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## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代考|GENERALIZED BURGERS’ EQUATION

Here we consider a generalization of the Hopf-Cole transformation. Can we connect the following NLPDEs
\begin{aligned} u_t+A(u) u_x &=u_{x x}, \ v_t+B(v) v_x &=v_{x x}, \end{aligned}
via the substitution
Substituting (3.29) into (3.28a) and imposing (3.28b) leads to the following determining equations:
$$u=F\left(v, v_x\right), \quad F_{v_x} \neq 0 ?$$
where $p=v_x$. Differentiating (3.30b) with respect to $p$ twice (using (3.30a)) gives
$$F_p^3 A^{\prime \prime}(F)=0 .$$

From (3.31) and (3.30a) we find
$$A(F)=c_1 F+c_2, \quad F=F_1(v) p+F_2(v),$$
where $c_1$ and $c_2$ are arbitrary constants and $F_1$ and $F_2$ are arbitrary functions. With these assignments, returning to (3.30b) and isolating coefficients with respect to $p$ gives
\begin{aligned} 2 F_1^{\prime}-c_1 F_1^2 &=0, \ F_1\left(B-c_1 F_2-c_2\right) &=0, \end{aligned}
which we conveniently solve as
$$F_1=\frac{-2}{c_1 v+c_3}, \quad B=c_1 F+c_2,$$
where $c_3$ is an additional arbitrary constant. The remaining equation in (3.30c) becomes
$$F_2^{\prime \prime}=0,$$
which we solve as
$$F_2=c_4 v+c_5 \text {. }$$
With appropriate translation and scaling of variables we can set the following: $c_1=1, c_2=$ $c_4=c_5=0$, and suppress the subscript in $c_3$. Thus, we have the following: solutions of
$$u_t+u u_x=u_{x x}$$
can be obtained via
$$u=-2 \frac{v_x}{v}+c v$$
where $v$ satisfies
$$v_t+c v v_x=v_{x x} .$$
If we set $c=0$ we get the Hopf-Cole transformation. It is interesting to note that if we set $c=1$, we get a transformation which gives rise to solutions of the same equation.

## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代考|KDV-MKDV CONNECTION

The Korteweg-deVries equation (KdV)
$$u_t+6 u u_x+u_{x x x}=0,$$
first introduced by Korteweg and DeVries [56] to model shallow water waves, is a remarkable NLPDE. It has a number of applications and possesses a number of special properties (see, for example, Miura [57]). In 1968, Robert Miura found this remarkable transformation [58]. He found that solutions of the $\mathrm{KdV}$ equation (3.40) can be found using solutions of the modified Korteweg-deVries $(\mathrm{MKdV})$ equation
$$v_t-6 v^2 v_x+v_{x x x}=0,$$
via the transformation
$$u=v_x-v^2,$$
which today is known as the Miura transformation. We ask whether it’s possible to connect two general $\mathrm{KdV}$-type equations.

## 数学代写非线性偏微分方程代写Nonlinear Partial Differential Equation代 考|GENERALIZED BURGERS’ EQUATION

$$u_t+A(u) u_x=u_{x x}, v_t+B(v) v_x \quad=v_{x x}$$

$$u=F\left(v, v_x\right), \quad F_{v_x} \neq 0 ?$$

$$F_p^3 A^{\prime \prime}(F)=0$$
$从(3.31)$ 和 $(3.30 \mathrm{a})$ 戈们发现
$$A(F)=c_1 F+c_2, \quad F=F_1(v) p+F_2(v)$$

$$2 F_1^{\prime}-c_1 F_1^2=0, F_1\left(B-c_1 F_2-c_2\right) \quad=0$$

$$F_1=\frac{-2}{c_1 v+c_3}, \quad B=c_1 F+c_2$$

$$F_2^{\prime \prime}=0,$$

$$F_2=c_4 v+c_5$$

$$u_t+u u_x=u_{x x}$$

$$u=-2 \frac{v_x}{v}+c v$$

$$v_t+c v v_x=v_{x x} .$$

## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代 考|KDV-MKDV CONNECTION

Korteweg-deVries 方程 (KdV)
$$u_t+6 u u_x+u_{x x x}=0,$$
Korteweg 和 DeVries [56] 首次引入用于模拟浅水波，是 个了不起的 NLPDE。它有许多应用并具有许侈特殊属性 (例如，参 见 Miura [57])。 1968 年，Robert Miura 发现了这种显着的转变 [58]。他发现解快方琲KdV方程 (3.40) 可以使用修改后的 Korteweg-deVries 的解来找到 $(\mathrm{MKdV})$ 方程
$$v_t-6 v^2 v_x+v_{x x x}=0$$

$$u=v_x-v^2$$

## 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。

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## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代考|COMPATIBILITY IN （2+1） DIMENSIONS

Up until now, we have considered compatibility of PDEs in $(1+1)$ dimensions. We now extend this idea and consider PDEs in $(2+1)$ dimensions. In this section we consider the compatibility between the $(2+1)$ dimensional reaction-diffusion equation
$$u_t=u_{x x}+u_{y y}+Q\left(u, u_x, u_y\right)$$
and the first-order partial differential equation
$$u_t=F\left(t, x, y, u, u_x, u_y\right) .$$
This section is based on the work of Arrigo and Suazo [39]. We will assume that $F$ in (2.222) is nonlinear in the first derivatives $u_x$ and $u_y$. The case where $F$ is linear in the first derivatives $u_x$ and $u_y$, is left as an exercise to the reader.

