# Parabolic 2. order partial differential equation

• MHB
• mathmari
In summary, the conversation is about finding the normal form of a second-order partial differential equation. The process involves determining the type of equation, finding the transformation variables, and substituting them into the equation to obtain the normal form. The final result should be in the form $u_{\xi \xi}=f(u,u_{\xi}, u_{\eta})$. After some discussion and calculations, it is confirmed that the result is correct.
mathmari
Gold Member
MHB
Hey!

I have to find the Normal form of the 2.order partial differential equation. I am not sure if my solution is correct..

The differential equation is:

$u_{xx}-4u_{xy}+4_{yy}-6u_x+12u_y-9u=0$

$a=1, b=-2, c=4$
$b^2-ac=4-4=0 \Rightarrow$ parabolic

$\frac{dy}{dx}=\frac{1}{a}(b \pm \sqrt{b^2-ac})$
$\frac{dy}{dx}=-2 \Rightarrow dy=-2dx \Rightarrow y=-2x+c \Rightarrow \xi=y+2x$

To find $\eta$ ,we know that the determinant of the Jacobi-Matrix should be different from $0$ .

$J=\begin{vmatrix} \xi_x & \xi_y\\ \eta_x & \eta_y \end{vmatrix}=\xi_x \eta_y-\xi_y \eta_x \neq 0$
$2 \eta_y-n_x \neq 0$

So we could take $\eta=y$, couldn't we?

In this case it is $2 \neq 0$.

$\partial_x=2\partial_{\xi}$

$\partial_y=\partial_{\xi}+\partial_{\eta}$

$\partial_{xx}=4\partial_{\xi \xi}$

$\partial_{yy}=\partial_{\xi \xi}+2 \partial_{\xi \eta}+\partial_{\eta \eta}$

$\partial_{xy}=2\partial_{\xi \xi}+2 \partial_{\xi \eta}$

By replacing this in the differential equation we have:

$u_{\eta \eta}+3u_{\eta}-\frac{9}{4}u=0$

Could you tell me if my result is correct?

Shouldn't it be in the form $u_{\xi \xi}=f(u,u_{\xi}, u_{\eta})$ ?
My result is in the form $u_{\eta \eta}=f(u,u_{\xi}, u_{\eta})$.. (Thinking)

mathmari said:
Hey!

I have to find the Normal form of the 2.order partial differential equation. I am not sure if my solution is correct..

The differential equation is:

$u_{xx}-4u_{xy}+4_{yy}-6u_x+12u_y-9u=0$

$a=1, b=-2, c=4$
$b^2-ac=4-4=0 \Rightarrow$ parabolic

$\frac{dy}{dx}=\frac{1}{a}(b \pm \sqrt{b^2-ac})$
$\frac{dy}{dx}=-2 \Rightarrow dy=-2dx \Rightarrow y=-2x+c \Rightarrow \xi=y+2x$

To find $\eta$ ,we know that the determinant of the Jacobi-Matrix should be different from $0$ .

$J=\begin{vmatrix} \xi_x & \xi_y\\ \eta_x & \eta_y \end{vmatrix}=\xi_x \eta_y-\xi_y \eta_x \neq 0$
$2 \eta_y-n_x \neq 0$

So we could take $\eta=y$, couldn't we?

In this case it is $2 \neq 0$.

$\partial_x=2\partial_{\xi}$

$\partial_y=\partial_{\xi}+\partial_{\eta}$

$\partial_{xx}=4\partial_{\xi \xi}$

$\partial_{yy}=\partial_{\xi \xi}+2 \partial_{\xi \eta}+\partial_{\eta \eta}$

$\partial_{xy}=2\partial_{\xi \xi}+2 \partial_{\xi \eta}$

By replacing this in the differential equation we have:

$u_{\eta \eta}+3u_{\eta}-\frac{9}{4}u=0$

Could you tell me if my result is correct?

Shouldn't it be in the form $u_{\xi \xi}=f(u,u_{\xi}, u_{\eta})$ ?
My result is in the form $u_{\eta \eta}=f(u,u_{\xi}, u_{\eta})$.. (Thinking)

Hey!

The calculations look good. :)

But I think you should assign the constant equation to the 2nd variable $\eta$.
As a result $\eta = C$ will correspond to the tangent line.

Consider that for a regular upright parabola $y=x^2$ the tangent line at the top is $y=0$.

You might use for instance:
\begin{cases}
\xi = x \\
\eta = 2x + y
\end{cases}

I like Serena said:
But I think you should assign the constant equation to the 2nd variable $\eta$.
As a result $\eta = C$ will correspond to the tangent line.

Consider that for a regular upright parabola $y=x^2$ the tangent line at the top is $y=0$.

You might use for instance:
\begin{cases}
\xi = x \\
\eta = 2x + y
\end{cases}

Using \begin{cases}
\xi = x \\
\eta = 2x + y
\end{cases} I found the following:$\partial_x=\partial_{\xi}+2 \partial_{\eta}$

$\partial_y=\partial_{\eta}$

$\partial_{xx}=\partial_{\xi \xi}+4 \partial_{\xi \eta} +4 \partial_{\eta \eta}$

$\partial_{yy}=\partial_{\eta \eta}$

$\partial_{xy}=\partial_{\xi \eta}+2 \partial_{\eta \eta}$

So substituting these at the differential equation we get:
$$u_{\xi \xi}-6u_{\xi}-9u=0$$

That is the form that it should be! (Yes)

Thank you for your help! (Smirk)

## What is a second-order partial differential equation?

A second-order partial differential equation is a mathematical equation that involves two independent variables and their partial derivatives up to the second order. It is commonly used to model physical phenomena such as heat flow, fluid dynamics, and wave propagation.

## What is a parabolic partial differential equation?

A parabolic partial differential equation is a type of second-order partial differential equation that describes phenomena that change over time and space, but the rate of change in one direction is much larger than the rate of change in the other directions. This results in a curved or parabolic shape in the solution.

## What is the physical significance of a parabolic partial differential equation?

Parabolic partial differential equations are commonly used in physics and engineering to describe diffusion processes, such as heat transfer and fluid flow. They also have applications in mathematical finance and population dynamics.

## What are some examples of parabolic partial differential equations?

The heat equation, the wave equation, and the diffusion equation are all examples of parabolic partial differential equations. These equations are used to model different physical phenomena, such as heat flow, wave propagation, and diffusion of particles or substances.

## How do you solve a parabolic partial differential equation?

There are various methods for solving parabolic partial differential equations, such as separation of variables, the method of characteristics, and numerical methods. Which method is used depends on the specific equation and its boundary conditions. In general, solving these equations requires knowledge of advanced mathematics and computational techniques.

• Differential Equations
Replies
1
Views
2K
• Differential Equations
Replies
2
Views
1K
• Differential Equations
Replies
1
Views
1K
• Quantum Physics
Replies
5
Views
646
• Differential Equations
Replies
1
Views
2K
• Differential Equations
Replies
1
Views
2K
• Differential Equations
Replies
7
Views
750
• Differential Equations
Replies
3
Views
2K
• Special and General Relativity
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
3
Views
906