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Math372课程简介
Course description
A Partial Differential Equation (PDE for short), is a differential equation involving derivatives with respect to more than one variable. These arise in numerous applications from various disciplines. A prototypical example is the `heat equation’, governing the evolution of temperature in a conductor.
Usually finding explicit solutions for even the simplest (LINEAR) PDE’s is a formidable task, which doesn’t always have a tractable solution. The mathematical study of PDE’s usually focuses on deducing properties of solutions, without use of an explicit solution formula. For instance, the fact that heat doesn’t collect at hot points is a consequence of the “Maximum principle”; a fundamental theorem about solutions to the heat equation, which also applies to solutions of a more general class of equations.
Prerequisites
This course will serve as a conceptual introduction to PDE’s differential equations, focussing more on studying properties of solutions and less on finding explicit (and horrendously complicated) solutions. It is aimed at undergraduate Math majors, however is suitable for students from Physics, Engineering and other disciplines who want to develop a more conceptual understanding of the subject.
References
Introduction to PDE by Walter Strauss. (Strongly recommended! Homework problems will be assigned from here.)
Basic Partial Differential Equations by Bleecker and Csordas.
Math372 Partial Differential Equations HELP（EXAM HELP， ONLINE TUTOR）
This exercise is partly reprinted from [Strauss], §1.1, exercise 3.
For each of the following partial differential equations, (1) write the associated differential operator, (2) state the order of the equation, and (3) whether it is linear, homogeneous. When it is nonlinear, justify.
(1) $\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=x^4$, where $u \equiv u(t, x)$
(2) $\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}+x u=0$, where $u \equiv u(t, x)$.
(3) $\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}+u \frac{\partial u}{\partial x}=0$, where $u \equiv u(t, x)$.
(4) $\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}+x^2=0$, where $u \equiv u(t, x)$.
(5) $i \frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}+\frac{u}{x}=0$, where $u \equiv u(t, x)$.
(6) $\frac{\partial u}{\partial x}\left(1+\left(\frac{\partial u}{\partial x}\right)^2\right)^{-\frac{1}{2}}+\frac{\partial u}{\partial y}\left(1+\left(\frac{\partial u}{\partial y}\right)^2\right)^{-\frac{1}{2}}=0$, where $u \equiv u(x, y)$.
(7) $\frac{\partial u}{\partial x}+e^y \frac{\partial u}{\partial y}=0$, where $u \equiv u(x, y)$.
(8) $i \frac{\partial u}{\partial t}+\frac{\partial^4 u}{\partial x^4}+\sqrt{1+\frac{\partial u}{\partial x}}=0$, where $u \equiv u(t, x)$.
This exercise is partly reprinted from [Strauss], §1.1, exercise 11.
(1) Verify that, for all pairs $(f, g)$ of differentiable functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$, the function $u(x, y)=f(x) g(y)$ is a solution to the PDE:
$$
u \frac{\partial^2 u}{\partial x \partial y}=\frac{\partial u}{\partial x} \frac{\partial u}{\partial y}
$$
(2) Find all the solutions to the PDE
$$
\frac{\partial u}{\partial x}+u=0
$$
where the unknown function $u$ is a function of two variables $u \equiv u(x, y)$. ‘How many’ solutions do you end up with ?
(3) Same question as (2), with the additional ‘boundary condition’ $u(0, y)=y^3$ for all $y \in \mathbb{R}$.
This exercise is partly reprinted from [Strauss], §1.2, exercise 4.
(1) Solve the partial differential equation
$$
-\left(1+x^2\right) \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0
$$
by using the method of characteristics, where the unknown function $u$ is a function of two variables $u \equiv u(x, y)$
(2) Draw several characteristic curves.
(3) Verify that the solutions you end up with indeed solve the considered PDE, by computing its partial derivatives, and evaluating
$$
\mathcal{L}(u)=-\left(1+x^2\right) \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}
$$
This exercise is reprinted from [Haberman], §12.2, exercise 5 .
Solve the partial differential equation
$$
\frac{\partial u}{\partial t}+c \frac{\partial u}{\partial x}=e^{2 x}
$$
of an unknown function of two variables $u \equiv u(t, x)$, where $c \in \mathbb{R}$ is a fixed parameter, and with the additional ‘initial condition’ $u(0, x)=f(x)$, where $f$ is a given function. Draw several characteristic curves.
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