# Can a distribution or delta function solve a NONlinear ODE or PDE

1. Sep 23, 2009

### zetafunction

the question is , can a delta function /distribution $$\delta (x-a)$$

solve a NOnlinear problem of the form $$F(y,y',y'',x)$$

the question is that in many cases you can NOT multiply a distribution by itself so you could not deal with Nonlinear terms such as $$(y)^{3}$$ or $$yy'$$

2. Sep 24, 2009

### trambolin

Suppose you kick a mass connected to a spring with a known nonlinear stiffness coefficient. And you measure the position, velocity and acceleration then you form your diff. equation. What should the right hand side should be?

Check out the term "impulse response" and you will get plenty of them.

The key is that Delta distribution makes sense under the integral sign.

3. Sep 24, 2009

### George Jones

Staff Emeritus
Yes, this often is impossible. In, for example, interacting quantum field theory, it is sometimes necessary to multiply distributions. This can be done if certain conditions are satisfied; see the work of Hormander and the book by Colombeau.
Since the LHS of this example involves second derivatives while the RHS is a delta funtion, the solution to the differential equation could be a continuous function (thus, not delta function-like) that has cusps (similar to the absolute value function). Differentiating once will give a jump discontinuity and differentiating again will give the delta function.