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# 数学代写|偏微分方程代考Partial Differential Equations代写|The Stimulated Raman Scattering Laser Model

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## 数学代写|偏微分方程代考Partial Differential Equations代写|The Stimulated Raman Scattering Laser Model

The interaction of two lasers within an electro-magnetic medium can be described by a system of partial differential equations. The method of spectroscopy referred to as stimulated Raman scattering can be roughly described as an extension of Maxwell’s equations and Figure 7.1. More specifically, let $M_1$ and $M_2$ be two mirrors which are coated in such a manner as to be transparent to an input or pump laser. The electric field for the pump laser will be denoted as $E_p$. Moreover, the coatings on the mirrors reflect the energy from a second input or Stokes laser within the aforementioned Raman gain medium. The electric field on the Stokes laser is represented by $E_s$. Finally, suppose that the mirror $M_1$ is $100 \%$ reflective of the Stokes laser while $M_2$ is less than $100 \%$ reflective (e.g., 95\%). The energy from the pump laser $E_p$ excites photons within the Raman gain medium that, in turn, transfer energy to the Stokes laser $E_s$. This energy is amplified by the back and forth reflection of the pump laser between the mirrors. With respect to the reflection rates mentioned above, energy from the pump laser (solid lines) is transmitted via the gain medium to the Stokes laser (dashed lines). The Stokes field then transmits energy through the second mirror $M_2$ at the leakage rate (in this case $5 \%$ ).

The two lasers undergo a nonlinear interaction within the gain medium that is modeled by the system of PDEs (7.1.1), see Newell and Moloney [53], Butcher and Cotter [15], and Penzkopfer, Laubereau, and Kaiser [60] for greater detail. The development presented here follows Costa [20].
$$\left.\begin{array}{l} \left(\frac{\partial}{\partial z}+\frac{1}{c} \frac{\partial}{\partial t}\right) E_p+\frac{i}{2 k_p} \Delta_{\perp} E_p=i \frac{\hbar \omega_p}{2 \varpi c} v\left(\omega_p\right) q_{p s} \sigma_{p s} E_s \ \left(\frac{\partial}{\partial z}+\frac{1}{c} \frac{\partial}{\partial t}\right) E_s+\frac{i}{2 k_s} \Delta_{\perp} E_s=i \frac{\hbar \omega_s}{2 \varpi c} v\left(\omega_s\right) q_{p s} \bar{\sigma}{p s} E_p \ \frac{\partial}{\partial t} \sigma{p s}+\beta_{p s} \sigma_{p s}=i q_{p s} E_p \bar{E}_s \end{array}\right}$$
This rather formidable looking set of equations employs the notation listed below.

## 数学代写|偏微分方程代考Partial Differential Equations代写|A Quasi–Implicit Finite Difference Scheme

The method utilized is implicit in the (linear) spatial derivative and explicit with respect to the nonlinear portion. First, the one-spatial dimensional case $\boldsymbol{\xi}=\xi \in\left[a_o, b_o\right] \subset \mathbb{R}$ is developed.
$$\delta_\tau^{+}\left(\boldsymbol{w}k^n\right)=C \delta{\xi}^2\left(\boldsymbol{w}k^{n+1}\right)+N\left(\boldsymbol{w}_k^n\right) \boldsymbol{w}_k^n$$ Equation (7.2.2) is referred to as the quasi-implicit method for the PDE (7.2.1). Throughout this chapter, use the notation $\lambda_p=\frac{(\Delta \xi)^2}{c_p \Delta \tau}, \lambda_s=\frac{(\Delta \xi)^2}{c_s \Delta \tau}, \zeta_p=\frac{(\Delta \xi)^2}{c_p}, \zeta_s=\frac{(\Delta \xi)^2}{c_s}, u_k^n=u\left(\xi_k, \tau_n\right)$, $v_k^n=v\left(\xi_k, \tau_n\right), \boldsymbol{u}^n=\left[u_1^n, u_2^n, \cdots, u_K^n\right]^T, \boldsymbol{v}^n=\left[v_1^n, v_2^n, \cdots, v_K^n\right]^T, I{K \times K}=$ the $K \times K$ identity matrix, $U^n=\operatorname{diag}\left{\left|u_1^n\right|^2,\left|u_2^n\right|^2, \cdots,\left|u_K^n\right|^2\right}$ and $V^n=\operatorname{diag}\left{\left|v_1^n\right|^2,\left|v_2^n\right|^2, \cdots,\left|v_K^n\right|^2\right}$ are $K \times K$ diagonal matrices, $A=\left[\begin{array}{ccccc}2 & -1 & 0 & \cdots & 0 \ -1 & 2 & -1 & \cdots & 0 \ 0 & -1 & 2 & \cdots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \cdots & 2\end{array}\right]$ -1 across the sub- and super-diagonals, and $A(\lambda)=A-i \cdot \lambda I_{K \times K}$. With this notation established, the quasi-implicit system (7.2.2) can be written as the pair of linear matrix equations
$$\left.\begin{array}{l} A\left(\lambda_p\right) \boldsymbol{u}^{n+1}=-i\left(\lambda_p I_{K \times K}-\zeta_p V^n\right) \boldsymbol{u}^n \ A\left(\lambda_s\right) \boldsymbol{v}^{n+1}=-i\left(\lambda_s I_{K \times K}+\zeta_s U^n\right) \boldsymbol{v}^n \end{array}\right} .$$

