1

In textbook, it says that

If a differential equation (DE) contains only ordinary derivatives of one or more unknown functions with respect to a single independent variable, then it is called an ODE; An equation involving partial derivatives of one or more unknown functions of two or more independent variables is called a partial DE (PDE).

Based on the definitions, it appears that the following DE does not belong to either type. Is my understanding correct? $$\frac{du(x)}{dx}+\frac{dv(y)}{dy}=u(x)+v(y),$$ where $u$ and $v$ are functions of independent variables $x$ and $y$, respectively. I know this DE might be weird from the mathematical modeling perspective, but I just want to check the definitions since I thought we only have ODEs and PDEs.

Chasel Weng
  • 186
  • 2
    isn't this equivalent to a pair of decoupled ODE's ? ($du/dx=u$, $dv/dy=v$) – Carlo Beenakker Sep 06 '23 at 11:37
  • Yes, it is. So, my question can be rephrased as "Is the sum of two ODEs an ODE?" – Chasel Weng Sep 06 '23 at 11:40
  • 1
    I think that a partial derivative is a special case of an ordinary derivative, when it is applied to a function of only one variable. With that in mind, clearly this is a partial differential equation, not an ordinary differential equation. – Ben McKay Sep 06 '23 at 12:43
  • Is it another way around? Ordinary derivative is a special case of partial derivative, when it is applied to a function of only one variable? From this viewpoint, every ODE is a PDE. However, the definitions in textbook tend to distinguish two cases. – Chasel Weng Sep 06 '23 at 13:05
  • Sorry, yes the other way around. – Ben McKay Sep 06 '23 at 13:23
  • You wrote: my question can be rephrased as "Is the sum of two ODEs an ODE?". I would answer yes, as soon as you accept vector valued ODEs. – DamienC Sep 06 '23 at 13:47
  • If you only care about the terminology, your example is a "non-local PDE" because it relates the values of the unknown functions at different points of the domain of the independent variable. Sometimes there is definite relation between the points, like $y=x-1$, this non-locality might be called a "delay". In this case you could also say ODE instead of PDE, because the unknown function still depends only on one variable. – Igor Khavkine Sep 06 '23 at 15:42
  • If the non-locality is expressed in terms of an integral, you would get an "integro-differential equation". All of these are anyway special cases of "functional equations". – Igor Khavkine Sep 06 '23 at 15:47
  • Since there are 2 variables, it is definitely not an ODE. We may see it as a PDE with an Ansatz on the unknown functions. Or also, a system of PDE's for 2 unknown functions u(x,y) , v(x,y), adding 2 more equations, $\partial u/\partial y=0$ and $\partial v/\partial x=0$. – Pietro Majer Sep 06 '23 at 17:15
  • 1
    Thanks everyone for your comments :)

    @PietroMajer Does ``an Ansatz for the unknown functions'' mean that "consider u(x) as u(x, y) and v(y) as v(x, y), even they are only functions of x and y, respectively. Then, we can rewrite du/dx as ∂/∂ and dv/dy as ∂v/∂, becoming a PDE"?

     – Chasel Weng Sep 07 '23 at 04:15
  • Exactly. You may think that equation as $\partial f/\partial x+\partial f/\partial y = f$ written for solutions of the special form $f(x,y)=u(x)+v(y)$ – Pietro Majer Sep 07 '23 at 05:04

0 Answers0