数学代写|数值方法作业代写numerical methods代考|CS514

statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods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|数值方法作业代写numerical methods代考|Vehicular pollution

Vehicular pollution is significantly when ozone harming substances and other destructive contaminations are discharged into the environment from vehicles. These unsafe toxins negatively affect both the ecosystem and the human body. Constituents of vehicular pollution includes carbon monoxide, oxides of nitrogen, particles of soot, metal and pollen, sulfur dioxide other dangerous pollutants.

From Fig. 2.5, you will see a few gases leaving the fumes of the vehicles. This is the thing that vehicular pollution is majorly about. Vehicular pollution is one of the significant reasons for air pollution in light of the fact that these gases leaving the vehicles go up into the air and thus pollute the air. Vehicular pollution is one of the most widely recognized types of pollution across the world since vehicles are fundamentally all over the place. Indeed, vehicular pollution is disregarded in light of the fact that it is from automobile. Each country experiences vehicular pollution. Iran is adjudged to have high urban contamination which includes vehicular pollution, and dust storms which is well known locally as “the 120-day wind.” Schools and different offices are a few times compelled to close down during the period while the specialists disseminate face masks to residents for wellbeing. In Nigeria, there have been deaths recorded due to indoor fossil-fuel generator emission. Gases released (and inhaled) in the rush hour traffic jammed at Bombay-India is reported as smoking 51 cigarettes every day. In other words, research in vehicular pollution does not only require measurement but a well designed numerical analysis.

数学代写|数值方法作业代写numerical methods代考|Gas flaring

Gas flares known as the burning of gas are created through different stages of oil and gas exploration. It is a main source of concern in oil producing countries as it releases significant amount of greenhouse gases. There are research works on how to convert this process for energy generation (ref). However, in a few developing countries, these gases are burned in air, thereby polluting the atmosphere and increasing the temperature of the geographical location.

Gas flaring is also defined as hydrocarbon harvesting and the procedure of combusting gas from wells. In recent times, it is regarded as a major environmental issue, contributing to approximately 150 billion meter cube.

There are three types of flaring: emergency, process, and product flaring. Emergency flaring occurs during compression failure from valve breakage. Process flaring occurs during petrochemical processes, and product flaring occurs during exploration.
There are different causes of gas flaring:
i. Natural gas carried to the surface but cannot be used as it is burned as a means of disposal
ii. Result of oil extraction
iii. Inadequate structure to put gas for industrialization
iv. Excess gas and oils after extraction
v. To avoid explosions caused by simply bottling up huge quantities of gases, flaring is used.

The effects of gas flaring includes acid rain, air pollution, influencing climate change, and reduced agricultural practice. Sulfur dioxide and nitrogen oxide emissions are the main factors of acid rain, which also are combined with atmospheric moisture to produce sulfuric acid and nitric acid, respectively. Acidification of lakes, ponds, and rivers affects both the aquatic and terrestrial organisms. Acid rain also quickens the deterioration of construction materials and paints. Flaring of gas results in the release of impurities, toxic substances that are harmful to humans. $\mathrm{CO}_2$ is produced when gas is not completely burned, and it the most toxic substance to human health. Environmental implications of this gas flaring are severe because it is such an inefficient and poor use of potential fuel that pollutes air. The effects of gas flaring on climate change are significant as it is also a form of fossil fuels burning. The main component of gas flaring is carbon dioxide. By emitting $\mathrm{CO}_2$, the major greenhouse gas, gas flaring contributes to global warming. The second major gas which contributes to greenhouse effect is methane, which is released when gas is vented without being burned. Gas flaring has been seen to affect agriculture as its pollutants are released into the atmosphere like nitrogen, carbon, sulfur oxides, particulate matter, and hydrogen sulfide. These pollutants deplete soil nutrients by acidifying the soil. Given the immense heat generated as well as the $\mathrm{pH}$ acid characteristics of the soil, there would not be any vegetation in the areas of gas flares. Temperature changes have a different effect on crops, including stunted growth, scotched plants, and withered young crops. Gas flaring has also negatively impacted upon human health due to the inhalation of toxic gases which are emitted during unfinished gas flare combustion. These gases have been connected to negative health challenges including cancer, neurological problems, reproductive issues, developmental disorders, children’s abnormalities, lung damage, and skin issues. As seen from the above, there are lots of gray area in numerically modeling to nowcast or forecast gas flares (Fig. 2.6).

数值方法代写

数学代写|数值方法作业代写numerical methods代考|Gas flaring

i. 天然气被带到地表，但不能使用，因为它作为一种处置手段燃烧
ii. 采油结果
iii．
iv.工业化气体结构不足 提取后过量的气体和油
v. 为了避免因简单地装瓶大量气体而引起的爆炸，使用了燃烧。

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|数值方法作业代写numerical methods代考|APM5333

statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods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|数值方法作业代写numerical methods代考|Statistical data treatment

There are various methods involved in the treatment of data, and one of the most common methods is the statistical method of treatment of data. When you apply a statistical approach to a data set in order to turn it from a list of meaningless numbers into useful output, this is known as the statistical treatment of data. Statistical method includes but not limited to; mean, median mode, range, standard deviation, conditional probability, range, distribution range, sampling, correlation, regression,and probability. There are some notable errors in data treatment, and using statistical techniques to classify potential outliers and errors is an important aspect of data processing. Statistical data treatment is one of the essential aspects of any experiment conducted today. It can be seen using any known statistical method to draw meaning from a set of given meaningless data sets. Statistical distribution can be classified into two groups. To begin with, one of them is considered to have discrete random variables, which means that each word includes a single numerical value. The second form of statistical distribution, which includes continuous random variables, is called a continuous random variable distribution (the data is known to take infinitely many values). Statistical data treatment often entails defining the data collection, and one of the most effective ways to do so is to use the measure of core tendencies such as the mean, mode, and median.

The core tendencies described above make it simple for any researcher to perform any research experiment and understand how the data set is concentrated. Central tendencies such as the standard deviation, range, and uncertainty help the researcher understand the data set’s distribution. Nevertheless, care should consistently be taken to assume that all data sets are the same and evenly distributed. Any of the above-mentioned central tendencies can be used to ensure that.

This method involves using some statistical methods to transform a given meaningless data into meaningful data sets. It involves the use of some statistical methods:

MEAN: In statistics, this is a key idea. It describes the characteristics of a statistical distribution. In a set of numbers, it is the most common value.

To measure it, take into account the figures of the relative multitude of terms and then divide by the number of terms. The mean of a collection of data can be determined in several ways. It can be determined using the arithmetic mean process, which involves dividing the total number of data sets by the sum of the total number of data sets. To find the mean, add all of the numbers in a set together, then divide the total by the total number of numbers. A dataset’s mean can also be calculated by a method known as the geometric mean, which is the $n$th root of the product of all numbers in the data set. It includes the volatility and compounding effects of returns. The arithmetic mean, also known as the mean or standard, is the sum of a set of values divided by the number of values in the group.

数学代写|数值方法作业代写numerical methods代考|Air pollution

According to World Health Organization (WHO), 9 out of 10 people breathe highly contaminated air. Air pollution can be defined as the presence or addition of harmful particulates (such as aerosols) or gases (such as greenhouse gases) to the atmosphere that are detrimental to the well-being of human beings and other living organisms and cause damage to the ozone and climate. Some examples of these harmful substances include chlorofluorocarbon $(\mathrm{CFC})$, ammonia, nitrogen oxide $\left(\mathrm{NO}_{\mathrm{x}}\right)$, carbon monoxide (CO), exhaust fumes (soot) etc. Air pollution can be classified under indoor and outdoor air pollution.

Air pollution is one of the biggest risk factors in the world as it causes up to 5 million deaths each year and is the cause of $9 \%$ of deaths around the world. In some developed countries, death rates have been on a decline due to the control and reduction measures of indoor air pollution such as improving proper ventilation, reducing the use of a fireplace. Also, the reduction of outdoor pollution through the enactment of laws and decrees that has strict implications on industrial emissions, anthropogenic emissions, and emissions from unconventional sources such as sewage. The unconventional sources are the new area of research as it is found to emit dangerous bioaerosols into the environment. Most of the bioaerosols are pathogenic. The anthropogenic emissions is the most common, and it can appear as one of the following.

