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• Statistical Computing 统计计算
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## 数学代写|偏微分方程代写partial difference equations代考|VERIFICATION AND VALIDATION

As discussed earlier, in the vast majority of situations, PDEs represent mathematical models of certain physical phenomena. Therefore, their solutions are worthwhile and of practical value only when they resemble reality, i.e., the observed physical phenomena. In order to lend credibility to the solution of the mathematical model PDEs, in our particular case – it is important to identify the errors that may be incurred in various stages of the modeling/solution process, and quantify and analyze them in a systematic manner. Only then can one have a certain level of confidence in the predicted results in the event that the solution to the mathematical model does not exactly match measured data or observed behavior of the system being modeled. In order to identify and quantify the errors associated with various stages of the modeling process, we begin with an overview of these stages. A flowchart depicting the various stages of the modeling process is shown in Fig. 1.17.

The analysis of any problem begins with a physical description of the real-life problem. This description usually constitutes some text and, perhaps, a few accompanying illustrations that describes the problem. For example, a car manufacturer may be interested in exploring the difference between two spoiler designs. In this case, the physical description will constitute typical range of air flow speeds over the spoiler and other physical details, along with illustrations detailing the car body and spoiler geometries. In addition, the description will include what the goal of the analysis is and what is sought. Is it the drag? Is it the structural deformation? Is it the wind noise?

The second step in the analysis is to create a physical model from the available physical description. The creation of the physical model entails making important decisions on what physical phenomena need to be included. These decisions are driven not only by the goals of the study, but also the feasibility of including or excluding a certain physical phenomenon based on the resources and time available to complete the task. For the specific example at hand, important questions that need to be addressed may include (a) is the unsteadiness of the flow important, or is it sufficient to find steady-state solutions? Perhaps, for noise predictions, considering unsteadiness is important, while for drag predictions, it is not. (b) Is it sufficient to model the problem as isothermal? (c) Is it important to account for the unsteady nature of the turbulence? (d) For noise and drag predictions, it is adequate to treat the spoiler as a rigid body? It should be noted that the answers to some of these questions may not be known a priori, and one may have to explore the various possibilities. This is where a survey of past work (literature review), in addition to the experience of the analyst, can assist in narrowing down the possibilities. In most disciplines, prior knowledge and training in the field aids in answering many of these questions with a sufficient level of confidence. In emerging fields, this step may be cited as one of the most challenging steps in the overall analysis process.

## 数学代写|偏微分方程代写partial difference equations代考|The Finite Difference Method

Although it is difficult to credit a particular individual with the discovery of what is known today as the finite difference method (FDM), it is widely believed that the seminal work of Courant, Friedrichs, and Lewy [1] in 1928 is the first published example of using five-point difference approximations to derivatives for solving the elliptic Laplace equation. Using the same method, the same researchers also proceeded to solve the hyperbolic wave equation in the same paper [1] and established the so-called CFL criterion for the stability of hyperbolic partial differential equations (PDEs). For additional details on the history of the FDM and how it influenced the development of the finite element method, the reader is referred to the excellent article by Thomée [2]. Even though the FDM has had several descendants since 1928, most of whom have, arguably, outshined the FDM in many regards, the FDM remains the method of choice for solving PDEs in many application areas in science and engineering. Its success can be attributed to its simplicity as well as its adaptability to PDEs of any form. Disciplines where it still finds prolific usage are computational heat transfer and computational electromagnetics – the so-called finite difference time domain (FDTD) method.

As discussed in Section 1.3.1, one of the important features of the FDM, in contrast with other popular methods for solving PDEs, is that the PDE is solved in its original form, i.e., without modifying the governing equation into an alternative form. In other words, the original differential form of the governing equation is satisfied at various locations within the computational domain. The solution, hence obtained, is called the strong form solution. In contrast, other popular methods, such as the finite volume method and the finite element method, provide the so-called weak form solution, in which a modified form, usually an integral form of the governing equation, is satisfied within the computational domain. Which type of solution is desirable depends of the underlying physical problem from which the governing equation was conceived. It is worth clarifying here that the words “strong” and “weak” should not be interpreted by the reader as “good” and “bad,” respectively. Further discussion of the pros and cons of these two types of solution is postponed until Chapter 6.

This chapter commences with a discussion of the basic procedure for deriving difference approximations and applying such approximations to solving PDEs. The fundamental concepts developed here will also be used later in Chapters 6 and 7, when the finite volume method is presented. The current chapter begins with simple one-dimensional differential equations (i.e., ODEs) and eventually extends the concepts to multidimensional problems (i.e., PDEs), including irregular geometry.

# 偏微分方程代写

## 有限元方法代写

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## 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。