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# 数学代写|偏微分方程作业代写Partial Differential Equations代考|The Method of Characteristics, Part III: General First-Order Equations

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## 数学代写|偏微分方程作业代写Partial Differential Equations代考|The Notation

The Notation. Your biggest foe in following this section is notation. If you can be in control of the notation and not let it get the better hand (i.e., confuse you to no end), you will be fine. The independent variables will be denoted by $x_{1}, \ldots, x_{N}$. We use $\mathbf{x}=\left(x_{1}, \ldots, x_{N}\right)$ to denote the full vector of independent variables which will lie in some potentially unbounded domain of $\mathbb{R}^{N}$. Perhaps, in an application, we would want to treat one of the independent variables as a temporal variable $t$; however, here for ease of notation, let us here simply denote all independent variables by $x_{i}$. We will look for a solution $u\left(x_{1}, \ldots, x_{N}\right)$ defined on some domain $\Omega \subset \mathbb{R}^{N}$ with data $g(\mathbf{x})$ supplied on some subset $\Gamma$ of the boundary of $\Omega$. Since $\Omega$ is $N$-dimensional, $\Gamma$ will be $(N-1)$-dimensional.

We must write down a generic form of the most general first-order PDE. This might seem like a mess but it is actually very easy. We write the equation involving the independent variables, the dependent variable $u$, and its first-order partial derivatives simply as
$$F(\nabla u(\mathbf{x}), u(\mathbf{x}), \mathbf{x})=0$$
for some choice of $C^{1}$ function $F: \mathbb{R}^{N} \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow \mathbb{R}$. The unknown solution $u\left(x_{1}, \ldots, x_{N}\right)$ will solve (2.28) at all $\mathbf{x}$ in its domain and satisfy $u(\mathbf{x})=g(\mathbf{x})$ on $\Gamma$.

It is convenient to denote the first two arguments of the function $F$ with variables $\mathbf{p}$ and $z$, respectively, where $\mathbf{p}$ denotes a vector in $\mathbb{R}^{N}$. For the PDE, we input $\nabla u(\mathbf{x})$ for $\mathbf{p}$ and $u(\mathbf{x})$ for $z$. Thus, we may write the PDE simply as
$$F(\mathbf{p}, z, \mathbf{x})=0 .$$
Note that a particular PDE is equivalent to specifying the function $F$. Hence, for a particular PDE, $F$ and its partial derivatives with respect to $p_{i}, z$ and $x_{i}$ are either given or can be readily computed. For example, the quasilinear PDE for $u\left(x_{1}, x_{2}, x_{3}\right)$
$$u u_{x_{1}}+u_{x_{1}} u_{x_{2}}+x_{1} u_{x_{3}}+u^{3}=x_{2} x_{3}$$
is equivalent to (2.29) with
$$F(\mathbf{p}, z, \mathbf{x})=F\left(p_{1}, p_{2}, p_{3}, z, x_{1}, x_{2}, x_{3}\right)=z p_{1}+p_{1} p_{2}+x_{1} p_{3}+z^{3}-x_{2} x_{3} .$$

## 数学代写|偏微分方程作业代写Partial Differential Equations代考|The Characteristic Equations

We proceed as before by parametrizing curves in the domain $\Omega$ of the solution by $s$. These characteristic curves (the projective characteristics) are denoted by $\mathbf{x}(s)$ and have components
$$\mathbf{x}(s)=\left(x_{1}(s), \ldots, x_{N}(s)\right), \quad s \in \mathbb{R} .$$
We will also denote derivatives with respect to $s$ as a dot above the variable. Assume that we have a $C^{2}$ solution $u$ to the PDE (2.28) on a domain $\Omega$. Consistent with our previous notation, define $z(s)$ and $\mathbf{p}(s)$ to denote, respectively, the values of the solution and the gradient of the solution along a characteristic $\mathbf{x}(s)$; i.e.,
$$z(s):=u(\mathbf{x}(s)) \quad \text { and } \quad \mathbf{p}(s):=\nabla u(\mathbf{x}(s)) .$$
Note again that $z(s)$ is scalar-valued whereas $\mathbf{p}(s)$ is vector-valued. We denote the component functions as
$$\mathbf{p}(s)=\left(p_{1}(s), \ldots, p_{N}(s)\right) .$$
The reader might wonder why we have introduced the new characteristic curves $p_{i}$ which keep track of partial derivatives along the projected characteristics $\mathbf{x}(s)$. In the previous cases of linear and quasilinear equations, we only needed $z(s)$ and $\mathbf{x}(s)$. However, for nonlinear equations the direction in which the projected characteristics should move can be tied to components of $\nabla u$. Hence it is imperative that we keep them in the picture.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|Linear and Quasilinear Equations in 𝑁 independent variables

