\right), \quad \mathbf{x} \in \Omega_\delta, 0<t \leq T
$$with the pseudo-Neumann boundary condition on \partial \Omega_\delta :$$
\phi(\mathbf{x}, t)=\phi(\mathrm{cp}(\mathbf{x}), t),
$$where \operatorname{cp}(\mathbf{x}) is a point on \mathcal{S}, which is closest to \mathbf{x} \in \partial \Omega_\delta [14]. For a sufficiently small \delta, \phi is constant in the direction normal to the surface. Thus, the Laplace-Beltrami operator in \Omega_\delta can be replaced by the standard Laplacian operator [14], i.e.,$$
\frac{\partial \phi(\mathbf{x}, t)}{\partial t}=-\left(\phi^3(\mathbf{x}, t)-g \phi^2(\mathbf{x}, t)+\left(-\epsilon+(1+\Delta)^2\right) \phi(\mathbf{x}, t)\right), \quad \mathbf{x} \in \Omega_\delta, 0<t \leq T
$$## 数学代写|数值分析代写numerical analysis代考|Numerical Method In this section, we propose an efficient linear second-order method for solving Equation (4) with the boundary condition (3). We discretize Equation (4) in \Omega=\left[-L_x / 2, L_x / 2\right] \times\left[-L_y / 2, L_y / 2\right] \times \left[-L_z / 2, L_z / 2\right] that includes \Omega_\delta. Let h=L_x / N_x=L_y / N_y=L_z / N_z be the uniform grid size, where N_x, N_y, and N_z are positive integers. Let \Omega^h=\left{\mathbf{x}{i j k}=\left(x_i, y_j, z_k\right) \mid x_i=-L_x / 2+i h, y_j=-L_y / 2+\right. j h, z_k=-L_z / 2+k h for \left.0 \leq i \leq N_x, 0 \leq j \leq N_y, 0 \leq k \leq N_z\right} be a discrete domain. Let \phi{i j k}^n be an approximation of \phi\left(\mathbf{x}{i j k}, n \Delta t\right), where \Delta t is the time step. Let \Omega\delta^h=\left{\mathbf{x}{i j k}|| \psi{i j k} \mid<\delta\right} be a discrete narrow band domain, where \psi is a signed distance function for the surface \mathcal{S}, and \partial \Omega_\delta^h=\left{\mathbf{x}{i j k}\left|I{i j k}\right| \nabla_h I_{i j k} \mid \neq\right. 0} are discrete domain boundary points, where \nabla_h I_{i j k}=\left(I_{i+1, j, k}-I_{i-1, j, k}, I_{i, j+1, k}-I_{i, j-1, k}, I_{i, j, k+1}-\right. \left.I_{i, j, k-1}\right) /(2 h). Here, I_{i j k}=0 if \mathbf{x}{i j k} \in \Omega\delta^h, and I_{i j k}=1, otherwise. We here split Equation (4) into the following subequations:$$
\begin{aligned}
& \frac{\partial \phi}{\partial t}=-\left(\phi^3-\epsilon \phi\right), \
& \frac{\partial \phi}{\partial t}=g \phi^2 \
& \frac{\partial \phi}{\partial t}=-(1+\Delta)^2 \phi .
\end{aligned}
$$Equations (5) and (6) are solved analytically and the solutions \phi_{i j k}^{n+1} are given as follows:$$
\phi_{i j k}^{n+1}=\frac{\phi_{i j k}^n}{\sqrt{\left(\phi_{i j k}^n\right)^2 / \epsilon+\left(1-\left(\phi_{i j k}^n\right)^2 / \epsilon\right) e^{-2 \epsilon \Delta t}}} \text { and } \phi_{i j k}^{n+1}=\frac{\phi_{i j k}^n}{1-g \Delta t \phi_{i j k}^n}
$$respectively. In addition, Equation (7) is solved using the Crank-Nicolson method:$$
\frac{\phi_{i j k}^{n+1}-\phi_{i j k}^n}{\Delta t}=-\frac{\left(1+\Delta_h\right)^2}{2}\left(\phi_{i j k}^{n+1}+\phi_{i j k}^n\right)
$$with the boundary condition on \partial \Omega_\delta^h :$$
\phi_{i j k}^n=\phi^n\left(\mathrm{cp}\left(\mathbf{x}_{i j k}\right)\right)
$$# 数值分析代考 ## 数学代写|数值分析代写numerical analysis代考|Swift–Hohenberg Type of Equation on a Narrow Band Domain 曲面上的 SH 型方程 \mathcal{S} 是 (谁) 给的$$
\frac{\partial \phi(\mathbf{x}, t)}{\partial t}=-\left(\phi^3(\mathbf{x}, t)-g \phi^2(\mathbf{x}, t)+\left(-\epsilon+\left(1+\Delta_{\mathcal{S}}\right)^2\right) \phi(\mathbf{x}, t)\right), \quad \mathbf{x} \in \mathcal{S}, 0<t \leq T

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。