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Mathematics LibreTexts

3.E: Classification (Exercises)

These are homework exercises to accompany Miersemann's "Partial Differential Equations" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Partial differential equations are differential equations that contains unknown multivariable functions and their partial derivatives. Prerequisite for the course is the basic calculus sequence.

Q3.1

Let \(\chi\): \({\mathbb{R}^n}\to {\mathbb{R}^1}\) in \(C^1\), \(\nabla\chi\not=0\). Show that for given \(x_0\in {\mathbb{R}^n}\) there is in a neighborhood of \(x_0\) a local diffeomorphism \(\lambda=\Phi(x)\), \(\Phi:\ (x_1,\ldots,x_n)\mapsto(\lambda_1,\ldots,\lambda_n)\), such that \(\lambda_n=\chi(x)\).

Q3.2

Show that the differential equation

$$a(x,y)u_{xx}+2b(x,y)u_{xy}+c(x,y)u_{yy}+\mbox{lower order terms}=0$$

is  elliptic  if \(ac-b^2>0\), parabolic if \(ac-b^2=0\) and hyperbolic if \(ac-b^2<0\).

Q3.3

Show that in the hyperbolic case there exists a solution of \(\phi_x+\mu_1\phi_y=0\), see equation (3.9), such that \(\nabla\phi\not=0\).

Hint: Consider an appropriate Cauchy initial value problem.

Q3.4

Show equation (3.4).

Q3.5

Find the type of

$$Lu:=2u_{xx}+2u_{xy}+2u_{yy}=0$$

and transform this equation into an equation with vanishing mixed derivatives by using the orthogonal mapping (transform to principal axis) \(x=Uy,\ U\) orthogonal.

Q3.6

Determine the type of the following equation at \((x,y)=(1,1/2)\).

$$Lu:=xu_{xx}+2yu_{xy}+2xyu_{yy}=0.$$

Q3.7

Find all \(C^2\)-solutions of

$$u_{xx}-4u_{xy}+u_{yy}=0.$$

Hint: Transform to principal axis and stretching of axis lead to the wave equation.

Q3.8

Oscillations of a beam are described by

\begin{eqnarray*}
w_x-{1\over E}\sigma_t&=& 0\\
\sigma_x-\rho w_t&=&0,
\end{eqnarray*}

where \(\sigma\) stresses, \(w\) deflection of the beam and \(E,\ \rho\) are positive constants.

  1. Determine the type of the system.
  2. Transform the system into two uncoupled equations, that is, \(w,\ \sigma\) occur only in one equation, respectively.
  3. Find non-zero solutions.

Q3.9

Find nontrivial solutions (\(\nabla \chi\not=0\)) of the characteristic equation to

$$x^2u_{xx}-u_{yy}=f(x,y,u,\nabla u),$$

where \(f\) is given.

Q3.10

Determine the type of

$$u_{xx}-xu_{yx}+u_{yy}+3u_x=2x,$$

where \(u=u(x,y)\).

Q3.11

Transform equation

$$u_{xx}+(1-y^2)u_{xy}=0,$$

\(u=u(x,y)\), into its normal form.

Q3.12

Transform  the Tricomi-equation

$$yu_{xx}+u_{yy}=0,$$

\(u=u(x,y)\), where \(y<0\), into its normal form.

Q3.13

Transform  equation

$$x^2u_{xx}-y^2u_{yy}=0,$$

\(u=u(x,y)\), into its normal form.

Q3.14

Show that

$$\lambda=\dfrac{1}{\left(1+|p|^2\right)^{3/2}},\ \ \Lambda=\dfrac{1}{\left(1+|p|^2\right)^{1/2}}.$$

are the minimum and maximum of eigenvalues of the matrix \((a^{ij})\), where

$$a^{ij}=\left(1+|p|^2\right)^{-1/2}\left(\delta_{ij}-\dfrac{p_ip_j}{1+|p|^2}\right).$$

Q3.15

Show that Maxwell equations are a hyperbolic system.

Q3.16

Consider Maxwell equations and prove that \(\text{div}\ E=0\) and \(\text{div}\ H=0\) for all \(t\) if these equations are satisfied for a fixed time \(t_0\).

Hint. \(\text{div}\ \text{rot} \ A=0\) for each \(C^2\)-vector field \(A=(A_1,A_2,A_3)\).

