Classification and standard formula of second order linear partial differential equations

Updated: 9 APR 2016

======== Method ========

For arbitrary two-element order homogeneous linear partial differential equations,

\ (a_{11}\dfrac{\partial^2u}{\partial x^2}+2a_{12}\dfrac{\partial^2 u}{\partial x\partial y}+a_{22}\dfrac{\partial^ 2 u}{\partial y^2}+b_1\dfrac{\partial u}{\partial x}+b_2\dfrac{\partial u}{\partial y}+cu=0\)

The method of finding the characteristic equation, determining the classification and translating it into the standard type is:

1. Cut: Only cares about the second part

\ (a_{11}\dfrac{\partial^2u}{\partial x^2}+2a_{12}\dfrac{\partial^2 u}{\partial x\partial y}+a_{22}\dfrac{\partial^ 2 u}{\partial y^2}=0\)

2. change: Convert the partial derivative into dx, dy

\ (\dfrac{\partial^2 u}{\partial x^2} \rightarrow (dy) ^2\)

\ (\dfrac{\partial^2 u}{\partial y^2} \rightarrow (dx) ^2\)

\ (\dfrac{\partial^2 u}{\partial x\partial y} \rightarrow \color{red}{\textbf{–}} (Dxdy) \) Pay attention to the minus sign!

\ (a_{11} (DY) ^2\color{red}{\textbf{–}}a_{12} (Dxdy) +a_{22} (DX) ^2=0\)

This is the characteristic equation .

3. points: Coefficients in the characteristic equation \ (a_{ij}\) may be a function of x, Y, even though it is still regarded as a common coefficient. This is a two-time equation that can write its \ (\delta\)

\ (\delta=a_{12}^2-a_{11}a_{22}\)

The symbol is discussed within the definition field of the equation,

\ (\delta>0\) elliptic (elliptic) equation, such as Laplace equation;

\ (\delta=0\) parabolic (parabolic) equation, such as dimensional thermal equation;

\ (\delta<0\) hyperbolic (hyperbolic) equation, such as a dimensional wave equation.

This is the classification of the equation. There are also mixed types.

4. anti: Solve the relationship between DY and DX, it may be simpler to use decomposition-type method. Get two ordinary differential equations (or one).

When we get two equations, we solve the relationship between two Y and X, and note that there is an integral constant in each of the two equations. The integral constant is used as the new coordinate (denoted by \ (\xi, \eta\)), and the two equations are written back, that is, the two coordinates are represented by x and Y. In this case, variable substitution is performed.

For a parabolic equation, an ordinary differential equation is obtained, and a coordinate is obtained, and the other coordinate can be arbitrarily assumed, as set as Y. Pay attention to do partial differential can not be replaced directly, need to use chain law calculation.

5. Guide: Calculates u about x, Y's first order, second derivative and mixed partial derivative, denoted by \ (\xi, \eta\).

6. substituting: The above partial derivative is put into the original equation, and a simplified equation with \ (\xi, \eta\) as the independent variable is obtained. This is the standard type of equation.

7. Solution: According to the standard solution, get the general solution about \ (\xi, \eta\). Substitution back to X, Y uses boundary conditions to solve.

======== principle ========

The reason why we write the characteristic equation is because here we want the original equation

\ (a_{11}\dfrac{\partial^2u}{\partial x^2}+2a_{12}\dfrac{\partial^2 u}{\partial x\partial y}+a_{22}\dfrac{\partial^ 2 u}{\partial y^2}+b_1\dfrac{\partial u}{\partial x}+b_2\dfrac{\partial u}{\partial y}+cu=0\)

Into

\ (a_{11}\dfrac{\partial^2u}{\partial \xi^2}+2a_{12}\dfrac{\partial^2 u}{\partial \xi\partial \eta}+A_{22}\dfrac{\ partial^2 u}{\partial \eta^2}+b_1\dfrac{\partial u}{\partial \xi}+b_2\dfrac{\partial u}{\partial \eta}+Cu=0\)

where variable substitution

\ (\xi=\varphi (x, y) \)

\ (\eta=\psi (x, y) \)

Classification and standard formula of second order linear partial differential equations