数学代写|偏微分方程|MATH 319 Introduction to Partial Differential Equations

Question 1 ( 10 marks).
(a) For which $a, b \in \mathbb{R}$ is $e^{a t} \sin (b x)$ is a solution to the heat equation
$$
u_{t}-k u_{x x}=0 ?
$$
(b) Prove that if $u$ is a $C^{3}$ function that solves the heat equation $u_{t}-k u_{x x}=0,$ then so does $x u_{x}+2 t u_{t}$

Solve the equation
$$
u_{x}+u_{y}+u=0, \quad u(x, 0)=x^{2}
$$

Question 3 (10 marks). Consider the following problem:
$$
u_{t}+x u_{x}=x, \quad u(x, 0)=0
$$
(a) Find the equation to the characteristic curves, and sketch some of them.
(b) Solve the equation by considering the restriction $u$ to the characteristic curves.

(10 marks).Use the method of subtraction to solve the heat equation with Dirichlet boundary condition on the half line:
$$
\begin{array}{ll}
u_{t}-k u_{x x}=0, & x>0, \quad t>0 \\
u(x, 0)=1-x, & x>0 \\
u(0, t)=\sin (t), & t>0
\end{array}
$$
(Give the solution as an integral over the domain $(0, \infty),$ you do not need to plug in the specific form of the function $S(x, t)$ or perform any simplification that come afterwards.)

(10 marks). (a) Let $u$ solves the initial value problem
$$
\begin{array}{l}
u_{t t}-c^{2} u_{x x}=0, \quad x \in \mathbb{R}, t \in \mathbb{R} \\
u(x, 0)=0, x \in \mathbb{R}, \quad u_{t}(x, 0)=\left\{\begin{array}{ll}
0 & x>0 \\
1 & x \leq 0
\end{array}\right.
\end{array}
$$
Sketch the region of $(x, t)$ for which $u(x, t)=0$. Justify your answer.
(b) Let $v$ solve the initial value problem
$$
\begin{array}{l}
v_{t t}-c^{2} v_{x x}=0, \quad x \in \mathbb{R}, t \in \mathbb{R} \\
v(x, 0)=\left\{\begin{array}{ll}
-1 & x<0 \\
1 & x \geq 0
\end{array} \quad v_{t}(x, 0)=0\right.
\end{array}
$$
Sketch the region of $(x, t)$ for which $v(x, t)=0$. Justify your answer.

Question 6 (4 marks). Consider the equation
$$
u_{t}-k u_{x x}-u=0
$$
(a) By letting $u(x, t)=v(x, t) e^{a t}$ for s suitable $a \in \mathbb{R},$ reduce the equation to a standard heat equation.
(b) Let $R=[0, l] \times[0, T]$ and $\Sigma=\{(x, t) \in \mathbb{R}: x=0$ or $x=l$ or $t=0\} .$ Prove that
$$
\max _{(x, t) \in R} u(x, t) \leq e^{T} \max _{(x, t) \in \Sigma} u(x, t)
$$

Question 7 (4 marks). Solve the wave equation on the half line with the Neumann boundary condition
$$
\begin{array}{l}
u_{t t}-c^{2} u_{x x}=0, \quad x>0, t>0 \\
u_{x}(0, t)=h(t), \quad t>0 \\
u(x, 0)=u_{t}(x, 0)=0, \quad x>0
\end{array}
$$
(Hint: Use the general formula $u(x, t)=f(x+c t)+g(x-c t)$. You only need to solve for formulas for $f$ and $g$ in terms of the function $h,$ i.e. you don’t need to discuss the different cases of the piecewise defined function.)