Q3.17

Assume a characteristic surface \(\mathcal{S}(t)\) in \(\mathbb{R}^3\) is defined by \(\chi(x,y,z,t)=const.\) such that \(\chi_t=0\) and \(\chi_z\not=0\). Show that \(\mathcal{S}(t)\) has a nonparametric representation \(z=u(x,y,t)\) with \(u_t=0\), that is \(\mathcal{S}(t)\) is independent of \(t\).

Q3.18

Prove formula (3.22) for the normal on a surface.

Q3.19

Prove formula (3.23) for the speed of the surface \(\mathcal{S}(t)\).

Q3.20

Write the Navier-Stokes system as a system of type (3.4.1).

Q3.21

Show that the following system (linear elasticity, stationary case of (3.4.1.1) in the two-dimensional case) is elliptic
$$
\mu\triangle u+(\lambda+\mu)\mbox{\ grad(div}\ u)+f=0,
$$
where \(u=(u_1,u_2)\). The vector \(f=(f_1,f_2)\) is given and
\(\lambda,\ \mu\) are positive constants.

Q3.22

Discuss the type of the following system in stationary gas dynamics (isentrop flow) in \(\mathbb{R}^2\).
\begin{eqnarray*}
\rho u u_x+\rho v u_y+ a^2\rho_x&=&0\\
\rho u v_x+\rho v v_y+ a^2\rho_y&=&0\\
\rho (u_x+v_y)+u\rho_x+ v\rho_y&=&0.
\end{eqnarray*}
Here are \((u,v)\) velocity vector, \(\rho\) density and
\(a=\sqrt{p'(\rho)}\) the sound velocity.

Q3.23

Show  formula 7. (directional derivative).

Hint: Induction with respect to \(m\).

Q3.24

Let \(y=y(x)\) be the solution of:
\begin{eqnarray*}
y'(x)&=&f(x,y(x))\\
y(x_0)&=&y_0,
\end{eqnarray*}
where \(f\) is  real analytic in a neighborhood of \((x_0,y_0)\in \mathbb{R}^2\).
Find the polynomial \(P\) of degree 2 such that
$$
y(x)=P(x-x_0)+O(|x-x_0|^3)
$$
as \(x\to x_0\).

Q3.25

Let \(u\) be the solution of
\begin{eqnarray*}
\triangle u&=&1\\
u(x,0)&=&u_y(x,0)=0.
\end{eqnarray*}
Find the polynomial \(P\) of degree 2 such that
$$
u(x,y)=P(x,y)+O((x^2+y^2)^{3/2})
$$
as \((x,y)\to(0,0)\).

Q3.26

Solve the Cauchy initial value problem
\begin{eqnarray*}
V_t&=&{Mr\over r-s-NV}(1+N(n-1)V_s)\\
V(s,0)&=&0.
\end{eqnarray*}

Hint: Multiply the differential equation with \((r-s-NV)\).

Q3.27

Write \(\triangle^2 u=-u\) as a system of first order.

Hint: \(\triangle^2 u\equiv\triangle(\triangle u)\).

Q3.28

Write the minimal surface equation

$${\partial\over\partial x}\left({u_x\over\sqrt{1+u_x^2+u_y^2}}\right)+{\partial\over\partial y}\left({u_y\over\sqrt{1+u_x^2+u_y^2}}\right)=0$$

as a system of first order.

Hint: \(v_1:= u_x/\sqrt{1+u_x^2+u_y^2},\ v_2:=u_y/\sqrt{1+u_x^2+u_y^2}.\)

Q3.29

Let \(f:\ \mathbb{R}^1\times\mathbb{R}^m\to\mathbb{R}^m\) be real analytic in \((x_0,y_0)\). Show that a real analytic solution in a neighborhood of \(x_0\) of the problem

\begin{eqnarray*}
y'(x)&=&f(x,y)\\
y(x_0)&=&y_0
\end{eqnarray*}

exists and is equal to the unique \(C^1[x_0-\epsilon, x_0+\epsilon]\)-solution from the Picard-Lindel\"of theorem, \(\epsilon>0\) sufficiently small.

Q3.30

Show (see the proof of Proposition A7)

$$\dfrac{\mu\rho(r-x_1-\ldots-x_n)}{\rho r-(\rho+mM)(x_1+\ldots+x_n)} <<\dfrac{\mu\rho r}{\rho r-(\rho+mM)(x_1+\ldots+x_n)}.$$

Hint: Leibniz's rule.

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