Author Topic: The definition of semilinear equation in 2.1 of textbook  (Read 177 times)

Luyu CEN

  • Jr. Member
  • **
  • Posts: 12
  • Karma: 0
    • View Profile
The definition of semilinear equation in 2.1 of textbook
« on: September 27, 2016, 05:22:03 PM »
In the textbook, it reads
Definition 2. If a=a(x,t) and b=b(x,t) equation is semilinear.
But in the previous section this should be linear equation with variable cofficients, right?
I think the next definition is consistent with what I know.
Definition 3. Furthermore if f is a linear function of u: f=c(x,t)u+g(x,t) original equation is linear.
In this case the last ODE is also linear.
But I don't understand this sentence. Is there an ODE?

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 1332
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: The definition of semilinear equation in 2.1 of textbook
« Reply #1 on: September 27, 2016, 06:09:54 PM »
In the textbook, it reads
Definition 2. If a=a(x,t) and b=b(x,t) equation is semilinear.
But in the previous section this should be linear equation with variable cofficients, right?
I think the next definition is consistent with what I know.
It may be linear, but generally semilinear:
$$a(x,t)u_t+b(x,t)u_x=f(x,t,u)$$
Quote
Definition 3. Furthermore if f is a linear function of u: f=c(x,t)u+g(x,t) original equation is linear.[/center]
In this case the last ODE is also linear.
But I don't understand this sentence. Is there an ODE?
After we found integral curves $x=x(t,c)$ satisfying $\frac{dt}{a(x,t)}=\frac{dx}{b(x,t)}$ we have  ODE $\frac{du}{dt}=\frac{df(x,t,u)}{a(x,t)}$ with $x=x(t,c)$ along integral lines. For semilinear PDE this ODE is non-linear, for linear PDE this ODE  is linear.