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Mathematics LibreTexts

6.E: Parabolic Equations (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.

Q6.1

Show that the solution \(u(x,t)\) given by Poisson's formula satisfies

$$
\inf_{z\in \mathbb{R}^n}\varphi(z)\le u(x,t)\le\sup_{z\in\mathbb{R}^n}\varphi(z)\ ,
$$
provided \(\varphi(x)\) is continuous and bounded on \(\mathbb{R}^n\).

Q6.2

Solve for given \(f(x)\) and \(\mu\in\mathbb{R}^1\) the initial value problem
\begin{eqnarray*}
u_t+u_x+\mu u_{xxx} &=& 0\quad \mbox{in}\quad \mathbb{R}^1\times\mathbb{R}^1_+\\
u(x,0) &=& f(x) \ .
\end{eqnarray*}

Q6.3

Show by using Poisson's formula:
(i) Each function \(f\in C([a,b])\) can be approximated uniformly by a sequence \(f_n\in C^\infty[a,b]\) .
(ii)  In (i) we can choose polynomials \(f_n\) (Weierstrass's approximation theorem).

Hint: Concerning (ii), replace the kernel \(K=exp(-{|y-x|^2\over4t})\) by a sequence of
Taylor polynomials in the variable
\(z=-\frac{|y-x|^2}{4t}\).

Q6.4

Let \(u(x,t)\) be a positive solution of
$$
u_t=\mu u_{xx},\ t>0,
$$
where \(\mu\) is a constant. Show that \(\theta:=-2\mu u_x/u\) is a solution of Burger's equation
$$
\theta_t+\theta\theta_x=\mu\theta_{xx},\ t>0.
$$

Q6.5

Assume \(u_1(s,t), ...,u_n(s,t)\) are solutions of  \(u_t=u_{ss}\). Show that  \(\prod_{k=1}^nu_k(x_k,t)\) is a solution of the
heat equation \(u_t-\triangle u=0\) in \(\mathbb{R}^n\times (0,\infty)\).

Q6.6

Let \(A\), \(B\) are real, symmetric and non-negative matrices. Non-negative means that all eigenvalues are non-negative. Prove that trace \((AB)\equiv\sum_{i,j=1}^na^{ij}b_{ij}\ge0\).

Hint: (i) Let \(U=(z_1,\ldots,z_n)\), where \(z_l\) is an orthonormal system of eigenvectors to the eigenvalues \(\lambda_l\) of
the matrix \(B\). Then
$$
X=U\left(\begin{array}{llcl}
\sqrt{\lambda_1} & 0&\cdots & 0\\
0 & \sqrt{\lambda_2}&\cdots&0\\
\cdots & \cdots& \cdots& \cdots\\
0&0&\cdots&\sqrt{\lambda_n}
\end{array}\right) U^T
$$
is a square root of \(B\). We recall that
$$
U^TBU=\left(\begin{array}{llcl}
\lambda_1 & 0&\cdots & 0\\
0 & \lambda_2&\cdots&0\\
\cdots& \cdots& \cdots& \cdots\\
0&0&\cdots&\lambda_n
\end{array}\right).
$$
(ii) trace \((QR)=\)trace \((RQ)\).
(iii) Let \(\mu_1(C),\ldots\mu_n(C)\) are the eigenvalues of a real symmetric \(n\times n\)-matrix. Then trace \(C=\sum_{l=1}^n\mu_l(C)\), which follows from the fundamental lemma of algebra:
\begin{eqnarray*}
\mbox{det}\ (\lambda I-C)&=&\lambda^n-(c_{11}+\ldots+c_{nn})\lambda^{n-1}+\ldots\\
&\equiv&(\lambda-\mu_1)\cdot\ldots\cdot(\lambda-\mu_n)\\
&=&\lambda^n-(\mu_1+\ldots+\mu_n)\lambda^{n+1}+\ldots
\end{eqnarray*}

Q6.7

Assume \(\Omega\) is bounded, \(u\) is a solution of the heat equation and  \(u\) satisfies the regularity assumptions of the maximum principle (Theorem 6.2). Show that \(u\) achieves its maximum and its minimum on \(S_T\).

Q6.8

Prove the following comparison principle: Assume \(\Omega\) is bounded and \(u,v\) satisfy the regularity assumptions of the maximum principle.
Then
\begin{eqnarray*}
u_t-\triangle u&\le&v_t-\triangle v\ \ \mbox{in}\ D_T\\
u&\le&v\ \ \mbox{on}\ S_T
\end{eqnarray*}
imply that \(u\le v\) in \(D_T\).

Q6.9

Show that the comparison principle implies the maximum principle.

Q6.10

Consider the boundary-initial value problem
\begin{eqnarray*}
u_t-\triangle u&=&f(x,t) \ \ \mbox{in}\ D_T\\
u(x,t)&=&\phi(x,t)\ \ \mbox{on}\ S_T,
\end{eqnarray*}
where \(f\), \(\phi\) are given.\\
Prove uniqueness in the class \(u,\ u_t,\ u_{x_ix_k}\in C(\overline{D_T})\).

