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Mathematics LibreTexts

6.E: Parabolic Equations (Exercises)

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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