Classification of PDEs

[Lizwi]

Hlw. Still I failing to understand the difference between Linear, semilinear and Quasilinear pde even after reading from Google.

 

Can anyone clarify for me please.

[studiot]

5 hours ago, Lizwi said:

Hlw. Still I failing to understand the difference between Linear, semilinear and Quasilinear pde even after reading from Google.

 

Can anyone clarify for me please.

Hi, Liz,

What does your textbook say  ?

I ask this because there are several different ways to approach this and I don't want to confuse you with a different one than you are used to.

does your textbook use

x²U_xx - yU_xy = U   Which is linear

and

x²U_xx - yU_xy = U² Which is non linear

 

or the same two equations written like this

[math]{x^2}\frac{{{\partial ^2}U}}{{\partial {x^2}}} - y\frac{{{\partial ^2}U}}{{\partial x\partial y}} = U[/math]

and

[math]{x^2}\frac{{{\partial ^2}U}}{{\partial {x^2}}} - y\frac{{{\partial ^2}U}}{{\partial x\partial y}} = {U^2}[/math]

 

Or possibly even like this. This is known as operator notation.

Note these are different eqautions from before.

[math]L\left[ U \right] = \left[ {\frac{{{\partial ^2}U}}{{\partial {x^2}}} - {c^2}\frac{{{\partial ^2}U}}{{\partial u\partial y}}} \right][/math]

which is linear
and

[math]L\left[ U \right] = \left[ {{{\left( {\frac{{\partial U}}{{\partial x}}} \right)}^2} - {c^2}{{\left( {\frac{{\partial U}}{{\partial y}}} \right)}^2}} \right][/math]

Which is non linear

 

It is important to be able to separate PDEs into linear and non linear as a first step.

 

 

 

Edited December 29, 2019 by studiot

[Lizwi]

They said " Linear equation is the one in which there is no products of the dependent variable and its derivative. 

[studiot]

24 minutes ago, Lizwi said:

They said " Linear equation is the one in which there is no products of the dependent variable and its derivative. 

This is completely true.

But it does not answer my question.

to explain quasi linear and semi linear we need to write out some example equations.

So which notation is your book using for the PDE ?

[Lizwi]

For example

Is the first equation linear or non linear?

[studiot]

1 hour ago, Lizwi said:

Is the first equation linear or non linear?

Thank you for your answer.

Note that all semi linear and quasi linear equations are actually non linear but they can often be reduced to a set of coupled or simultaneous linear ones.

You first equation is non linear because

u is the dependent variable
x, t are the independent variables

so we only look at u and its derivatives.

The first term is linear because the first derivative (of u) is not multiplied by another function of u or its derivatives.

The second term is linear because the second derivative (of u) is not multiplied by another function of u or its derivatives.

But the third term is non linear because it contains u²  .

 

Have you identified the dependent and independent variables in the other equations and the non linear parts ?

 

[Lizwi]

Yes, I understand the variables.

 

Thanks, your explanation is very clear. I think I can classify PDEs now according to their linearity.