### 数学代写|有限元方法代写Finite Element Method代考|Approximation functions

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method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。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Approximation functions

In this section we discuss the properties of the set of approximation functions $\left{\phi_i\right}$ and $\phi_0$ used in the n-parameter Ritz solution in Eq. (2.5.4). First, we note that $u_n$ must satisfy only the specified essential boundary conditions of the problem, since the specified natural boundary conditions are included in the variational problem in Eq. (2.5.1). The particular form of $u_n$ in Eq. (2.5.4) facilitates satisfaction of specified boundary conditions. To see this, suppose that the approximate solution is sought in the form
$$u_n=\sum_{j=1}^n c_j \phi_j(x)$$
and suppose that the specified essential boundary condition is $u\left(x_0\right)=u_0$. Then $u_n$ must also satisfy the condition $u_n\left(x_0\right)=u_0$ at a boundary point $x=x_0$
$$\sum_{j=1}^n c_j \phi_j\left(x_0\right)=u_0$$
Since $c_j$ are unknown parameters to be determined, it is not easy to choose $\phi_j$ (x) such that the above relation holds for all $c_j$. If $u_0=0$, then we can select all $\phi_j$ such that $\phi_j\left(x_0\right)=0$ and satisfy the condition $u_n\left(x_0\right)=0$. By writing the approximate solution $u_n$ in the form Eq. (2.5.4), a sum of a homogeneous part $\sum c_j \phi_j(x)$ and a nonhomogeneous part $\phi_0(x)$, we require $\phi_0(x)$ to satisfy the specified essential boundary conditions while the homogeneous part vanishes at the same boundary point where the essential boundary condition is specified. This follows from
$$\begin{gathered} u_n\left(x_0\right)=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+\phi_0\left(x_0\right) \ u_0=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+u_0 \rightarrow \sum_{j=1}^n c_j \phi_j\left(x_0\right)=0 \end{gathered}$$
which is satisfied, for arbitrary $c_j$, by choosing $\phi_j\left(x_0\right)=0$.

## 数学代写|有限元方法代写Finite Element Method代考|The Method of Weighted Residuals

As noted in Section 2.4.3, one can always write the weighted-integral form of a differential equation, whether the equation is linear or nonlinear (in the dependent variables). The weak form can be developed if the equations are second-order or higher, even if they are nonlinear.

The weighted-residual method is a generalization of the Galerkin method in that the weight functions can be chosen from an independent set of functions, and it requires only the weighted-integral form to determine the parameters. Since the latter form does not include any of the specified boundary conditions of the problem, the approximation functions must be selected such that the approximate solution satisfies all of the specified boundary conditions. In addition, the weight functions can be selected independently of the approximation functions, but are required to be linearly independent so that the resulting algebraic equations are linearly independent.
We discuss the general method of weighted residuals first, and then consider certain special cases that are known by specific names (e.g., the Galerkin method, the collocation method, the least-squares method and so on). Although a limited use of the weighted-residual method is made in this book, it is informative to have a knowledge of this class of methods for use in the formulation of certain nonlinear problems and non-self-adjoint problems.

The method of weighted residuals can be described in its generality by considering the operator equation
$$A(u)=f \text { in } \Omega$$
where $A$ is an operator (linear or nonlinear), often a differential operator, acting on the dependent variable $u$, and $f$ is a known function of the independent variables. Some examples of such operators are given below.
$$A(u)=-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u$$
$$A(u)=\frac{d^2}{d x^2}\left(b \frac{d^2 u}{d x^2}\right)$$
$$A(u)=-\left[\frac{\partial}{\partial x}\left(k_x \frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\left(k_y \frac{\partial u}{\partial y}\right)\right]$$
$$A(u)=-\frac{d}{d x}\left(u \frac{d u}{d x}\right)$$
$$A(u, v)=u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+\frac{\partial^2 u}{\partial x^2}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)$$
For an operator $A$ to be linear in its arguments, it must satisfy the relation
$$A(\alpha u+\beta v)=\alpha A(u)+\beta A(v)$$
for any scalars $\alpha$ and $\beta$ and dependent variables $u$ and $v$. It can be easily verified that all operators in Eq. (2.5.52), except for those in (4) and (5), are linear. When an operator does not satisfy the condition in Eq. (2.5.53), it is said to be nonlinear.

## 数学代写|有限元方法代写Finite Element Method代考|Approximation functions

$$u_n=\sum_{j=1}^n c_j \phi_j(x)$$

$$\sum_{j=1}^n c_j \phi_j\left(x_0\right)=u_0$$

$$\begin{gathered} u_n\left(x_0\right)=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+\phi_0\left(x_0\right) \ u_0=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+u_0 \rightarrow \sum_{j=1}^n c_j \phi_j\left(x_0\right)=0 \end{gathered}$$

## 数学代写|有限元方法代写Finite Element Method代考|The Method of Weighted Residuals

$$A(u)=f \text { in } \Omega$$

$$A(u)=-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u$$
$$A(u)=\frac{d^2}{d x^2}\left(b \frac{d^2 u}{d x^2}\right)$$
$$A(u)=-\left[\frac{\partial}{\partial x}\left(k_x \frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\left(k_y \frac{\partial u}{\partial y}\right)\right]$$
$$A(u)=-\frac{d}{d x}\left(u \frac{d u}{d x}\right)$$
$$A(u, v)=u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+\frac{\partial^2 u}{\partial x^2}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)$$

$$A(\alpha u+\beta v)=\alpha A(u)+\beta A(v)$$

## 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。