Burning of fossil fuels: Most of the air pollution takes place due to the burning of fossil fuels. Over the years, the burning of fossil fuels has been almost inevitable because fossil fuels have been one of the major sources of energy, electricity, and power generation. In the United States, fossil fuel consumption has nearly tripled within the last 50 years. When these fuels are burnt, they release harmful gases such as carbon monoxide, i.e., a greenhouse gas which is unhealthy to living organisms. Though there is a new crusade under the aegis of sustainable development goals for the promotion of clean environment through the adoption of renewable energy sources, the use of fossil fuel is still on the increase due to many factors such international politics, governmental inadequacies, corruption, and existing employments relating to fossil fuel (Fig. 2.1).

Combustion of fossil fuels is considered a major source of the increased $\mathrm{CO}_2$. The amount of $\mathrm{CO}_2$ produced per equivalent energy unit varies depending on the fuel-gas produces less than oil, and oil produces less than coal (Fig. 2.2). There are other sources of $\mathrm{CO}_2$ production aside from fossil fuel as presented in Fig. 2.2.
Aside from the air pollution from fossil fuel, the pollutants in fuel include mercury, arsenic, and sulfur in coal; sulfur, vanadium, and nickel in oil; and sulfur in gas. These pollutants in the form of heavy metals are an extended danger of fossil fuel burning.

Wildfire: Climate change is causing an increase in forest wildfires. These wildfires have a high contribution in pollution. Wildfires could also be caused by burning of farm stubble. When these fires are ignited, they cause smog and these smog could lead to difficulty in breathing (Fig. 2.3).

数值方法代写

数学代写|数值方法作业代写numerical methods代考|Statistical data treatment

MEAN：在统计学中，这是一个关键思想。它描述了统计分布的特征。在一组数字中，它是最常见的值。

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|数值方法作业代写numerical methods代考|MATH340-002

statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods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|数值方法作业代写numerical methods代考|Mathematical technique

This is a techniquc that involves the use of mathematical theories, formulae, and mathematical manipulation. Sōme of these māthematicāl prōcessses include::

I. Regression analysis: This is an analysis used to evaluate the relationship between two or more set of numerical data. When using this technique, we look for a correlation between the dependent numerical data and any number of independent variables that might have an effect on these numerical data. The aim of regression analysis is to estimate how one or more variables might impact the dependent numerical data, in order to identify trends and patterns. This was used specifically for prediction and forecasting future trends. It is also important to note that regression analysis only helps to determine whether or not there is a relationship between a set of numerical set of data, and it does not say anything about the cause or effect.
II. Factor analysis: This is a technique used to reduce a large set of variables to a smaller number of variahles. It works on the idea of multiple separate, observable variables correlate with each other because they are all associated with an underlying set. This is useful not only because it reduces variable in a particular set of numerical data into smaller understandable variables, but it also helps to uncover hidden patterns.
III. Time series analysis: This is a statistical technique used to identify numerical data using time interval. It records and separate data into groups based on the data that have similar time interval or the time created.

Numerical analysis is mostly needed to solve engineering problems that result into equations that cannot be solved analytically with simple formulas. Some applications are listed here:
a. Modern applications and computer software: Most sophisticated numerical analysis software is embedded in popular software packages, e.g., spreadsheet programs.

b. Business applications: Modern businesses these days make much use of optimization methods in deciding what or how to allocate a resource most efficiently, such as inventory control, scheduling, budgeting, and investment strategies.

数学代写|数值方法作业代写numerical methods代考|Computational technique

This is a technique that involves the use of AI systems such as the computer system. This involves using programmed codes, encoded scripts formulas to arrange and present numerical data in an organized manner meaningful to interpret and use. There are a lot of programming software created to solve this problem. Some of the best ones include these:

I. Analytica: This is a software created and developed by Lumina Decision Systems for receiving/retrieving, analyzing, and communicating numerical data. It uses hierarchical influence diagrams for visual creation and view of models, intelligent arrays for working on multidimensional data.
II. MATLAB: Matrix Laboratory is a proprietary multi-paradigm programming language and numeric computing working environment developed by MathWorks. MATLAB makes it possible for matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. MATLAB is made for the source purpose of numerical data treatment.
III. FlexPro: This is a software designed for the analysis and presentation of scientific and technical data. This software was created by the Weisang GmbH team. It was designed to run Microsoft windows. FlexPro can analyze large amount of data with high sampling rates. All data to be analyzed are stored in an object database. FlexPro has a built-in programming language, FPScript, which is optimized to carry out data analysis and support direct operations on non-scalar objects such as vectors and matrices as well as composed data structures like signal series.
IV. FreeMat: A free open-source numerical data treatment environment and programming language, similar to MATLAB.
V. jLab: This is a numerical computational environment created with a Java software and interface.

数值方法代写

数学代写|数值方法作业代写numerical methods代考|Mathematical technique

：现代应用程序和计算机软件：最复杂的数值分析软件嵌入在流行的软件包中，例如电子表格程序。

数学代写|数值方法作业代写numerical methods代考|Computational technique

I. Analytica：这是由 Lumina Decision Systems 创建和开发的软件，用于接收/检索、分析和交流数值数据。它使用分层影响图来可视化创建和查看模型，使用智能数组来处理多维数据。

V. jLab：这是一个使用 Java 软件和界面创建的数值计算环境。

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|数值方法作业代写numerical methods代考|MATH861

statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods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|数值方法作业代写numerical methods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
$$\left|f-S_n f\right|_2 \leq C 2^{-n m}\left|f^{(m)}\right|_2, \quad \forall f \in W_2^m(\mathbb{R}), n \in \mathbb{N} .$$
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 $B_{2,2}^5$. We refer the reader to Reference [32] for more details. Hence, we impose the following decay condition such that
$$N_f=\sum_{\ell=1}^r \sum_{j \geq 0} \sum_{k \in \mathbb{Z}} 2^{s j}\left|\left\langle f, \psi_{j, k}^{\ell}\right)\right|<\infty,$$
where $s \geq-1$.

数学代写|数值方法作业代写numerical methods代考|Numerical Performance and Illustrative Examples

Based on the method presented in this paper, we solve the following examples using the quasi-affine tight framelets constructed in Section 3.2. The computations associated with these examples were obtained using Mathematica software.
Example 6. We consider the Гredholm integral equation of 2 nd kind defined by:
$$u(x)=1+\int_{-1}^1\left(x t+x^2 t^2\right) u(t) d t .$$
The exact solution is $u(x)=1+\frac{10}{9} x^2$.
In Tables 1 and 2 the absolute error $e_n$ for different values of $n$ and the numerical values of $u_n(x)$ when $n=2$ are computed, respectively. Using quasi-affine Haar framelet system, Figures 7 and 8 demonstrated the graphs of the exact and approximate solutions and Figure 9 demonstrated the graphs of $E_n(x)$ for different values of $n$. For the case of $B_2$-UEP, Figure 10 demonstrated the graphs of the exact and approximate solutions for different values of $n$.

$$u(x)=\mathbf{e}^x-\frac{\mathrm{e}^{x+1}-1}{x+1}+\int_0^1 \mathrm{e}^{x t} u(t) d t .$$
The exact solution is $u(x)=\mathbf{e}^x$.
In Tables 3 and 4 the absolute error $e_n$ for different values of $n$ and the numerical values of $u_n(x)$ when $n=2$ are computed, respectively. Some illustration for the graphs of the exact and approximate solutions are depicted in Figure 11.

