# Non-linear operator

1. Sep 24, 2009

### Mark_Twain_MO

Hello,

Could anyone help explain why

$$\partial$$/$$\partial$$x[K(x,u)$$\partial$$u/$$\partial$$u]

is not a linear operator?

2. Sep 24, 2009

### Coto

Are you sure you have that written down right?

$$\frac{\partial u}{\partial u} = 1$$

3. Sep 24, 2009

### Mark_Twain_MO

Sorry, I meant:

$$\partial$$/$$\partial$$x[K(x,u)$$\partial$$u/$$\partial$$x]

I understand that

$$\partial$$/$$\partial$$x[K(x)$$\partial$$u/$$\partial$$x]

is a linear operator, but I do not get why making K a function of both x and u should non-linearize this operator.

4. Sep 24, 2009

### Coto

The best way to test quickly for a linear operator is just to plug in the definition... that is put in

$$\lambda _1 u_1 + \lambda _2 u_2$$

to your operator above. Use the rules of differentiation and see what comes out. This would be a good exercise.

It seems there might have to be some conditions on K(x, u) for the above to be non-linear. I'm out the door myself, so hope this helps.

5. Sep 24, 2009

### g_edgar

If K(x,u) = u, then what is your operator? Is it linear?

6. Sep 26, 2009

### HallsofIvy

Staff Emeritus
It's not linear because you have a function of the dependent variable, u, K(x,u), multiplying the derivative of u, $\partial u/\partial x$.

(But why are you using the partial derivative if u is a function only of x? Is there another term in the equation with another independent variable?)

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