3.E: Classification (Exercises)
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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 χ: Rn→R1 in C1, ∇χ≠0. Show that for given x0∈Rn there is in a neighborhood of x0 a local diffeomorphism λ=Φ(x), Φ: (x1,…,xn)↦(λ1,…,λn), such that λn=χ(x).
Q3.2
Show that the differential equation
a(x,y)uxx+2b(x,y)uxy+c(x,y)uyy+lower order terms=0
is elliptic if ac−b2>0, parabolic if ac−b2=0 and hyperbolic if ac−b2<0.
Q3.3
Show that in the hyperbolic case there exists a solution of ϕx+μ1ϕy=0, see equation (3.9), such that ∇ϕ≠0.
Hint: Consider an appropriate Cauchy initial value problem.
Q3.4
Show equation (3.4).
Q3.5
Find the type of
Lu:=2uxx+2uxy+2uyy=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:=xuxx+2yuxy+2xyuyy=0.
Q3.7
Find all C2-solutions of
uxx−4uxy+uyy=0.
Hint: Transform to principal axis and stretching of axis lead to the wave equation.
Q3.8
Oscillations of a beam are described by
wx−1Eσt=0σx−ρwt=0,
where σ stresses, w deflection of the beam and E, ρ are positive constants.
- Determine the type of the system.
- Transform the system into two uncoupled equations, that is, w, σ occur only in one equation, respectively.
- Find non-zero solutions.
Q3.9
Find nontrivial solutions (∇χ≠0) of the characteristic equation to
x2uxx−uyy=f(x,y,u,∇u),
where f is given.
Q3.10
Determine the type of
uxx−xuyx+uyy+3ux=2x,
where u=u(x,y).
Q3.11
Transform equation
uxx+(1−y2)uxy=0,
u=u(x,y), into its normal form.
Q3.12
Transform the Tricomi-equation
yuxx+uyy=0,
u=u(x,y), where y<0, into its normal form.
Q3.13
Transform equation
x2uxx−y2uyy=0,
u=u(x,y), into its normal form.
Q3.14
Show that
λ=1(1+|p|2)3/2, Λ=1(1+|p|2)1/2.
are the minimum and maximum of eigenvalues of the matrix (aij), where
aij=(1+|p|2)−1/2(δij−pipj1+|p|2).
Q3.15
Show that Maxwell equations are a hyperbolic system.
Q3.16
Consider Maxwell equations and prove that div E=0 and div H=0 for all t if these equations are satisfied for a fixed time t0.
Hint. div rot A=0 for each C2-vector field A=(A1,A2,A3).
Q3.17
Assume a characteristic surface S(t) in R3 is defined by χ(x,y,z,t)=const. such that χt=0 and χz≠0. Show that S(t) has a nonparametric representation z=u(x,y,t) with ut=0, that is 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 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
μ△u+(λ+μ)\ grad(div u)+f=0,
where u=(u1,u2). The vector f=(f1,f2) is given and
λ, μ are positive constants.
Q3.22
Discuss the type of the following system in stationary gas dynamics (isentrop flow) in R2.
ρuux+ρvuy+a2ρx=0ρuvx+ρvvy+a2ρy=0ρ(ux+vy)+uρx+vρy=0.
Here are (u,v) velocity vector, ρ density and
a=√p′(ρ) 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:
y′(x)=f(x,y(x))y(x0)=y0,
where f is real analytic in a neighborhood of (x0,y0)∈R2.
Find the polynomial P of degree 2 such that
y(x)=P(x−x0)+O(|x−x0|3)
as x→x0.
Q3.25
Let u be the solution of
△u=1u(x,0)=uy(x,0)=0.
Find the polynomial P of degree 2 such that
u(x,y)=P(x,y)+O((x2+y2)3/2)
as (x,y)→(0,0).
Q3.26
Solve the Cauchy initial value problem
Vt=Mrr−s−NV(1+N(n−1)Vs)V(s,0)=0.
Hint: Multiply the differential equation with (r−s−NV).
Q3.27
Write △2u=−u as a system of first order.
Hint: △2u≡△(△u).
Q3.28
Write the minimal surface equation
∂∂x(ux√1+u2x+u2y)+∂∂y(uy√1+u2x+u2y)=0
as a system of first order.
Hint: v1:=ux/√1+u2x+u2y, v2:=uy/√1+u2x+u2y.
Q3.29
Let f: R1×Rm→Rm be real analytic in (x0,y0). Show that a real analytic solution in a neighborhood of x0 of the problem
y′(x)=f(x,y)y(x0)=y0
exists and is equal to the unique C1[x0−ϵ,x0+ϵ]-solution from the Picard-Lindel\"of theorem, ϵ>0 sufficiently small.
Contributors and Attributions
Integrated by Justin Marshall.