数值方法代写

数学代写|数值方法作业代写numerical methods代考|Error Analysis

$$1-\left|\hat{h}0(\xi)\right|^2=\mathcal{O}\left(|\cdot|{ }^{2 m}\right),$$ 截断框架表示的精度阶数也会增加。 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，其中函数的规律性 $f$ 低且不满足所需 的平滑顺序。这使得从功能分析方面难以确定近似阶数。取而代之的是，假设解函数满足具有 Besov空间小波 特征的衰减条件 $B{2,2}^5$. 我们请读者参考参考文献 [32] 了解更多详细信息。因此，我们强加以下衰减条件，使得
$$N_f=\sum_{\ell=1}^r \sum_{j \geq 0} \sum_{k \in \mathbb{Z}} 2^{s j}\left|\left\langle f, \psi_{j, k}^{\ell}\right)\right|<\infty$$

数学代写|数值方法作业代写numerical methods代考|Numerical Performance and Illustrative Examples

$$u(x)=1+\int_{-1}^1\left(x t+x^2 t^2\right) u(t) d t .$$

$$u(x)=\mathbf{e}^x-\frac{\mathrm{e}^{x+1}-1}{x+1}+\int_0^1 \mathrm{e}^{x t} u(t) d t .$$

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。

数学代写|数值方法作业代写numerical methods代考|CS3513

statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods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|数值方法作业代写numerical methods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
$$|\left. f\right|{L^2(\mathbb{R})}=\left(\int{\mathbb{R}}|f(x)|^2\right)^{1 / 2}<\infty .$$ 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
$$\widehat{f}(\xi)=\int_{\mathbb{R}} f(t) \mathrm{e}^{-i \xi \xi} d t, \xi \in \mathbb{R},$$
and its inverse is
$$f(x)=\frac{1}{2 \pi} \int_{\mathbb{R}} \widehat{f}(\xi) \mathrm{e}^{i \zeta \tilde{\zeta} x} d \xi, x \in \mathbb{R} .$$
Similarly, we can define the Fourier series for a sequence $h \in \ell_2(\mathbb{Z})$ by
$$\widehat{h}(\xi)=\sum_{k \in \mathbb{Z}} h[k] \mathrm{e}^{-i \zeta \xi k}$$

数学代写|数值方法作业代写numerical methods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
$$\sum_{s, \ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} m_{j, k, p, q}^{\ell, s}=g_{p, q,} \quad p, q \in \mathbb{Z},$$
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
$$g_{p, q}=\sum_{s=1}^r \int_a^b f(x) \psi_{p, q}^s(x) d x, \quad p, q \in \mathbb{Z} .$$

数值方法代写

数学代写|数值方法作业代写numerical methods代考|Preliminary Results

[16]。Daubechies 等人。构造的框架 $L^2(\mathbb{R})$ 基于函数的膨胀和平移[17]。这些论文和其他论文在接下来的几年 里推动了小波和框架理论的巨大发展。

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

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

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

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

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

数学代写|数值方法作业代写numerical methods代考|Solving Fredholm Integral Equation via Tight Framelets

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

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

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

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

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

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

$$g_{p, q}=\sum_{s=1}^r \int_a^b f(x) \psi_{p, q}^s(x) d x, \quad p, q \in \mathbb{Z}$$

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|数值方法作业代写numerical methods代考|MATH131

statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods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|数值方法作业代写numerical methods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 $\Omega_\delta$ :
$$\frac{\partial \phi(\mathbf{x}, t)}{\partial t}=-\left(\phi^3(\mathbf{x}, t)-g \phi^2(\mathbf{x}, t)+\left(-\epsilon+(1+\Delta \mathcal{S})^2\right) \phi(\mathbf{x}, t)\right), \quad \mathbf{x} \in \Omega_\delta, 0<t \leq T$$
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|数值方法作业代写numerical methods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:
$$\phi_{i j k}^{n+1}=\frac{\phi_{i j k}^n}{\sqrt{\left(\phi_{i j k}^n\right)^2 / \epsilon+\left(1-\left(\phi_{i j k}^n\right)^2 / \epsilon\right) e^{-2 \epsilon \Delta t}}} \quad \text { and } \quad \phi_{i j k}^{n+1}=\frac{\phi_{i j k}^n}{1-g \Delta t \phi_{i j k}^n}$$ 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 $\partial \Omega_\delta^h$ :
$$\phi_{i j k}^n=\phi^n\left(\mathrm{cp}\left(\mathbf{x}_{i j k}\right)\right) .$$

数值方法代写

数学代写|数值方法作业代写numerical methods代考|Swift–Hohenberg Type of Equation on a Narrow Band Domain

$$\frac{\partial \phi(\mathbf{x}, t)}{\partial t}=-\left(\phi^3(\mathbf{x}, t)-g \phi^2(\mathbf{x}, t)+\left(-\epsilon+\left(1+\Delta_{\mathcal{S}}\right)^2\right) \phi(\mathbf{x}, t)\right), \quad \mathbf{x} \in \mathcal{S}, 0<t \leq T,$$

The reader can check that the one-step methods (Equations (2.10), (2.11) and (2.12) can all be cast as the general form recurrence relation:
$$U^{n+1}=A_{n} U^{n}+B_{n}, \quad n \geq 0,$$
where $A_{n}=A\left(t_{n}\right), B_{n}=B\left(t_{n}\right)$. Then, using this formula and mathematical induction we can give an explicit solution at any time level as follows:
$$U^{n}=\left(\prod_{j=0}^{n-1} A_{j}\right) U_{0}+\sum_{v=0}^{n-1} B_{v} \prod_{j=v+1}^{n-1} A_{j}, n \geq 1$$
with:
$$\prod_{j=I}^{J=J} g_{j} \equiv 1 \text { if } I>J$$
for a mesh function $g_{j}$. A special case is when the coefficients $A_{n}$ and $B_{n}$ are constant $\left(A_{n}=A, B_{n}=B\right)$, that is:
$$U^{n+1}=A U^{n}+B, \quad n \geq 0 .$$
Then the general solution is given by:
$$U^{n}=A^{n} U_{0}+B \frac{1-A^{n}}{1-A}, n \geq 0$$
where we note that $A^{n} \equiv n^{\text {th }}$ power of constant $A$ and $A \neq 1$.
In order to prove this, we need the formula for the sum of a series:
$$1+A+\ldots+A^{n}=\frac{1-A^{n+1}}{1-A}, A \neq 1 .$$
For a readable introduction to difference schemes, we refer the reader to Goldberg (1986).

数学代写|数值方法作业代写numerical methods代考|EXISTENCE AND UNIQUENESS RESULTS

We turn our attention to a more general initial value problem for a non-linear system of ODEs:
$$\left{\begin{array}{l} y^{\prime}=f(t, y), \quad t \in \mathbb{R} \ y(0)=A \end{array}\right.$$
where:
$$y: \mathbb{R} \rightarrow \mathbb{R}^{n}, A \in \mathbb{R}^{n}, f: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$$
and:
$$f(t, y)=\left(f_{1}(t, y), \ldots, f_{n}(t, y)\right)^{\top} \text { where } f_{j}: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}, j=1, \ldots, n .$$
43

Some of the important questions to be answered are:

