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# 数学代写|偏微分方程代考Partial Differential Equations代写|Duhamel’s Principle for the One-Dimensional Wave Equation

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Duhamel’s Principle for the One-Dimensional Wave Equation

Duhamel’s Principle is the method to obtain solutions to non-homogeneous linear evolution equations like the wave equation, heat equation, and vibrating plate equation.

Application of Duhamel’s Principle: Finite String Problem If $U(x, t, s)$ is the solution to the problem
$$U_{t t}-c^2 U_{x x}=0,(x, t) \in(0, L) \times(0, \infty)$$
with boundary conditions $U(0, t, s)=0, U(L, t, s)=0, t>0, s>0$ and with initial conditions $U(x, 0, s)=0, U_t(x, 0, s)=f(x, s), s>0$, then $u(x, t)$ defined by $u(x, t)=\int_0^t U(x, t-\theta, \theta) d \theta$ is the solution to the non-homogeneous problem
$$u_{t t}-c^2 u_{x x}=f(x, t),(x, t) \in(0, L) \times(0, \infty)$$
with boundary conditions $u(0, t)=0, u(L, t)=0, t>0$ and with initial conditions $u(x, 0)=0, u_t(x, 0)=0$.

## 数学代写|偏微分方程代考Partial Differential Equations代写|ONE-DIMENSIONAL HEAT EQUATION

We shall solve the one-dimensional heat equation $\frac{\partial u}{\partial t}=c^2 \frac{\partial^2 u}{\partial x^2}$ for some practical approach, initial conditions, and boundary condition. We assume that two ends $x=0$ and $x=L$ of the rod are insulated. Therefore, the temperature at two ends $x=0$ and $x=L$ of the rod is zero. So that, boundary conditions are $u(0, t)=0$, $u(L, t)=0$ for all $t$ and the initial temperature in the rod is $f(x)$.

We shall determine the solution of the temperature $u(x, t)$ of the heat equation which satisfying initial and boundary conditions.
Let us assume, $u(x, t)=X(x) T(t)$ be a solution to the heat equation.
Hence, it satisfies the heat equation.
Differentiate $u(x, t)=X(x) T(t)$ with respect to $x$ and $t$
$$\frac{\partial u}{\partial x}=X^{\prime}(x) T(t), \frac{\partial u}{\partial t}=X(x) T^{\prime}(t)$$

$$\frac{\partial^2 u}{\partial t^2}=X(x) T^{\prime \prime}(t)$$
Substituting above derivatives in $\frac{\partial u}{\partial t}=c^2 \frac{\partial^2 u}{\partial x^2}$
$$X(x) T^{\prime}(t)=c^2 X^{\prime \prime}(x) T(t)$$
separates the variables
$$\frac{T^{\prime}(t)}{c^2 T(t)}=\frac{X^{\prime \prime}(x) .}{X(x)}$$
Since $x$ and $t$ are independent variables; therefore, $\frac{T^{\prime}(t)}{c^2 T(t)}=\frac{X^{\prime \prime}(x)}{X(x)}$ can hold only when each side equal to some constant, say $k$
$$\begin{gathered} \frac{T^{\prime}(t)}{c^2 T(t)}=\frac{X^{\prime \prime}(x)}{X(x)}=k \ \frac{X^{\prime \prime}(x)}{X(x)}=k \text { and } \frac{T^{\prime}(t)}{c^2 T(t)}=k \ \therefore X^{\prime \prime}(x)-k X(x)=0 \text { and } T^{\prime}(t)-k c^2 T(t)=0 \ \therefore D^2 X-k X=0 \text { and } D T-k c^2 T=0 . \end{gathered}$$

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Duhamel’s Principle for the One-Dimensional Wave Equation

Duhamel原理是求解波动方程、热方程、振动板方程等非齐次线性演化方程的方法。

Duhamel原理的应用:有限弦问题如果$U(x, t, s)$是问题的解
$$U_{t t}-c^2 U_{x x}=0,(x, t) \in(0, L) \times(0, \infty)$$

$$u_{t t}-c^2 u_{x x}=f(x, t),(x, t) \in(0, L) \times(0, \infty)$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|ONE-DIMENSIONAL HEAT EQUATION

$$\frac{\partial u}{\partial x}=X^{\prime}(x) T(t), \frac{\partial u}{\partial t}=X(x) T^{\prime}(t)$$

$$\frac{\partial^2 u}{\partial t^2}=X(x) T^{\prime \prime}(t)$$

$$X(x) T^{\prime}(t)=c^2 X^{\prime \prime}(x) T(t)$$

$$\frac{T^{\prime}(t)}{c^2 T(t)}=\frac{X^{\prime \prime}(x) .}{X(x)}$$

$$\begin{gathered} \frac{T^{\prime}(t)}{c^2 T(t)}=\frac{X^{\prime \prime}(x)}{X(x)}=k \ \frac{X^{\prime \prime}(x)}{X(x)}=k \text { and } \frac{T^{\prime}(t)}{c^2 T(t)}=k \ \therefore X^{\prime \prime}(x)-k X(x)=0 \text { and } T^{\prime}(t)-k c^2 T(t)=0 \ \therefore D^2 X-k X=0 \text { and } D T-k c^2 T=0 . \end{gathered}$$

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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。