Is a PDE non-linear if it has a term containing the product of a function with its derivative (over either time or space)?

kapalani

What is the physical significance of the Burgers' equation? Is it used in graphics anywhere?

dsaksena

edit:

from wiki I see burgers' equation is used in fluid mechanics to use viscosity make a fluid move (like we did with pendulum, maybe)

kmcrane

@kapalani: Yes, if a function gets multiplied with its derivative in a differential equation, then it is nonlinear. (You can think of this as the "0th derivative" times one of the other derivatives.)

kmcrane

@kapalani: Burgers' equation can be thought of as a simple model equation for advection, and can be helpful for understanding the (more challenging) Navier-Stokes equations, for instance.

dsaksena

@kmcrane, trying to visualize this,
assume a simple u = x + t (which is linear)

u' = t

uu' = xt + t^2 --- not linear anymore

for

u = x (which is linear)

u' = 1

uu' = x

so we need u to be in 2 variables for this non-linearity?

kmcrane

@dsaksena: Not 100% sure I understand what you're asking, but here's a nonlinear ODE (depends only on one variable): $\tfrac{d}{dt} u(t) = u(t)^2$.

dsaksena

Professor I was trying to prove by an example that the term uu' is non linear (what you answered above to @kapalani)

I took u(x,t) to be a linear function as an example and showed it becomes non-linear if it was linear in 2 variables (x,t) but not if it was only linear in x (u(x)).

I think it might be incomplete and my assumption is too much a special case and not in the context of the differential equations but just functions I'll think more on this later.

kmcrane

@dsaksena: Ok! Let me know if you want to chat more.

Is a PDE non-linear if it has a term containing the product of a function with its derivative (over either time or space)?

What is the physical significance of the Burgers' equation? Is it used in graphics anywhere?

edit:

from wiki I see burgers' equation is used in fluid mechanics to use viscosity make a fluid move (like we did with pendulum, maybe)

@kapalani: Yes, if a function gets multiplied with its derivative in a differential equation, then it is nonlinear. (You can think of this as the "0th derivative" times one of the other derivatives.)

@kapalani: Burgers' equation can be thought of as a simple model equation for advection, and can be helpful for understanding the (more challenging) Navier-Stokes equations, for instance.

@kmcrane, trying to visualize this, assume a simple u = x + t (which is linear)

u' = t

uu' = xt + t^2 --- not linear anymore

for

u = x (which is linear)

u' = 1

uu' = x

so we need u to be in 2 variables for this non-linearity?

@dsaksena: Not 100% sure I understand what you're asking, but here's a nonlinear ODE (depends only on one variable): $\tfrac{d}{dt} u(t) = u(t)^2$.

Professor I was trying to prove by an example that the term uu' is non linear (what you answered above to @kapalani)

I took u(x,t) to be a linear function as an example and showed it becomes non-linear if it was linear in 2 variables (x,t) but not if it was only linear in x (u(x)).

I think it might be incomplete and my assumption is too much a special case and not in the context of the differential equations but just functions I'll think more on this later.

@dsaksena: Ok! Let me know if you want to chat more.