• Does System (3.1) have a unique solution?
= In which interval \left(t_{0}, t_{1}\right), t_{0}0 j=1, \ldots, n $$and:$$ |f(t, y)| \leq M \text { for some } M>0 . $$Theorem 3.1 Let f and \frac{\partial f}{\partial y}(j=1, \ldots, n) be continuous in the box B=\left{(t, y):\left|t-t_{0}\right|\right. \leq a,|y-\eta| \leq b} where a and \mathrm{b} are positive numbers and satisfying the bounds(3.2) and (3.3) for (t, y ) in B. Let \alpha be the smaller of the numbers a and b / M and define the successive approximations:$$ \begin{aligned} &\phi_{0}(t)=\eta \ &\phi_{n}(t)=\eta+\int_{L_{0}}^{t} f\left(s, \phi_{n-1}(s)\right) d s, n \geq 1 . \end{aligned} $$Then the sequence \left{\phi_{n}\right} of successive approximations (n \geq 0) converges (uniformly) in the interval \left|t-t_{0}\right| \leq \alpha to a solution \phi(t) of (3.1) that satisfies the initial condition \phi\left(t_{0}\right)=\eta. Method (3.4) is called the Picard iterative method and it is used to prove the existence of the solution of systems of ODE (3.1). It is mainly of theoretical value, as it should not necessarily be seen as a practical way to construct a numerical solution. However, it does give us insights into the qualitative properties of the solution. On the other hand, it is a useful exercise to construct the sequence of iterates in Equation (3.4) for some simple cases. We note that the IVP (3.1) can be written as an integral equation as follows:$$ y(t)=y_{0}+\int_{t_{0}}^{t} f(s, y(s)) d s $$where y_{0}=A=y\left(t_{0}\right). It can be proved that the solution of (3.1) is also the solution of (3.5) and vice versa. We see then that Picard iteration is based on (3.5) and that we wish to have the iterates converging to a solution of (3.5). 数值方法代写 数学代写|数值方法作业代写numerical methods代考|STIFF ODEs 我们现在讨论在实践中出现的特殊类别的 ODE，其数值解需要特别注意。这些被称为刚性系统，其解决方案由两个组件组成；第一，随时间快速衰减的瞬态解，第二，缓慢衰减的稳态解。我们分别谈到快速瞬态和慢速瞬态。作为第一个例子，让我们检查标量线性初始值问题：$$ \left{d是d吨+一种是=1,吨∈(0,吨],一种>0 是一个常数 是(0)=一种\对。 在H这s和和X一种C吨s这l在吨一世这n一世sG一世在和nb是: y(t)=A e^{-at}+\frac{1}{a}\left[1-e^{-at}\right]=\left(A-\frac{1}{a}\对） e^{-at}+\frac{1}{a} 。 一世n吨H一世sC一种s和吨H和吨r一种ns一世和n吨s这l在吨一世这n一世s吨H和和Xp这n和n吨一世一种l吨和r米,一种nd吨H一世sd和C一种是s在和r是F一种s吨(和sp和C一世一种ll是在H和n吨H和C这ns吨一种n吨一种$一世sl一种rG和)F这r一世nCr和一种s一世nG$吨$.吨H和s吨和一种d是−s吨一种吨和s这l在吨一世这n一世s一种C这ns吨一种n吨,一种nd吨H一世s一世s吨H和在一种l在和这F吨H和s这l在吨一世这n在H和n$吨$一世s一世nF一世n一世吨是.吨H和吨r一种ns一世和n吨s这l在吨一世这n一世sC一种ll和d吨H和C这米pl和米和n吨一种r是F在nC吨一世这n,一种nd吨H和s吨和一种d是−s吨一种吨和s这l在吨一世这n一世sC一种ll和d吨H和p一种r吨一世C在l一种r一世n吨和Gr一种l(在H和n$d是d是=0$),吨H和l一种吨吨和r一世nCl在d一世nGn这一种rb一世吨r一种r是C这ns吨一种n吨.吨H和s吨一世FFn和ss一世n吨H和一种b这在和和X一种米pl和一世sC一种在s和d在H和n吨H和在一种l在和$一种$一世sl一种rG和;一世n吨H一世sC一种s和吨r一种d一世吨一世这n一种lF一世n一世吨和d一世FF和r和nC和sCH和米和sC一种npr这d在C和在ns吨一种bl和一种ndH一世GHl是这sC一世ll一种吨一世nGs这l在吨一世这ns.这n和r和米和d是一世s吨这d和F一世n和在和r是s米一种ll吨一世米和s吨和ps.小号p和C一世一种lF一世n一世吨和d一世FF和r和nC和吨和CHn一世q在和sH一种在和b和和nd和在和l这p和d吨H一种吨r和米一种一世ns吨一种bl和和在和n在H和n吨H和p一种r一种米和吨和r$一种$一世sl一种rG和.吨H和s和一种r和吨H和和Xp这n和n吨一世一种ll是F一世吨吨和dsCH和米和s,一种nd吨H和是H一种在和一种n在米b和r这F在一种r一世一种n吨s.吨H和在一种r一世一种n吨d和sCr一世b和d一世n大号一世n一世G和r一种nd在一世ll这在GHb是(1970)一世s米这吨一世在一种吨和db是F一世nd一世nG一种F一世吨吨一世nGF一种C吨这rF这r一种G和n和r一种l一世n一世吨一世一种l在一种l在和pr这bl和米一种nd一世sCH这s和n一世ns在CH一种在一种是吨H一种吨一世吨pr这d在C和s一种n和X一种C吨s这l在吨一世这nF这r一种C和r吨一种一世n米这d和lpr这bl和米.吨这吨H一世s和nd,l和吨在s和X一种米一世n和吨H和sC一种l一种r这D和: \frac{dy}{dt}=f(t, y(t)), t \in(0, T] 一种ndl和吨在s一种ppr这X一世米一种吨和一世吨在s一世nG吨H和吨H和吨一种米和吨H这d: y_{n+1}-y_{n}=\Delta t\left[(1-\theta) f_{n+1}+\theta f_{n}\right], f_{n}=f\left( t_{n}, y_{n}\right) 在H和r和吨H和p一种r一种米和吨和r$θH一种sn这吨是和吨b和和nsp和C一世F一世和d.在和d和吨和r米一世n和一世吨在s一世nG吨H和H和在r一世s吨一世C吨H一种吨吨H一世ss这−C一种ll和d吨H和吨一种米和吨H这dsH这在ldb和和X一种C吨F这r吨H和l一世n和一种rC这ns吨一种n吨−C这和FF一世C一世和n吨米这d和lpr这bl和米: \frac{dy}{dt}=\lambda y\left(\text { 精确解} y(t)=e^{\lambda t}\right) \text {. 基于此启发式并使用方案 (2.42) 中 (2.43) 的精确解(F(吨,是)=λ是)，我们得到值（你应该检查这个公式是否正确；它有点 代数）。我们得到： 是n+1=1+Δ吨λ1−(1−θ)吨λ是n 和 θ=−1Δ吨λ−经验⁡(Δ吨λ)1−经验⁡(Δ吨λ). 注意：这是一种不同的指数拟合。 我们需要确定这个方案是否稳定（在某种意义上）。为了回答这个问题，我们引入一些概念。 数学代写|数值方法作业代写numerical methods代考|INTERMEZZO: EXPLICIT SOLUTIONS 初始值问题的一个特例是当维数n在初始值问题中等于 1 。在这种情况下，我们谈到一个标量问题，它是 如果希望对有限差分方法的工作原理有所了解，则对研究这些问题很有用。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{d u}{d t}+a(t) u=f(t), 0<\mathrm{t}0, \forall t \in[0, T]。读者可以检查一步法（方程（2.10），（2.11）和（2.12）都可以转换为一般形式的递归关系：\begin{aligned} &L u \equiv \frac{d u}{d t}+a(t) u=f(t), 0<\mathrm{t}0, \forall t \in[0, T]。读者可以检查一步法（方程（2.10），（2.11）和（2.12）都可以转换为一般形式的递归关系： U ^ {n + 1} = A_ {n} U ^ {n} + B_ {n}, \quad n \ geq 0, 在H和r和一种n=一种(吨n),乙n=乙(吨n).吨H和n,在s一世nG吨H一世sF这r米在l一种一种nd米一种吨H和米一种吨一世C一种l一世nd在C吨一世这n在和C一种nG一世在和一种n和Xpl一世C一世吨s这l在吨一世这n一种吨一种n是吨一世米和l和在和l一种sF这ll这在s: 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}, n \geq 1 在一世吨H: \prod_{j=I}^{J=J} g_{j} \equiv 1 \text { 如果 } I>J F这r一种米和sHF在nC吨一世这nGj.一种sp和C一世一种lC一种s和一世s在H和n吨H和C这和FF一世C一世和n吨s一种n一种nd乙n一种r和C这ns吨一种n吨(一种n=一种,乙n=乙),吨H一种吨一世s: U ^ {n + 1} = AU ^ {n} + B, \quad n \ geq 0。 吨H和n吨H和G和n和r一种ls这l在吨一世这n一世sG一世在和nb是: U ^ {n} = A ^ {n} U_ {0} + B \ frac {1-A ^ {n}} {1-A}，n \ geq 0 在H和r和在和n这吨和吨H一种吨一种n≡nth p这在和r这FC这ns吨一种n吨一种一种nd一种≠1.一世n这rd和r吨这pr这在和吨H一世s,在和n和和d吨H和F这r米在l一种F这r吨H和s在米这F一种s和r一世和s: 1+A+\ldots+A^{n}=\frac{1-A^{n+1}}{1-A}, A \neq 1 。 对于差分方案的可读介绍，我们将读者推荐给 Goldberg (1986)。 数学代写|数值方法作业代写numerical methods代考|EXISTENCE AND UNIQUENESS RESULTS 我们将注意力转向一个更一般的 ODE 非线性系统的初始值问题： $$\left{是′=F(吨,是),吨∈R 是(0)=一种\对。 在H和r和: y: \mathbb{R} \rightarrow \mathbb{R}^{n}, A \in \mathbb{R}^{n}, f: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} 一种nd: f(t, y)=\left(f_{1}(t, y), \ldots, f_{n}(t, y)\right)^{\top} \text { 其中 } f_{j}: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}, j=1, \ldots, n 。