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|The Stimulated Raman Scattering Laser Model

$$\left.\begin{array}{l} \left(\frac{\partial}{\partial z}+\frac{1}{c} \frac{\partial}{\partial t}\right) E_p+\frac{i}{2 k_p} \Delta_{\perp} E_p=i \frac{\hbar \omega_p}{2 \varpi c} v\left(\omega_p\right) q_{p s} \sigma_{p s} E_s \ \left(\frac{\partial}{\partial z}+\frac{1}{c} \frac{\partial}{\partial t}\right) E_s+\frac{i}{2 k_s} \Delta_{\perp} E_s=i \frac{\hbar \omega_s}{2 \varpi c} v\left(\omega_s\right) q_{p s} \bar{\sigma}{p s} E_p \ \frac{\partial}{\partial t} \sigma{p s}+\beta_{p s} \sigma_{p s}=i q_{p s} E_p \bar{E}_s \end{array}\right}$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|A Quasi–Implicit Finite Difference Scheme

$$\delta_\tau^{+}\left(\boldsymbol{w}k^n\right)=C \delta{\xi}^2\left(\boldsymbol{w}k^{n+1}\right)+N\left(\boldsymbol{w}k^n\right) \boldsymbol{w}_k^n$$式(7.2.2)被称为PDE(7.2.1)的准隐式方法。在本章中，使用符号$\lambda_p=\frac{(\Delta \xi)^2}{c_p \Delta \tau}, \lambda_s=\frac{(\Delta \xi)^2}{c_s \Delta \tau}, \zeta_p=\frac{(\Delta \xi)^2}{c_p}, \zeta_s=\frac{(\Delta \xi)^2}{c_s}, u_k^n=u\left(\xi_k, \tau_n\right)$, $v_k^n=v\left(\xi_k, \tau_n\right), \boldsymbol{u}^n=\left[u_1^n, u_2^n, \cdots, u_K^n\right]^T, \boldsymbol{v}^n=\left[v_1^n, v_2^n, \cdots, v_K^n\right]^T, I{K \times K}=$$K \times K单位矩阵，U^n=\operatorname{diag}\left{\left|u_1^n\right|^2,\left|u_2^n\right|^2, \cdots,\left|u_K^n\right|^2\right}和V^n=\operatorname{diag}\left{\left|v_1^n\right|^2,\left|v_2^n\right|^2, \cdots,\left|v_K^n\right|^2\right}是K \times K对角线矩阵，A=\left[\begin{array}{ccccc}2 & -1 & 0 & \cdots & 0 \ -1 & 2 & -1 & \cdots & 0 \ 0 & -1 & 2 & \cdots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \cdots & 2\end{array}\right] -1横跨子对角线和超对角线，和A(\lambda)=A-i \cdot \lambda I{K \times K}。建立了这个符号后，拟隐系统(7.2.2)可以写成一对线性矩阵方程$$ \left.\begin{array}{l} A\left(\lambda_p\right) \boldsymbol{u}^{n+1}=-i\left(\lambda_p I_{K \times K}-\zeta_p V^n\right) \boldsymbol{u}^n \ A\left(\lambda_s\right) \boldsymbol{v}^{n+1}=-i\left(\lambda_s I_{K \times K}+\zeta_s U^n\right) \boldsymbol{v}^n \end{array}\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。