Sample questions

1.

Solve the one-dimensional initial value problem using a Fourier series solution.

$$\begin{eqnarray*}u_{tt}-u_{xx} &=& 0 \quad \mbox{for} \quad 0<x<\pi \quad \mbox{... ...i \\ u(0,t) &=& 0,\quad u_x(\pi,t)=0 \quad \mbox{for} t \ge 0. \end{eqnarray*}$$

2.

Given the first order partial differential equation

$$u u_x + u_y=0, \quad u(x,0)=h(x)$$

(a)

Is the initial curve a characteristic of the problem?

(b)

Use the method of characteristics to find a solution

(c)

Find a general solution

(d)

Impose the boundary condition on the general solution to obtain a solution to this problem.

(e)

Discuss the global existence and uniqueness of these solutions in terms of properties of the function h.

3.

Find the solution of the initial value problem

$$\begin{eqnarray*}u_{tt} &=& \Delta u \\ u(x,0) &=& g(x) = x_1^2+x_2^2,\quad u_t(x,0)=h(x)= 0 \end{eqnarray*}$$

(a)

by means of Kirchhoff's formula in three dimension

$$u(x,t)=\frac{1}{4 \pi}\frac{\partial}{\partial t} \left( t \... ... + \frac{t}{4 \pi} \int_{\vert\xi\vert} h(x+ct\xi) dS_{\xi}$$

(b)

and by means of the solution for the two dimensional wave equation

$$\begin{eqnarray*}u(x_1,x_2,t)&=&\frac{1}{4 \pi}\frac{\partial}{\partial t} \lef... ...x_2+ct\xi_2) d\xi_1 d\xi_2} {\sqrt{1-\xi_1^2-\xi_2^2}} \right) \end{eqnarray*}$$

4.

Consider the initial/boundary value problem with forcing term

$$\begin{eqnarray*}u_{tt} &=& \Delta u + f(x,t) \quad \mbox{for} x\in \Omega\sub... ...box{for}\quad x \in \partial\Omega \quad \mbox{and} \quad t>0. \end{eqnarray*}$$

Use Duhamel's principle and an expansion of f in eigenfunctions to obtain a formal solution.

5.

Consider the following Initial Value Problem (IVP):

$$\begin{array}{rcl} \frac{\partial^2u}{\partial t^2}-c^2\Delt... ...l u}{\partial t}(x,0)&=&h(x),\ x\in{\bf {\rm R}}^n. \end{array}$$ (1)

Prove the uniqueness and continuous dependence parts of the following theorem: Suppose g, h have compact support. Then () has a unique solution that depends continuously on the initial data g, h. (You do not have to prove the existence of a solution.)

6.

Consider again the IVP (). Prove the following theorem: Suppose u is a C² solution of (), and suppose that for some $x_0\in{\bf {\rm R}}^n$ and some t₀>0, g and h are both identically zero on the set

$$\left\{x\in{\bf {\rm R}}^n\,:\,\vert\vert x-x_0\vert\vert\le ct_0\right\}.$$

Then u(x₀,t₀)=0.

7.

Let $\Omega$ be a smooth, bounded domain in ${\bf {\rm R}}^n$.

(a)

Suppose $u\in C^2(\Omega)$ satisfies $\Delta u\ge 0$ in $\Omega$. Prove that

$$y\in\Omega\Rightarrow u(y)<\sup_{x\in\Omega}u(x).$$

(b)

Suppose u,v are two function in $C^2(\Omega)\cap C(\overline{\Omega})$, and

$$\Delta u=f,\ \Delta v=f$$

both hold in $\Omega$, where $f\in C(\Omega)$. Prove that if

$$\vert u(x)-v(x)\vert\le\epsilon,\ \forall x\in\partial\Omega$$

holds, then

$$\vert u(x)-v(x)\vert\le\epsilon,\ \forall x\in\overline{\Omega}$$

also holds.

8.

Let $\Omega$ be a smooth, bounded domain in ${\bf {\rm R}}^n$.

(a)

Write down a well-posed Initial-Boundary Value Problem (IBVP) for the heat equation.

(b)

State and prove a maximum principle for your (IBVP).

(c)

Use the maximum principle to prove uniqueness and continuous dependence on the initial conditions for your IBVP.

9.