The previous cases of linear, semilinear, and quasilinear equations all fall into our general framework. In fact, now we can consider these equations in $N$ independent variables, denoted here by the vector $\mathbf{x}$.
A general linear equation takes the form
$$F(\nabla u, u, \mathbf{x})=\mathbf{a}(\mathbf{x}) \cdot \nabla u(\mathbf{x})+b(\mathbf{x}) u(\mathbf{x})+c(\mathbf{x})=0,$$
for some vector-valued function $\mathbf{a}$ and scalar-valued functions $b$ and $c$ of $\mathbf{x}$. In our notation, we have
$$F(\mathbf{p}, z, \mathbf{x})=\mathbf{a}(\mathbf{x}) \cdot \mathbf{p}+b(\mathbf{x}) z+c(\mathbf{x})$$
which means $\nabla_{\mathbf{p}} F=\mathbf{a}(\mathbf{x})$. Hence the characteristic equations (2.36) for $\mathbf{x}$ and $z$ are
$$\dot{\mathbf{x}}(s)=\mathbf{a}(\mathbf{x}(s)), \quad \dot{z}(s)=\mathbf{a}(\mathbf{x}(s)) \cdot \mathbf{p}(s) .$$
However, the PDE tells us that $\mathbf{a}(\mathbf{x}(s)) \cdot \mathbf{p}(s)=-b(\mathbf{x}(s)) z(s)-c(\mathbf{x}(s))$. Hence, the $z$ and $\mathbf{x}$ equations form a closed system
$$\left{\begin{array}{l} \dot{\mathbf{x}}(s)=\mathbf{a}(\mathbf{x}(s)), \ \dot{z}(s)=-b(\mathbf{x}(s)) z(s)-c(\mathbf{x}(s)), \end{array}\right.$$
and we do not need $\mathbf{p}$ or its equations to find $u$. Of course, one could write down the $\mathbf{p}$ equations but they would be redundant and will be automatically satisfied by the solution $u$ along the characteristics $\mathbf{x}$. A very important observation about the ODEs for $z$ and $\mathbf{x}$ in (2.37) is that they are decoupled; we can first solve for $\mathbf{x}(s)$ and then solve for $z(s)$. Another way of saying this is that the structure of the PDE alone (i.e., the function $F)$ tells us exactly what the characteristics $\mathbf{x}(s)$ are.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|THE NOTATION

F(∇在(X),在(X),X)=0

F(p,和,X)=0.

F(p,和,X)=F(p1,p2,p3,和,X1,X2,X3)=和p1+p1p2+X1p3+和3−X2X3.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|THE CHARACTERISTIC EQUATIONS

X(s)=(X1(s),…,Xñ(s)),s∈R.

p(s)=(p1(s),…,pñ(s)).

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|LINEAR AND QUASILINEAR EQUATIONS IN 𝑁 INDEPENDENT VARIABLES

F(∇在,在,X)=一种(X)⋅∇在(X)+b(X)在(X)+C(X)=0,

F(p,和,X)=一种(X)⋅p+b(X)和+C(X)

X˙(s)=一种(X(s)),和˙(s)=一种(X(s))⋅p(s).

$$\left{X˙(s)=一种(X(s)), 和˙(s)=−b(X(s))和(s)−C(X(s)),\对。$$