Compatibility between (2.221) and (2.222) gives rise to the compatibility equation constraints
Eliminating the $x$ and $y$ derivatives in (2.223b) and (2.223c) by $(i)$ cross differentiation and (ii)
$\begin{array}{ll}2 F_{u p}+\left(F_p-Q_p\right) F_{p p}+\left(F_q-Q_q\right) F_{p q}=0, & \quad(2.224 \mathrm{a}) \ 2 F_{u q}+\left(F_p-Q_p\right) F_{p q}+\left(F_q-Q_q\right) F_{q q}=0 . & (2.224 \mathrm{~b})\end{array}$

## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代考|COMPATIBILITY IN （2+1） DIMENSIONS

\begin{aligned} &\text { Further, eliminating } F_{u p} \text { and } F_{u q} \text { by again }(i) \text { cross differentiation and (ii ) imposing (2.223a) } \ &\text { gives rise to } \ &\qquad \begin{array}{r} \left(2 F_{p p}-Q_{p p}+Q_{q q}\right) F_{p p}+2\left(F_{p q}-Q_{p q}\right) F_{p q}=0, \ \left(Q_{p p}-Q_{q q}\right) F_{p q}+2 Q_{p q} F_{q q}=0 . \end{array} \end{aligned}
Solving (2.223a), (2.225a) and (2.225b) for $F_{p p}, F_{p q}$ and $F_{q q}$ gives rise to two cases:
(i) $F_{p p}=F_{p q}=F_{q q}=0$,
(ii) $\quad F_{p p}=\frac{1}{2}\left(Q_{p p}-Q_{q q}\right), \quad F_{p q}=Q_{p q}, \quad F_{q q}=\frac{1}{2}\left(Q_{q q}-Q_{p p}\right)$
As we are primarily interested in compatible equations that are more general than quasilinear, we omit the first case. If we require that the three equations in (2.226b) be compatible, then to within equivalence transformations of the original equation, $Q$ satisfies
$$Q_{p p}+Q_{q q}=0 .$$
Using (2.227), we find that (2.226b) becomes
$$F_{p p}=Q_{p p}, \quad F_{p q}=Q_{p q}, \quad F_{q q}=Q_{q q},$$
from which we find that
$$F=Q(u, p, q)+X(t, x, y, u) p+Y(t, x, y, u) q+U(t, x, y, u),$$
where $X, Y$ and $U$ are arbitrary functions. Substituting (2.229) into (2.224a) and (2.224b) gives
$$\begin{gathered} 2 Q_{u p}+X Q_{p p}+Y Q_{p q}+2 X_u=0 \ 2 Q_{u q}+X Q_{p q}+Y Q_{q q}+2 Y_u=0 \end{gathered}$$
while (2.223b) and (2.223c) become (using (2.227) and (2.230))
\begin{aligned} &(X p+Y q+2 U) Q_{p p}+(X q-Y p) Q_{p q}+2\left(X_x-Y_y\right)=0 \ &(X q-Y p) Q_{p p}-(X p+Y q+2 U) Q_{p q}-2\left(X_y+Y_x\right)=0 \end{aligned}

## 数学代写非线性偏微分方程代写Nonlinear Partial Differential Equation代 考|COMPATIBILITY IN $(2+1)$ DIMENSIONS

$$u_t=u_{x x}+u_{y y}+Q\left(u, u_x, u_y\right)$$

$$u_t=F\left(t, x, y, u, u_x, u_y\right) .$$

(2.221) 和 $(2.222)$ 之间的相容性昌致相容性方程

$$2 F_{u p}+\left(F_p-Q_p\right) F_{p p}+\left(F_q-Q_q\right) F_{p q}=0, \quad(2.224 \mathrm{a}) 2 F_{u q}+\left(F_p-Q_p\right) F_{p q}+\left(F_q-Q_q\right) F_{q q}=0 . \quad(2.224 \mathrm{~b})$$

## 数学代写非线性偏微分方程代写Nonlinear Partial Differential Equation代考|COMPATIBILITY IN (2+1) DIMENSIONS

Further, eliminating $F_{u p}$ and $F_{u q}$ by again $(i)$ cross differentiation and (ii ) imposing (2.223a) $\quad$ gives rise to $\quad\left(2 F_{p p}-Q_{p p}+Q_{q q}\right) F_{p p}+2($