Q6.11

Assume \(u,\ v_1,\ v_2\in C^2(D_T)\cap C(\overline{D_T})\), and \(u\) is a solution of the previous boundary-initial value problem and \(v_1\), \(v_2\) satisfy
\begin{eqnarray*}
(v_1)_t-\triangle v_1&\le&f(x,t)\le (v_2)_t-\triangle v_2 \ \ \mbox{in}\ D_T\\
v_1&\le&\phi\le v_2\ \ \mbox{on}\ S_T.
\end{eqnarray*}
Show that (inclusion theorem)
$$
v_1(x,t)\le u(x,t)\le v_2(x,t)\ \ \mbox{on}\ \overline{D_T}.
$$

Q6.12

Show by using the comparison principle: let \(u\) be a sufficiently regular solution of
\begin{eqnarray*}
u_t -\triangle u &=& 1 \quad\mbox{in}\quad D_T\\
u &=& 0 \quad\mbox{on}\quad S_T,
\end{eqnarray*}
then \(0\le u(x,t)\le t \quad\mbox{in}\quad D_T\).

Q6.13

Discuss the result of Theorem 6.3 for the case
$$
Lu=\sum_{i,j=1}^n a_{ij}(x,t)u_{x_ix_j}+\sum_i^nb_i(x,t)u_{x_i}+c(x,t)u(x,t).
$$

Q6.14

Show that
$$
u(x,t)=\sum_{n=1}^\infty c_ne^{-n^2t}\sin(nx),
$$
where
$$
c_n={2\over \pi}\int_0^\pi\ f(x)\sin(nx)\ dx,
$$
is a solution of the initial-boundary value problem
\begin{eqnarray*}
u_t&=&u_{xx},\ x\in(0,\pi),\ t>0,\\
u(x,0)&=&f(x),\\
u(0,t)&=&0,\\
u(\pi,t)&=&0,
\end{eqnarray*}
if \(f\in C^4({\mathbb R})\) is odd with respect to \(0\) and \(2\pi\)-periodic.

Q6.15

(i) Find the solution of the diffusion problem \(c_t=Dc_{zz}\) in \(0\le z\le l$, $0\le t<\infty\), \(D=const.>0\), under the boundary conditions \(c_z(z,t)=0\) if \(z=0\) and \(z=l\) and with the given initial concentration

\[ c(z,0)=c_0(z):= \left\{\begin{array}{r@{\quad\mbox{if}\quad}l}
               c_0=const. & 0\le z\le h\\
               0 & h<z\le l.
               \end{array} \right. \]

(ii) Calculate \(\lim_{t\to\infty}\ c(z,t)\).

Q6.16

Solve the initial-boundary value problem (rotationally symmetric solution in a ball):
 find \(c(r,t)\) on \((0,R)\times(0,\infty)\) of
\begin{eqnarray}
\label{equr}
\frac{\partial c}{\partial t}&=&\frac{1}{r^2}\frac{\partial}{\partial r}\left(Dr^2\frac{\partial c}{\partial r}\right)-kc\\
\label{initial}
c(r,0)&=&h(r),\ 0<r<R,\\
\label{bc1r}
c(R,t)&=&c_0\qquad \mbox{(boundary condition)},\\
\label{bc2r}
\sup_{0<r<R,\;0<t<T}|c(r,t)|&<&\infty \qquad \mbox{(boundary condition)},
\end{eqnarray}
where \(T>0\) is fixed, \(k\), \(c_0\), \(D\) are positive constants,  and
$$
h(r)=\left\{\begin{array}{r@{\quad:\quad}l}
0&0<r<R_0\\
c_0\frac{r-R_0}{R-R_0}&  R_0<r<R
\end{array} \right. ,
$$
where \(0<R_0<R\) and \(R_0\) close to \(R\).

Q6.17

Prove the Black-Scholes formula for an European put option.

Hint: Put-call parity.

Q6.18

Prove the put-call parity for European options
$$
C(S,t)-P(S,t)=S-Ee^{-r(T-t)}
$$
by using the following uniqueness result: Assume \(W\) is a solution of (6.5.1) under the side conditions \(W(S,T)=0\), \(W(0,t)=0\) and \(W(S,t)=O(S)\) as \(S\to\infty\), uniformly on \(0\le t\le T\). Then \(W(S,t)\equiv 0\).

Q6.19

Prove that a solution \(V(S,t)\) of the initial-boundary value problem (6.5.1) in \(\Omega\) under the side conditions (i) \(V(S,T)=0\), \(S\ge0\), (ii) \(V(0,t)=0\), \(0\le t\le T\), (iii) \(\lim_{S\to\infty}V(S,t)=0\) uniformly in \(0\le t\le T\), is uniquely determined in the class \(C^2(\Omega)\cap C(\overline{\Omega})\).

Q6.20

Prove that a solution \(V(S,t)\) of the initial-boundary value problem (6.5.1) in \(\Omega\), under the side conditions (i) \(V(S,T)=0\), \(S\ge0\), (ii) \(V(0,t)=0\), \(0\le t\le T\), (iii) \(V(S,t)=S+o(S)\) as \(S\to\infty\), uniformly on \(0\le t\le T\), satisfies \(|V(S,t)|\le c S\) for all \(S\ge 0\) and \(0\le t\le T\).

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