Author Topic: Classification of PDEs (Chapter Problems 1) clarification  (Read 1340 times)

Tristan Fraser

  • Full Member
  • ***
  • Posts: 30
  • Karma: 11
    • View Profile
Classification of PDEs (Chapter Problems 1) clarification
« on: January 07, 2018, 08:08:17 PM »

From the textbook, we defined ( I might be paraphrasing)

Quote
"Equations of the form $Lu = f(X) $ where Lu is a linear partial differential expression with respect to an unknown function u is called a Linear Equation. The equation is homogenous if $ f  = 0 $ and inhomogenous otherwise
and used the example of $ Lu := a_{11}u_{xx} + 2a_{12}u_{xy} +a_{22}u_{yy} + a_{1}u_{x} + a_{2}u_{y} + au = f(x) $ (i) . We also know it is Non linear if it cannot be expressed in the form (i).

This leads me to my question as Problem 1 asks us to classify equations (1-10), if they are linear homogenous, inhomogenous or , non linear
and for two of them:
$ u_{t} + uu_{x} = 0 $ (2) and $ u_{t} + uu_{x} + u = 0$ (4).

Paraphrasing the text: 
Quote
If $Lu = f(x) $ , and both the coefficients and right hand side expression depend on lower order derivatives, then the equation is Quasilinear

So does this mean that eqns (2) and (4) are Quasilinear then? Is that wholly seperate from the options of Linear Homogenous, Linear Inhomogenous and Nonlinear, or are Quasilinear and Semilinear sub-catagories of those?

« Last Edit: January 08, 2018, 03:26:16 PM by Victor Ivrii »

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2553
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #1 on: January 07, 2018, 11:08:26 PM »
Quasinear are subclass of Nonlinear , and Semilinear are subclass of Quasilinear


It will not be an error to call $u_{xx}+u_{yy}=u^2$ nonlinear or quasilinear, but it is in fact semilinear (linear expression, plus non-linear expression, depending only on lower order terms).

It will not be an error to call $u_{x}+u^2u_{y}=0$ nonlinear, but it is in fact quasilinear.

These problems ask for the most precise classification

Jingxuan Zhang

  • Elder Member
  • *****
  • Posts: 106
  • Karma: 20
    • View Profile
Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #2 on: January 08, 2018, 01:36:02 PM »
The term $uu_{x}$ determines $(2),(4)$ to be both Quasilinear.
« Last Edit: January 08, 2018, 03:25:09 PM by Victor Ivrii »

Zhuoyan Zou

  • Newbie
  • *
  • Posts: 1
  • Karma: 0
    • View Profile
Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #3 on: January 11, 2018, 06:46:47 PM »
I have some more questions regarding classification; from the homework questions:

(1) ut+ux-u2=0
(2) ut2-ux2-1=0
(3) xux+yuy+zuz=0

My answers are semi-linear for all of them since the coefficients are not dependent on u, and they are not linear operator. Please help me if I am wrong. I am not quite sure about the definition of semi-linear.
Quote
Equations of the type (11) where the right-hand expression f depend on the lower-order derivatives are called semilinear.
  What does it exactly mean the right-hand expression f? For example, for (1), if f=u2, then it's not semi-linear.

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2553
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #4 on: January 11, 2018, 07:12:19 PM »
Quote
What does it exactly mean the right-hand expression?
Quote
It is only for semi-linear. That you can move some terms containing lower order derivatives only (including function) to the right, so on the left only linear part remains.




Jaisen Kuhle

  • Full Member
  • ***
  • Posts: 20
  • Karma: 2
    • View Profile
Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #5 on: January 12, 2018, 02:19:54 PM »
I have some more questions regarding classification; from the homework questions:

(1) ut+ux-u2=0
(2) ut2-ux2-1=0
(3) xux+yuy+zuz=0

My answers are semi-linear for all of them since the coefficients are not dependent on u, and they are not linear operator. Please help me if I am wrong. I am not quite sure about the definition of semi-linear.
Quote
Equations of the type (11) where the right-hand expression f depend on the lower-order derivatives are called semilinear.
  What does it exactly mean the right-hand expression f? For example, for (1), if f=u2, then it's not semi-linear.

(1) ut+ux-u2=0

Semi-Linear because of u2

(2) ut2-ux2-1=0

Fully Non-Linear because of ut2, so not quasi or semi linear.

3) xux+yuy+zuz=0

Linear Homogenous with variable coefficients.

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2553
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Classification of PDEs (Chapter Problems 1) clarification
« Reply #6 on: January 12, 2018, 05:01:16 PM »
Please, do use MathJax which is supported and is far superior to html in displaying mathematical snippets.