(i) $F_{p p}=F_{p q}=F_{q q}=0$
(ii) $\quad F_{p p}=\frac{1}{2}\left(Q_{p p}-Q_{q q}\right), \quad F_{p q}=Q_{p q}, \quad F_{q q}=\frac{1}{2}\left(Q_{q q}-Q_{p p}\right)$

$$Q_{p p}+Q_{q q}=0 .$$

$$F_{p p}=Q_{p p}, \quad F_{p q}=Q_{p q}, \quad F_{q q}=Q_{q q},$$

$$F=Q(u, p, q)+X(t, x, y, u) p+Y(t, x, y, u) q+U(t, x, y, u),$$

$$2 Q_{u p}+X Q_{p p}+Y Q_{p q}+2 X_u=02 Q_{u q}+X Q_{p q}+Y Q_{q q}+2 Y_u=0$$

$$(X p+Y q+2 U) Q_{p p}+(X q-Y p) Q_{p q}+2\left(X_x-Y_y\right)=0 \quad(X q-Y p) Q_{p p}-(X p+Y q+2 U) Q_{p q}-2\left(X_y+Y_x\right)=0$$

## 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。

Posted on Categories:Nonlinear Partial Differential Equation, 数学代写, 非线性偏微分方程

## avatest™帮您通过考试

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## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代考|Compatibility

We start our discussion by first solving the NPDE
$$x u_x-u_y^2=2 u,$$
subject to the boundary condition
$$u(x, x)=0 .$$
As with most introductory courses in partial differential equations (see, for example, [37]) we use the method of characteristics. Here, we define $F$ as
$$F=x p-q^2-2 u .$$
The characteristic equations become
\begin{aligned} &x_s=F_p=x, \ &y_s=F_q=-2 q, \ &u_s=p F_p+q F_q=x p-2 q^2, \ &p_s=-F_x-p F_u=p, \ &q_s=-F_y-q F_u=2 q . \end{aligned}
In order to solve the PDE (2.1) we will need to solve the system (2.4). As (2.1) has a boundary condition (BC), we will create BCs for the system (2.4). In the $(x, y)$ plane, the line $y=x$ is the boundary where $u$ is defined. To this, we associate a boundary in the $(r, s)$ plane. Given the flexibility, we can choose $s=0$ and connect the two boundaries via $x=r$. Therefore, we have
$$x=r, \quad y=r, \quad u=0 \text { when } s=0 .$$
To determine $p$ and $q$ on $s=0$, it is necessary to consider the initial condition $u(x, x)=0$. Differentiating with respect to $x$ gives
$$u_x(x, x)+u_y(x, x)=0 .$$
From the original PDE (2.1)
$$x u_x(x, x)-u_y^2(x, x)=0 .$$
If we denote $p_0=u_x(x, x)$ and $q_0=u_y(x, x)$, then (2.6) and (2.7) become
$$p_0+q_0=0, \quad r p_0-q_0^2=0 .$$

## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代考|CHARPIT’S METHOD

Obtaining exact solutions to NLPDEs such as (2.1) can be a difficult task as we are required to solve equations such as (2.4)! The difficulty is not so much in solving these characteristic equations but in eliminating the five arbitrary functions that appear upon integration. So we ask, is it possible to obtain exact solutions another way?
Consider the PDE
This integrates to give
$$u_y=-y .$$
$$u=-\frac{y^2}{2}+f(x)$$
and substitution into the original PDE (2.1) gives
$$x f^{\prime}=2 f .$$
This ODE is solved giving
$$f=c x^2$$
which from (2.22) leads to
$$u=c x^2-\frac{y^2}{2}$$
and the initial condition (2.2) gives $c=1 / 2$, and leads to the solution given in (2.20).
Consider the PDE
$$u_x=x$$

## 数学代写|非线性偏微分方程代写Nonlinear Partial Differential Equation代 孝|Compatibility

$$x u_x-u_y^2=2 u$$

$$u(x, x)=0$$

$$F=x p-q^2-2 u$$

$$x_s=F_p=x, \quad y_s=F_q=-2 q, u_s=p F_p+q F_q=x p-2 q^2, \quad p_s=-F_x-p F_u=p, q_s=-F_y-q F_u=2 q .$$

$$x=r, \quad y=r, \quad u=0 \text { when } s=0 .$$

$$u_x(x, x)+u_y(x, x)=0 .$$

$$x u_x(x, x)-u_y^2(x, x)=0 .$$

$$p_0+q_0=0, \quad r p_0-q_0^2=0 .$$

## 数学代写非线性偏微分方程代写Nonlinear Partial Differential Equation代 考|CHARPIT’S METHOD

$$u_y=-y .$$
$$u=-\frac{y^2}{2}+f(x)$$

$$x f^{\prime}=2 f .$$

$$f=c x^2$$
$从(2.22)$ 导致
$$u=c x^2-\frac{y^2}{2}$$

$$u_x=x$$

## 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。