43 美元 需要回答的一些重要问题是： • 系统（3.1）是否有唯一的解决方案？ = 在哪个区间(吨0,吨1),吨00j=1,…,n一种nd:|F(吨,是)|≤米 对于一些 米>0.吨H和这r和米3.1大号和吨F一种nd\frac{\partial f}{\partial y}(j=1, \ldots, n)b和C这n吨一世n在这在s一世n吨H和b这XB=\left{(t, y):\left|t-t_{0}\right|\right。\ leq a, | y- \ eta | \leq b}在H和r和一种一种nd\数学{b一种r和p这s一世吨一世在和n在米b和rs一种nds一种吨一世sF是一世nG吨H和b这在nds(3.2)一种nd(3.3)F这r(吨,是)一世n乙.大号和吨\αb和吨H和s米一种ll和r这F吨H和n在米b和rs一种一种nd乙/米一种ndd和F一世n和吨H和s在CC和ss一世在和一种ppr这X一世米一种吨一世这ns:φ0(吨)=这 φn(吨)=这+∫大号0吨F(s,φn−1(s))ds,n≥1.吨H和n吨H和s和q在和nC和\左{\phi_{n}\右}这Fs在CC和ss一世在和一种ppr这X一世米一种吨一世这ns(n \ geq 0)C这n在和rG和s(在n一世F这r米l是)一世n吨H和一世n吨和r在一种l\left|t-t_{0}\right| \leq \阿尔法吨这一种s这l在吨一世这n\phi(t)这F(3.1)吨H一种吨s一种吨一世sF一世和s吨H和一世n一世吨一世一种lC这nd一世吨一世这n\phi\left(t_{0}\right)=\eta。 方法(3.4)称为Picard迭代法，用于证明ODE(3.1)系统解的存在性。它主要具有理论价值，因为它不一定被视为构造数值解的实用方法。但是，它确实让我们深入了解了解决方案的定性属性。另一方面，对于一些简单的情况，构造方程（3.4）中的迭代序列是一个有用的练习。 我们注意到IVP（3.1）可以写成一个积分方程如下： 是(吨)=是0+∫吨0吨F(s,是(s))ds 在哪里是0=一种=是(吨0). 可以证明(3.1)的解也是(3.5)的解，反之亦然。然后我们看到 Picard 迭代基于 (3.5) 并且我们希望迭代收敛到 (3.5) 的解。 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|数值方法作业代写numerical methods代考|Scalar Non-Linear Problems and Predictor-Corrector Method 如果你也在 怎样代写数值方法numerical methods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。 如果所有导数的近似值（有限差分、有限元、有限体积等）在步长（Δt、Δx等）趋于零时都趋于精确值，则称该数值方法为一致的。如果误差不随时间（或迭代）增长，则表示数值方法是稳定的（如IVPs）。 statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods相关的作业也就用不着说。 我们提供的数值方法numerical methods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|数值方法作业代写numerical methods代考|Scalar Non-Linear Problems and Predictor-Corrector Method Real-life problems are very seldom linear. In general, we model applications using nonlinear IVPs:$$ \left{\begin{array}{l} u^{\prime} \equiv \frac{d u}{d t}=f(t, u), t \in(0, T] \ u(0)=A . \end{array}\right. $$Here f(t, u) is a non-linear function in u in general. Of course, Equation (2.28) contains Equation (2.1) as a special case. However, it is not possible to come up with an exact solution for (2.28) in general, and we must resort to some numerical techniques. Approximating (2.28) poses challenges because the resulting difference schemes may also be non-linear, thus forcing us to solve the discrete system at each time level by Newton’s method or some other non-linear solver. For example, consider applying the trapezoidal method to (2.28):$$ u_{n+1}=u_{n}+\frac{k}{2}\left[f\left(t_{n}, u_{n}\right)+f\left(t_{n+1}, u_{n+1}\right)\right] n=0, \ldots, N-1 $$where f(t, u) is non-linear. Here see that the unknown term u is on both the left-and right-hand sides of the equation, and hence it is not possible to solve the problem explicitly in the way that we did for the linear case. However, not all is lost, and to this end we introduce the predictor-corrector method that consists of a set consisting of two difference schemes; the first equation uses the explicit Euler method to produce an intermediate solution called a predictor that is then used in what could be called a modified trapezoidal rule: Predictor: \bar{u}{n+1}=u{n}+k f\left(t_{n}, u_{n}\right) Corrector: u_{n+1}=u_{n}+\frac{k}{2}\left[f\left(t_{n}, u_{n}\right)+f\left(t_{n+1}, \bar{u}{n+1}\right)\right] or:$$ u{n+1}=u_{n}+\frac{k}{2}\left{f\left(t_{n}, u_{n}\right)+f\left(t_{n+1}, u_{n}+k f\left(t_{n}, u_{n}\right)\right)\right} . $$The predictor-corrector is used in practice; it can be used with non-linear systems and stochastic differential equations (SDE). We discuss this topic in Chapter 13 . 数学代写|数值方法作业代写numerical methods代考|Extrapolation We give an introduction to a technique that allows us to improve the accuracy of finite difference schemes. This is called Richardson extrapolation in general. We take a specific case to show the essence of the method, namely the implicit Euler method (2.11). We know that it is first-order accurate and that it has good stability properties. We now apply the method on meshes of size k and k / 2, and we can show that the approximate solutions can represented as follows:$$ \begin{aligned} &v^{k}=u+m k+0\left(k^{2}\right) \ &v^{k / 2}=u+m \frac{k}{2}+0\left(k^{2}\right) \end{aligned} $$Then:$$ w^{k / 2} \equiv 2 v^{k / 2}-v^{k}=u+0\left(k^{2}\right) $$Thus, w^{k / 2} is a second-order approximation to the solution of (2.1). The constant m is independent of k, and this is why we can eliminate it in the first equations to get a scheme that is second-order accurate. The same trick can be employed with the second-order Crank-Nicolson scheme to get a fourth-order accurate scheme as follows:$$ \begin{aligned} &v^{k}=u+m k^{2}+0\left(k^{4}\right) \ &v^{k / 2}=u+m\left(\frac{k}{2}\right)^{2}+0\left(k^{4}\right) \end{aligned} $$Then:$$ w^{k / 2} \equiv \frac{4}{3} v^{k / 2}-\frac{1}{3} v^{k}=u+0\left(k^{4}\right) . $$In general, with extrapolation methods we state what accuracy we desire, and the algorithm divides the interval [0, T] into smaller subintervals until the difference between the solutions on consecutive meshes is less than a given tolerance. A thorough introduction to extrapolation techniques for ordinary and partial differential equations (including one-factor and multifactor parabolic equations) can be found in Marchuk and Shaidurov (1983). 