Consider the boundary value problem:

$$\begin{eqnarray*}Lu = [ -\frac{d}{dx}(p(x)\frac{d}{dx}) + q(x)]u = f \\ u(0) = u'(\pi) = 0 \end{eqnarray*}$$

(a)

Find the quadratic functional I(v) associated with this BVP and identify the energy in v from the energy inner product a(v,w).

(b)

Suppose that a function u minimizes I(v) over the whole admissible space

$$H_0^1 = \{w \in H^1 : w \,\,\,\mbox{{\rm satisfies essential boundary conditions}}\}.$$

Let S^h be any closed subspace of H₀¹ and let u^h minimize I(v^h) in S^h. Show that the function u^h also minimizes the energy in S^h:

$$a(u-u^h,u-u^h) = {\rm min}\{a(u-v^h,u-v^h) \, : \, v^h \in S^h \}$$

(c)

Show that the function u^h is the projection of u onto S^h with respect to the energy inner product. That is, the error u-u^h is orthogonal to S^h.

(d)

Show that the function u^h satisfies the discrete weak form of the BVP:

$$a(u^h,v^h) = (f,v^h) \,\,\,\,\, \mbox{{\rm for all}} \,\, v^h \in S^h$$

10.

Consider a bilinear quadrilateral finite element with Cartesian coordinates (2,1), (4,1), (a,b) and (2,4).

(a)

Find the isoparametric coordinate transformations, $x=x(\xi,\eta)$ and $y=y(\xi,\eta)$.

(b)

Determine any algebraic conditions on the location of point (a,b) in order to ensure an invertible transformation.

11.

Show that in a mesh of bilinear quadrilateral finite elements, continuity of the interpolation functions at the vertices does not necessarily imply continuity of the interpolation functions throughout the whole domain.

12.

Given the boundary value problem:

$$\begin{eqnarray*}-\mbox{\boldmath$\nabla$ } \cdot (\mbox{\boldmath$A$ } \cdot \n... ...) \cdot \mbox{\boldmath$n$ } = t_0 \,\,\, {\rm on} \,\, \Gamma_2 \end{eqnarray*}$$

where f, a₀₀ and the components of the matrix $\mbox{\boldmath$A$ }=[a_{ij}]$are functions of x and y in a two-dimensional domain $\Omega$, and u₀and t₀ are known functions on portions $\Gamma_1$ and $\Gamma_2$ of the boundary. (Note: $\Gamma_1$ and $\Gamma_2$ are open sets which do not intersect and the union of their closures is the closure of $\partial\Omega$.) Find the weak formulation of this boundary value problem. Specify all continuity and integrability requirements (i.e. the function spaces to which all functions belong).

13.

Consider the IVP

$$\begin{eqnarray*}\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}&=&... ... -\infty<x<\infty,\ t>0,\\ u(x,0)&=&\phi(x),\ -\infty<x<\infty. \end{eqnarray*}$$

(a)

Write down two finite difference schemes for this problem, one that is conditionally stable and one that unconditionally stable.

(b)

Use von Neumann analysis to prove that the two schemes you present really have the specified stability properties.

14.

Consider solving Poisson's equation on the unit square:

$$\begin{eqnarray*}-\Delta u&=&f(x,y),\ (x,y)\in\O=\left\{(x,y)\,:\,0<x<1,0<y<1\right\},\\ u&=&0,\ (x,y)\in\partial\O. \end{eqnarray*}$$

(a)

Write down the finite difference scheme for this BVP using the standard five-point stencil for the Laplacian.

(b)

Write down the finite element equations for the same BVP using Lagrange triangles on a regular mesh like the one shown below.

The nodes in the mesh should be labeled from left to right, beginning with the bottom row. (The above mesh has 128 triangles on an $8\times 8$ grid. You should assume 2n² triangles on an $n\times n$ grid.)

(c)

What are the similarities and differences between the two schemes? Discuss the accuracy of each method.

15.

Consider the following initial-boundary value problem (IBVP) for the wave equation:

$$\begin{eqnarray*}\frac{\partial^2u}{\partial t^2}-c^2\frac{\partial^2u}{\partial... ...0,t)&=&0,\ t>0,\\ \frac{\partial u}{\partial x}(L,t)&=&0,\ t>0. \end{eqnarray*}$$

(a)

Write down a finite difference scheme that is second-order in both time and space.

(b)

Explain how the initial and boundary conditions are incorporated into your scheme.

(c)

Give the stability condition for your scheme.