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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], §1.2, exercise 10.
(1) Solve the ordinary differential equation
$$
\frac{d x}{d s}(s)+x(s)=e^{3 s+2 c_0}
$$
where $c \in \mathbb{R}$ is an arbitrary constant.
(2) Deduce from (1) the solution to the partial differential equation
$$
\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+u=e^{x+2 y}
$$
of an unknown function $u \equiv u(x, y)$ of two variables, with the ‘boundary condition’ $u(x, 0)=0$.
This exercise is optional, and is reprinted from [Strauss], $\S 1.2$, exercise 7.
(1) Solve the partial differential equation
$$
y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial x}=0
$$
of an unknown function $u \equiv u(x, y)$ of two variables, with the ‘boundary condition’ $u(0, y)=y^3$.
(2) In which region of the $x y$ plane is the solution uniquely determined?
This exercise is partly reprinted from [Strauss], §1.6, Exercises $1-2$.
(1) What are the types of the following two second-order PDE ?
(a) $\frac{\partial^2 u}{\partial x^2}+\frac{\partial u}{\partial x}-3 \frac{\partial^2 u}{\partial x \partial y}+\frac{\partial u}{\partial y}-\frac{\partial^2 u}{\partial y^2}=0$.
(b) $9 \frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial x \partial y}+3 \frac{\partial^2 u}{\partial y^2}+\frac{\partial u}{\partial x}=0$.
(2) Find the regions of the $(x, y)$ plane where the PDE:
$$
(1+x) \frac{\partial^2 u}{\partial x^2}+2 x y \frac{\partial^2 u}{\partial x \partial y}+y^2 \frac{\partial^2 u}{\partial y^2}=0
$$
of a function $u \equiv u(x, y)$ is elliptic, parabolic, and hyperbolic. Sketch those regions.
This exercise is reprinted from [Strauss], §1.6, Exercise 6.
Consider the partial differential equation:
$$
3 \frac{\partial u}{\partial y}+\frac{\partial^2 u}{\partial x \partial y}=0
$$
where the unknown $u \equiv u(x, y)$ is a function of two real variables.
(1) What is the order of this PDE? What is its type?
(2) Find the general solution of this PDE. (Hint: make the change of unknown function $v:=\frac{\partial u}{\partial y}$, and solve first the corresponding PDE of the unknown function $v)$.
(3) If this equation is supplemented with the boundary conditions:
$$
\forall x \in \mathbb{R}, u(x, 0)=e^{-3 x}, \text { and } \frac{\partial u}{\partial y}(x, 0)=0 ;
$$
does it admit a solution? Is this solution unique?
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