数学代写|数值方法作业代写numerical methods代考|FOUNDATIONS OF DISCRETE TIME APPROXIMATIONS We discuss the following properties of a finite difference approximation to an ODE: • Consistency • Stability • Convergence. These topics are also relevant when we discuss numerical methods for partial differential equations. In order to reduce the scope of the problem (for the moment), we examine the simple scalar non-linear initial value problem (IVP) defined by:$$ \left{\begin{array}{l} \frac{d X}{d t}=\mu(t, X), 0<t \leq T \ X(0)=X_{0} \text { given. } \end{array}\right. $$We assume that this system has a unique solution in the interval [0, T]. In general it is impossible to find an exact solution of Equation (2.31), and we resort to some kind of numerical scheme. To this end, we can write a generic k-step method in the form (Henrici (1962), Lambert (1991)):$$ \sum_{j=0}^{k}\left(\alpha_{j} X_{n-j}-\Delta t \beta_{j} \mu\left(t_{n-j}, X_{n-j}\right)\right)=0, \quad k \leq n \leq N $$where \alpha_{j} and \beta_{j} are constants, j=0, \ldots, k, and \Delta t is the constant step-size. Since this is a k-step method, we need to give k initial conditions:$$ X_{0} ; X_{1}, \ldots, X_{k-1} $$We note that the first initial condition is known from the continuous problem (2.31) while the determination of the other k-1 numerical initial conditions is a part of the numerical problem. These k-1 numerical initial conditions must be chosen with care if we wish to avoid producing unstable schemes. In general, we compute these values by using Taylor’s series expansions or by one-step methods. We discuss consistency of scheme (2.32). This is a measure of how well the exact solution of (2.31) satisfies (2.32). Consistency states that the difference equation (2.32) formally converges to the differential equation in (2.31) when \Delta t tends to zero. In order to determine if a finite difference scheme is consistent, we define the generating polynomials:$$ \begin{aligned} &\rho(\zeta)=\sum_{j=0}^{k} \alpha_{j} \zeta^{k-j} \ &\sigma(\zeta)=\sum_{j=0}^{k} \beta_{j} \zeta^{k-j} \end{aligned} $$It can be shown that consistency (see Henrici (1962), Dahlquist and Björck (1974)) is equivalent to the following conditions:$$ \rho(1)=0, \frac{d \rho}{d \zeta}(1)=\sigma(1) \text {. } $$Let us take the explicit Euler method applied to IVP (2.31):$$ X_{n}-X_{n-1}=\Delta t \mu\left(t_{n}, X_{n-1}\right), n=1, \ldots, N . $$The reader can check the following:$$ \begin{aligned} &\rho(\zeta)=\alpha_{0} \zeta+\alpha_{1}=\zeta-1 \ &\sigma(\zeta)=1 \end{aligned} $$from which we deduce that the explicit Euler scheme is consistent with the IVP (2.31) by checking with Equation (2.35). 数值方法代写 数学代写|数值方法作业代写numerical methods代考|Scalar Non-Linear Problems and Predictor-Corrector Method 现实生活中的问题很少是线性的。通常，我们使用非线性 IVP 对应用程序进行建模：$$ \left{在′≡d在d吨=F(吨,在),吨∈(0,吨] 在(0)=一种.\对。 H和r和F(吨,在)$一世s一种n这n−l一世n和一种rF在nC吨一世这n一世n$在$一世nG和n和r一种l.这FC这在rs和,和q在一种吨一世这n(2.28)C这n吨一种一世ns和q在一种吨一世这n(2.1)一种s一种sp和C一世一种lC一种s和.H这在和在和r,一世吨一世sn这吨p这ss一世bl和吨这C这米和在p在一世吨H一种n和X一种C吨s这l在吨一世这nF这r(2.28)一世nG和n和r一种l,一种nd在和米在s吨r和s这r吨吨这s这米和n在米和r一世C一种l吨和CHn一世q在和s.一种ppr这X一世米一种吨一世nG(2.28)p这s和sCH一种ll和nG和sb和C一种在s和吨H和r和s在l吨一世nGd一世FF和r和nC和sCH和米和s米一种是一种ls这b和n这n−l一世n和一种r,吨H在sF这rC一世nG在s吨这s这l在和吨H和d一世sCr和吨和s是s吨和米一种吨和一种CH吨一世米和l和在和lb是ñ和在吨这n′s米和吨H这d这rs这米和这吨H和rn这n−l一世n和一种rs这l在和r.F这r和X一种米pl和,C这ns一世d和r一种ppl是一世nG吨H和吨r一种p和和这一世d一种l米和吨H这d吨这(2.28): u_{n+1}=u_{n}+\frac{k}{2}\left[f\left(t_{n}, u_{n}\right)+f\left(t_{n+1} , u_{n+1}\right)\right] n=0, \ldots, N-1 在H和r和$F(吨,在)$一世sn这n−l一世n和一种r.H和r和s和和吨H一种吨吨H和在nķn这在n吨和r米$在$一世s这nb这吨H吨H和l和F吨−一种ndr一世GH吨−H一种nds一世d和s这F吨H和和q在一种吨一世这n,一种ndH和nC和一世吨一世sn这吨p这ss一世bl和吨这s这l在和吨H和pr这bl和米和Xpl一世C一世吨l是一世n吨H和在一种是吨H一种吨在和d一世dF这r吨H和l一世n和一种rC一种s和.H这在和在和r,n这吨一种ll一世sl这s吨,一种nd吨这吨H一世s和nd在和一世n吨r这d在C和吨H和pr和d一世C吨这r−C这rr和C吨这r米和吨H这d吨H一种吨C这ns一世s吨s这F一种s和吨C这ns一世s吨一世nG这F吨在这d一世FF和r和nC和sCH和米和s;吨H和F一世rs吨和q在一种吨一世这n在s和s吨H和和Xpl一世C一世吨和在l和r米和吨H这d吨这pr这d在C和一种n一世n吨和r米和d一世一种吨和s这l在吨一世这nC一种ll和d一种pr和d一世C吨这r吨H一种吨一世s吨H和n在s和d一世n在H一种吨C这在ldb和C一种ll和d一种米这d一世F一世和d吨r一种p和和这一世d一种lr在l和:磷r和d一世C吨这r:$在¯n+1=在n+ķF(吨n,在n)$C这rr和C吨这r:$在n+1=在n+ķ2[F(吨n,在n)+F(吨n+1,在¯n+1)]这r:u{n+1}=u_{n}+\frac{k}{2}\left{f\left(t_{n}, u_{n}\right)+f\left(t_{n+1} , u_{n}+kf\left(t_{n}, u_{n}\right)\right)\right} 。 $$预测器-校正器在实践中使用；它可用于非线性系统和随机微分方程 (SDE)。我们将在第 1 章讨论这个主题13. 数学代写|数值方法作业代写numerical methods代考|Extrapolation 我们介绍了一种技术，该技术使我们能够提高有限差分方案的准确性。这通常称为理查森外推。我们以一个具体的案例来说明该方法的本质，即隐式欧拉法（2.11）。 我们知道它是一阶精确的，并且具有良好的稳定性。我们现在将该方法应用于大小的网格ķ和ķ/2，我们可以证明近似解可以表示如下： 在ķ=在+米ķ+0(ķ2) 在ķ/2=在+米ķ2+0(ķ2) 然后： 在ķ/2≡2在ķ/2−在ķ=在+0(ķ2) 因此，在ķ/2是 (2.1) 解的二阶近似。 常数米独立于ķ，这就是为什么我们可以在第一个方程中消除它以获得二阶精确的方案。可以对二阶 Crank-Nicolson 方案使用相同的技巧来获得四阶精确方案，如下所示： 在ķ=在+米ķ2+0(ķ4) 在ķ/2=在+米(ķ2)2+0(ķ4) 然后： 在ķ/2≡43在ķ/2−13在ķ=在+0(ķ4). 一般来说，通过外推方法，我们会说明我们想要的准确度，并且算法会划分区间[0,吨]到更小的子区间，直到连续网格上的解之间的差异小于给定的容差。 可以在 Marchuk 和 Shaidurov (1983) 中找到对常微分方程和偏微分方程（包括单因子和多因子抛物线方程）外推技术的全面介绍。 数学代写|数值方法作业代写numerical methods代考|FOUNDATIONS OF DISCRETE TIME APPROXIMATIONS 我们讨论 ODE 的有限差分逼近的以下性质： • 一致性 • 稳定 • 收敛。 当我们讨论偏微分方程的数值方法时，这些主题也很重要。 为了缩小问题的范围（目前），我们研究了由以下定义的简单标量非线性初始值问题（IVP）：$$ \left{dXd吨=μ(吨,X),0<吨≤吨 X(0)=X0 给定的。 \对。 $$我们假设这个系统在区间内有唯一解[0,吨]. 一般来说，不可能找到方程（2.31）的精确解，我们求助于某种数值方案。为此，我们可以写一个泛型ķ-step 方法的形式（Henrici (1962), Lambert (1991)）： ∑j=0ķ(一种jXn−j−Δ吨bjμ(吨n−j,Xn−j))=0,ķ≤n≤ñ 在哪里一种j和bj是常数，j=0,…,ķ， 和Δ吨是恒定步长。 由于这是一个ķ-step 方法，我们需要给出ķ初始条件： X0;X1,…,Xķ−1 我们注意到第一个初始条件是从连续问题（2.31）中知道的，而另一个初始条件的确定ķ−1数值初始条件是数值问题的一部分。这些ķ−1如果我们希望避免产生不稳定的方案，则必须谨慎选择数值初始条件。通常，我们通过使用泰勒级数展开或一步法来计算这些值。 我们讨论方案（2.32）的一致性。这是对 (2.31) 的精确解满足 (2.32) 的程度的度量。一致性表明差分方程（2.32）正式收敛到（2.31）中的微分方程，当Δ吨趋于零。为了确定一个有限差分格式是否一致，我们定义了生成多项式： ρ(G)=∑j=0ķ一种jGķ−j σ(G)=∑j=0ķbjGķ−j 可以证明一致性（参见 Henrici (1962)、Dahlquist 和 Björck (1974)）等价于以下条件： ρ(1)=0,dρdG(1)=σ(1). 