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数学代写|偏微分方程代考Partial Differential Equations代写|Alternating Direction Implicit Methods

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数学代写|偏微分方程代考Partial Differential Equations代写|Alternating Direction Implicit Methods

The strategy is to advance the time step first in the $\xi_1$-direction while holding the grid about the $\xi_2$ variable fixed. Then switch or alternate directions by advancing the time step in $\xi_2$ while holding $\xi_1$ fixed. This is the long-established alternating direction method. By making the linear portion of the partial differential equation (that is the Laplacian) implicit in time and the nonlinear portion explicit in time, the alternating direction quasi-implicit method (ADQI) can be described by (7.5.2)
$$\left.\begin{array}{rl} \delta_\tau^{+}\left(\boldsymbol{w}{j, k}^n\right) & =C \delta_1^2\left(\boldsymbol{w}{j, k}^{n+1}\right)+N\left(\boldsymbol{w}{j, k}^n\right) \boldsymbol{w}{j, k}^n \ \delta_\tau^{+}\left(\boldsymbol{w}{j, k}^{n+1}\right) & =C \delta_2^2\left(\boldsymbol{w}{j, k}^{n+2}\right)+N\left(\boldsymbol{w}{j, k}^{n+1}\right) \boldsymbol{w}{j, k}^{n+1} \end{array}\right} .$$
It is left to the (brave-hearted) reader to extend the proofs of Theorems 7.5 (Stability) and 7.6 (Consistency) to the two-dimensional case. These results along with the Lax Equivalence Theorem, will lead to the bound from Theorem 7.6 on the temporal space $\Delta \tau$ which ensures that the $A D Q I$ method (7.5.2) is convergent.

数学代写|偏微分方程代考Partial Differential Equations代写|Definitions, Notation, and Key Conditions

The following symbols and notation will be used throughout this chapter. The symbols $X, Y$, and $Z$ represent separable Hilbert spaces with $X$ continuously embedded in $Z$. The symbol $\sigma(A)$ is the set of all eigenvalues of the linear operator $A$, also called the spectrum of $A$. The symbol $\sigma_n(\mathbb{R})$ is defined to be the set of all $n \mathrm{x} n$ matrices with real coefficients whose eigenvalues are not purely imaginary: $\sigma_n(\mathbb{R})=\left{A \in \operatorname{Mat}{n \times n}(\mathbb{R}): \lambda \in \sigma(A) \Rightarrow \operatorname{Re}(\lambda) \neq 0\right}$. Some of the other notations used in this section are as follows. $\Theta(X, Y)$ is the set of all continuous maps from $X$ into $Y$ $\partial^m(X, Y)$ is the set of all $m$-times continuously Fréchet-differentiable maps from $X$ into $Y$ $Q(X, Y)$ is the set of all linear maps from $X$ into $Y$ $\mathscr{ఠ}*^0(X, Y)={f \in \mathscr{C}(X, Y): f$ is uniformly bounded and uniformly continuous $}$
$f^{\prime}(u) v$ is the Fréchet derivative of the operator $f(u)$ evaluated at $v$
$\mathscr{S}(X, Y)=\left{f \in \mathscr{C}^1(X, Y): f\right.$ is bounded and $f(0)=0=f^{\prime}(0) v$ for all $\left.v \in \operatorname{Dom}\left[f^{\prime}(0)\right]\right}$
$\mathscr{B}_X(0, r)=\left{x \in X:|x|_X<r\right} \equiv$ the sphere of radius $r$ in $X$-space
$$H^1([0, T] ; Z)=\left{f: \int_0^T\left(|f(t)|_Z^2+\left|\frac{\partial f}{\partial t}(t)\right|_Z^2\right) d t<\infty\right}$$
Additional definitions of function spaces are found in the Introduction. The Hartman-Grobman Theorem for ordinary differential equations is now stated with respect to the Banach space $E$.

偏微分方程代写

数学代写|偏微分方程代考Partial Differential Equations代写|Alternating Direction Implicit Methods

$$\left.\begin{array}{rl} \delta_\tau^{+}\left(\boldsymbol{w}{j, k}^n\right) & =C \delta_1^2\left(\boldsymbol{w}{j, k}^{n+1}\right)+N\left(\boldsymbol{w}{j, k}^n\right) \boldsymbol{w}{j, k}^n \ \delta_\tau^{+}\left(\boldsymbol{w}{j, k}^{n+1}\right) & =C \delta_2^2\left(\boldsymbol{w}{j, k}^{n+2}\right)+N\left(\boldsymbol{w}{j, k}^{n+1}\right) \boldsymbol{w}{j, k}^{n+1} \end{array}\right} .$$

数学代写|偏微分方程代考Partial Differential Equations代写|Definitions, Notation, and Key Conditions

$f^{\prime}(u) v$ 是这个操作者的导数吗 $f(u)$ 评估于 $v$

$\mathscr{S}(X, Y)=\left{f \in \mathscr{C}^1(X, Y): f\right.$ 是有界的 $f(0)=0=f^{\prime}(0) v$ 对所有人 $\left.v \in \operatorname{Dom}\left[f^{\prime}(0)\right]\right}$

$\mathscr{B}_X(0, r)=\left{x \in X:|x|_X<r\right} \equiv$ 半径球 $r$ 在 $X$-space
$$H^1([0, T] ; Z)=\left{f: \int_0^T\left(|f(t)|_Z^2+\left|\frac{\partial f}{\partial t}(t)\right|_Z^2\right) d t<\infty\right}$$

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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。