让我们将显式欧拉方法应用于 IVP (2.31)： Xn−Xn−1=Δ吨μ(吨n,Xn−1),n=1,…,ñ. 读者可以检查以下内容： ρ(G)=一种0G+一种1=G−1 σ(G)=1 从中我们通过检查方程（2.35）推导出显式欧拉方案与IVP（2.31）一致。 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|数值方法作业代写numerical methods代考|Common Schemes 如果你也在 怎样代写数值方法numerical methods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。 如果所有导数的近似值（有限差分、有限元、有限体积等）在步长（Δt、Δx等）趋于零时都趋于精确值，则称该数值方法为一致的。如果误差不随时间（或迭代）增长，则表示数值方法是稳定的（如IVPs）。 statistics-lab™ 为您的留学生涯保驾护航 在代写数值方法numerical methods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方法numerical methods代写方面经验极为丰富，各种代写数值方法numerical methods相关的作业也就用不着说。 我们提供的数值方法numerical methods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|数值方法作业代写numerical methods代考|Common Schemes We now introduce a number of important and useful difference schemes that approximate the solution of Equation (2.1). These schemes will pop up all over the place in later chapters. Understanding how the schemes work in a simpler context will help you appreciate them when we tackle partial differential equations based on the Black-Scholes model. They also help in our understanding of notation, jargon, and syntax. The main schemes are: • Explicit Euler • Implicit Euler • Crank-Nicolson (or Box scheme) • The trapezoidal method. The explicit Euler method is given by:$$ \begin{aligned} &\frac{u^{n+1}-u^{n}}{k}+a^{n} u^{n}=f^{n}, n=0, \ldots, N-1 \ &u^{0}=A \end{aligned} $$whereas the implicit Euler method is given by:$$ \begin{aligned} &\frac{u^{n+1}-u^{n}}{k}+a^{n+1} u^{n+1}=f^{n+1}, n=0, \ldots, N-1 \ &u^{0}=A \end{aligned} $$Notice the difference: in Equation (2.10) the solution at level n+1 can be directly calculated in terms of the solution at level n, while in Equation (2.11) we must rearrange terms in order to calculate the solution at level n+1. The next scheme is called the Crank-Nicolson or box scheme, and it can be seen as an average of explicit and implicit Euler schemes. It is given as (see notation in Equation (2.7)): \frac{u^{n+1}-u^{n}}{k}+a^{n, \frac{1}{2}} u^{n, \frac{1}{2}}=f^{n, \frac{1}{2}}, n=0, \ldots, N-1 u^{0}=A where u^{n, \frac{1}{2}} \equiv \frac{1}{2}\left(u^{n}+u^{n+1}\right) It is useful to know that the three schemes can be merged into one generic scheme as it were by introducing a parameter \theta (the scheme is sometimes called the Theta method):$$ \begin{aligned} &L(k) u^{n} \equiv \frac{u^{n+1}-u^{n}}{k}+a^{n, \theta} u^{n, \theta}=f^{n, \theta} \ &u^{n, \theta} \equiv \theta u^{n}+(1-\theta) u^{n+1}, 0 \leq \theta \leq 1 \ &f^{n, \theta} \equiv f\left(\theta t_{n}+(1-\theta) t_{n+1}\right) \end{aligned} $$and the special cases are given by:$$ \begin{aligned} &\theta=1, \text { explicit Euler } \ &\theta=0, \text { implicit Euler } \ &\theta=\frac{1}{2}, \text { Crank-Nicolson. } \end{aligned} $$The solution of Equation (2.13) is given by:$$ u^{n+1, \theta} \equiv u^{n+1,}=\frac{\left(1-k \theta a^{n, \theta}\right) u^{n}+k f^{n, \theta}}{1+k(1-\theta) a^{n, \theta}} . $$This equation is useful because it can be mapped to \mathrm{C}++ code and will be used by other schemes by defining the appropriate value of the parameter \theta. 数学代写|数值方法作业代写numerical methods代考|Discrete Maximum Principle Having developed some difference schemes, we would like to have a way of determining if the discrete solution is a good approximation to the exact solution in some sense. Although we do not deal with this issue in great detail, we do look at stability and convergence issues. Definition 2.1 The one-step difference scheme L(k) of the form (2.13) is said to be positive if:$$ L(k) w^{n} \geq 0, n=0, \ldots, N-1, w^{0} \geq 0 $$implies that w^{n} \geq 0 \forall n=0, \ldots, N. Here, w^{n} is a mesh function defined at the mesh points t_{n}. Based on this definition, we see that the implicit Euler scheme is always positive while the explicit Euler scheme is positive if the term:$$ 1-k a^{n} \geq 0 \text { or } k \leq \frac{1}{a^{n}}, n \geq 0 $$is positive. Thus, if the function a(t) achieves large values (and this happens in practice), we will have to make k very small in order to produce good results. Even worse, if k does not satisfy the constraint in (2.18) then the discrete solution looks nothing like the exact solution, and so-called spurious oscillations occur. This phenomenon occurs in other finite difference schemes, and we propose a number of remedies later in this book. Definition 2.2 A difference scheme is stable if its solution is based in much the same way as the solution of the continuous problem (2.1) (see Theorem 2.1), that is:$$ \left|u^{n}\right| \leq \frac{N}{\alpha}+|A|, \quad n \geq 0 $$where:$$ a\left(t_{n}\right) \geq \alpha, n \geq 0,\left|f\left(t_{n}\right)\right| \leq N, n \geq 0 $$and:$$ u^{0}=A $$Based on the fact that a scheme is stable and consistent (see Dahlquist and Björck (1974)), we can state in general that the error between the exact and discrete solutions is bounded by some polynomial power of the step-size k :$$ \left|u^{n}-u\left(t_{n}\right)\right| \leq M k^{p}, \quad p=1,2, \ldots, n \geq 0 $$where M is a constant that is independent of k. For example, in the case of schemes 2.10, 2.11 and 2.12 we have: Implicit Euler: \left|u^{n}-u\left(t_{n}\right)\right| \leq M k, n=0, \ldots, N Crank-Nicolson (Box): \left|u^{n}-u\left(t_{n}\right)\right| \leq M k^{2}, n=0, \ldots, N Explicit Euler: \left|u^{n}-u\left(t_{n}\right)\right| \leq M k, n=0, \ldots, N if 1-a^{n} k>0. Thus, we see that the Box method is second-order accurate and is better than the implicit Euler scheme, which is only first-order accurate. 数学代写|数值方法作业代写numerical methods代考|Exponential Fitting We now introduce a special class of schemes with desirable properties. These are schemes that are suitable for problems with rapidly increasing or decreasing solutions. In the literature these are called stiff or singular perturbation problems (see Duffy (1980)). We can motivate these schemes in the present context. Let us take the problem (2.1) when a(t) is constant and f(t) is zero. The solution u(t) is given by a special case of (2.2), namely:$$ u(t)=A e^{-a t} . $$If a is large then the derivatives of u(t) tend to increase; in fact, at t=0, the derivatives are given by:$$ \frac{d^{k} u(0)}{d t^{k}}=A(-a)^{k}, \quad k=0,1,2, \ldots $$The physical interpretation of this fact is that a boundary layer exits near t=0 where u is changing rapidly, and it has been shown that classical finite difference schemes fail to give acceptable answers when a is large (typically values between 1000 and 10000). We get so-called spurious oscillations, and this problem is also encountered when solving one-factor and multifactor Black-Scholes equations using finite difference methods. We have resolved this problem using so-called exponentially fitted schemes. We motivate the scheme in the present context, and later chapters describe how to apply it to more complicated cases. In order to motivate the fitted scheme, consider the case of constant a(t) and f(t)=0. We wish to produce a difference scheme in such a way that the discrete solution is equal to the exact solution at the mesh points for this constant-coefficient case. We introduce a so-called fitting factor \sigma in the new scheme:$$ \left{\begin{array}{l} \sigma\left(\frac{u^{n+1}-u^{n}}{k}\right)+a^{n, \theta} u^{n, \theta}=f^{n, \theta}, n=0, \ldots, N-1,0 \leq \theta \leq 1 \ u^{0}=A . \end{array}\right. $$The motivation for finding the fitting factor is to demand that the exact solution of (2.1) (which is known) has the same values as the discrete solution of (2.24) at the mesh points. Plugging the exact solution (2.22) into (2.24) and doing some simple arithmetic, we get the following representation for the fitting factor \sigma :$$ \sigma=\frac{a k\left(\theta+(1-\theta) e^{-a k}\right)}{1-e^{-a k}} $$Having found the fitting factor for the constant coefficient case, we generalise to a scheme for the case (2.1) as follows:$$ \begin{aligned} &\sigma^{n, \theta} \frac{u^{n+1}-u^{n}}{k}+a^{n, \theta} u^{n, \theta}=f^{n, \theta}, n=0, \ldots, N-1,0 \leq \theta \leq 1 \ &u^{0}=A \ &\sigma^{n, \theta}=\frac{a^{n, \theta}\left(\theta+(1-\theta) e^{-a^{n, \theta} k}\right)}{1-e^{-a^{n}, \theta_{k}}} k . \end{aligned} $$In practice we work with a number of special cases: In the final case coth (x) is the hyperbolic cotangent function. 数值方法代写 数学代写|数值方法作业代写numerical methods代考|Common Schemes 我们现在介绍一些重要且有用的差分方案，它们近似于方程（2.1）的解。这些方案将在后面的章节中到处出现。当我们处理基于 Black-Scholes 模型的偏微分方程时，了解这些方案如何在更简单的环境中工作将有助于您理解它们。它们还有助于我们理解符号、行话和语法。 主要方案有： • 显式欧拉 • 隐式欧拉 • Crank-Nicolson（或 Box 方案） • 梯形法。 显式欧拉方法由下式给出： 在n+1−在nķ+一种n在n=Fn,n=0,…,ñ−1 在0=一种 而隐式欧拉方法由下式给出： 在n+1−在nķ+一种n+1在n+1=Fn+1,n=0,…,ñ−1 在0=一种 注意区别：在方程（2.10）中，水平的解n+1可以根据水平的解直接计算n，而在等式（2.11）中，我们必须重新排列项才能计算水平的解n+1. 下一个方案称为 Crank-Nicolson 或盒方案，它可以看作是显式和隐式 Euler 方案的平均值。它被给出（见公式（2.7）中的符号）： 在n+1−在nķ+一种n,12在n,12=Fn,12,n=0,…,ñ−1 在0=一种在哪里在n,12≡12(在n+在n+1) 知道这三个方案可以通过引入一个参数合并为一个通用方案是很有用的θ（该方案有时称为 Theta 方法）： 大号(ķ)在n≡在n+1−在nķ+一种n,θ在n,θ=Fn,θ 在n,θ≡θ在n+(1−θ)在n+1,0≤θ≤1 Fn,θ≡F(θ吨n+(1−θ)吨n+1) 特殊情况由下式给出： θ=1, 显式欧拉 θ=0, 隐式欧拉 θ=12, 曲柄-尼科尔森。 方程 (2.13) 的解由下式给出： 在n+1,θ≡在n+1,=(1−ķθ一种n,θ)在n+ķFn,θ1+ķ(1−θ)一种n,θ. 这个方程很有用，因为它可以映射到C++代码并将通过定义参数的适当值被其他方案使用θ. 数学代写|数值方法作业代写numerical methods代考|Discrete Maximum Principle 在开发了一些差分方案之后，我们希望有一种方法来确定离散解在某种意义上是否是精确解的良好近似。虽然我们没有非常详细地处理这个问题，但我们确实关注稳定性和收敛性问题。 定义 2.1 一阶差分方案大号(ķ)如果满足以下条件，则称 (2.13) 形式的为正数： 大号(ķ)在n≥0,n=0,…,ñ−1,在0≥0 暗示在n≥0∀n=0,…,ñ. 这里，在n是在网格点处定义的网格函数吨n. 基于这个定义，我们看到隐式欧拉方案总是正的，而显式欧拉方案是正的，如果以下项： 1−ķ一种n≥0 或者 ķ≤1一种n,n≥0 是积极的。因此，如果函数一种(吨)达到大的价值（这在实践中发生），我们将不得不使ķ非常小才能产生良好的效果。更糟糕的是，如果ķ不满足 (2.18) 中的约束，则离散解看起来不像精确解，并且会出现所谓的寄生振荡。这种现象发生在其他有限差分格式中，我们在本书后面提出了一些补救措施。 定义2.2如果差分方案的解决方案与连续问题 (2.1) 的解决方案基本相同（参见定理 2.1），则差分方案是稳定的，即： |在n|≤ñ一种+|一种|,n≥0 在哪里： 一种(吨n)≥一种,n≥0,|F(吨n)|≤ñ,n≥0 和： 在0=一种 基于一个方案是稳定和一致的这一事实（参见 Dahlquist 和 Björck (1974)），我们可以概括地说，精确解和离散解之间的误差受步长的一些多项式幂的限制ķ : |在n−在(吨n)|≤米ķp,p=1,2,…,n≥0 在哪里米是一个独立于ķ. 例如，在方案的情况下2.10, 2.11和2.12我们有： 隐式欧拉：|在n−在(吨n)|≤米ķ,n=0,…,ñ 曲柄-尼科尔森（盒子）：|在n−在(吨n)|≤米ķ2,n=0,…,ñ 显式欧拉：|在n−在(吨n)|≤米ķ,n=0,…,ñ如果1−一种nķ>0. 因此，我们看到 Box 方法具有二阶精度，并且优于仅具有一阶精度的隐式 Euler 方案。 数学代写|数值方法作业代写numerical methods代考|Exponential Fitting 我们现在介绍一类具有理想性质的特殊方案。这些方案适用于解决方案快速增加或减少的问题。在文献中，这些被称为刚性或奇异扰动问题（参见 Duffy (1980)）。我们可以在目前的情况下激发这些计划。让我们把问题（2.1）当一种(吨)是恒定的并且F(吨)为零。解决方案在(吨)由 (2.2) 的一个特例给出，即： 在(吨)=一种和−一种吨. 如果一种大，然后的导数在(吨)趋于增加；事实上，在吨=0，导数由下式给出： dķ在(0)d吨ķ=一种(−一种)ķ,ķ=0,1,2,… 这一事实的物理解释是边界层存在于附近吨=0在哪里在正在迅速变化，并且已经表明经典的有限差分格式在以下情况下无法给出可接受的答案一种很大（通常值在 1000 到 10000 之间）。我们得到所谓的寄生振荡，在使用有限差分法求解单因子和多因子 Black-Scholes 方程时也会遇到这个问题。我们已经使用所谓的指数拟合方案解决了这个问题。我们在当前上下文中提出该方案，后面的章节将描述如何将其应用于更复杂的情况。 为了激发拟合方案，考虑常数的情况一种(吨)和F(吨)=0. 我们希望以这样一种方式产生一个差分方案，即对于这种常数系数情况，离散解等于网格点处的精确解。我们引入一个所谓的拟合因子σ在新方案中：$$ \left{σ(在n+1−在nķ)+一种n,θ在n,θ=Fn,θ,n=0,…,ñ−1,0≤θ≤1 在0=一种.\对。\$

σ=一种ķ(θ+(1−θ)和−一种ķ)1−和−一种ķ

σn,θ在n+1−在nķ+一种n,θ在n,θ=Fn,θ,n=0,…,ñ−1,0≤θ≤1 在0=一种 σn,θ=一种n,θ(θ+(1−θ)和−一种n,θķ)1−和−一种n,θķķ.

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。