How to recognize nonlinearity

[fisico30]

how to recognize nonlinearity...

hello Forum!

in an ordinary or partial differential equation, a nonlinear term is recognizable if the dependent variable y is present in the following form:

1) y^2, y^3, log(y), e^y...
2) If the derivatives of y (which can be of any order) are raised to a power higher than one.
3) if y or any of its derivatives are either multiplied by each other, or by functions of y.

I am unsure about this case:

Take the definition of acceleration a = $$dv/dt$$, which is a linear term if present in an equation with v being the dependent variable.
But this ratio can also be written as the product $$v$$ $$\frac{dv}{dx}$$ , which appear to be nonlinear in v...

What is wrong?
thanks for any help.

 

[Mapes]

Hello fisico30!

There's no contradiction. It's a linear diff eq when considering the relationship between v and t, nonlinear for v and x. As a result, for constant acceleration we have

$$at=v-v_0$$

$$2ax=v^2-v_0^2$$

In a similar way, one equation is linear and one is quadratic, but it's the same v.

 

[arildno]

To belabor the point Mapes made:
A diff.eq might perfectly well be non-linear with respect to some set of variables (dependent, and our independent), but, when suitably re-formulated with respect to ANOTHER set of variables, its translated form is a linear diff.eq.

Thus, "linear" and "non-linear" are terms that only possesses meaning with respect to a given set of variables, in which the diff. eq is expressed.

 

[fisico30]

Thanks both Mapes and arildno.

That is very clear.
It seems that a differential equation trying to describe a physical phenomenon via a dependent variable can really be of different types depending on the domain of existence and on the independent variables.

A PDE like Navier-Stokes is a prime example: for v(x,y,x,t) there is a $$\frac{dv}{dt}$$ term which is linear and the $$\{v}$$$$\frac{dv}{dx}$$ which is nonlinear...

Also,
1) For linear PDEs, there is a distinction between parabolic, hyperbolic, and elliptic and depending on B^2-4AC. Is there a similar distinction for nonlinear equations?

2) can you tell from just analyzing a nonlinear term in an equation what type of behavior the equation/phenomenon has?

3)The word nonlinearity, to me, suggests "mixing". Is that a correct interpretation? How can I see that in a simple nonlinear equation? Does the the nonlinear equation has to be a vector equation with coupling between components to see that? What if it is just a scalar nonlinear ODE. What type of mixing, if any, occurs then?

with gratitude,
fisico30

 

[arildno]

Thanks both Mapes and arildno.

That is very clear.
It seems that a differential equation trying to describe a physical phenomenon via a dependent variable can really be of different types depending on the domain of existence and on the independent variables.

A PDE like Navier-Stokes is a prime example: for v(x,y,x,t) there is a $$\frac{dv}{dt}$$ term which is linear and the $$\{v}$$$$\frac{dv}{dx}$$ which is nonlinear...

Also,
1) For linear PDEs, there is a distinction between parabolic, hyperbolic, and elliptic and depending on B^2-4AC. Is there a similar distinction for nonlinear equations?

Not to my knowledge.

2) can you tell from just analyzing a nonlinear term in an equation what type of behavior the equation/phenomenon has?

Very occasionally. Most often, no.

For example, one nasty feature with non-linear equations is that the may exhibit singularities dependent upon initial conditions. This is not the case with linear diff.eqs, whose singularities (if any!) will always coincide with the singularities in the coefficient functions.

3)The word nonlinearity, to me, suggests "mixing". Is that a correct interpretation? How can I see that in a simple nonlinear equation? Does the the nonlinear equation has to be a vector equation with coupling between components to see that? What if it is just a scalar nonlinear ODE. What type of mixing, if any, occurs then?

with gratitude,
fisico30

Very good question!

In order to answer this, we need the concept of an "differential operator":

We can regard an operator as "doing something" to our unkown function:

Suppose we have a first-order diff eq like:
$$\frac{dy}{dx}+A(x)y=B(x)$$
Now, we can think of
$$L=\frac{d}{dx}+A(x)$$ as something which is "done to" (or sort of "multiplied with") our function "Y", so that we may rewrite our diff.eq as:
$$L(y)=B(x)$$
(Note the similarity of L-notation with "normal" function notation!)

We can then state the condition for "linearity":
A diff.eq is "linear" if and only if its associated operator L, for constants a, b, and functions y(x), z(x) follws the principle:
$$L(ay+bz)=aL(y)+bL(z)$$
That is, the output of an operator applied to a linear combination of two functions should be the same linear combination of the "operated" output of the two functions involved.

To see that the given example IS linear, we have:
$$L(ay+bz)=\frac{d}{dx}(ay(x)+bz(x))+A(x)*(ay(x)+b(z(x))=a(\frac{dy}{dx}+A(x)y)+b(\frac{dz}{dx}+A(x)z)=aL(y)+bL(z)$$
Thus, linearity of operator is shown!

Let us not take another diff.eq:
$$\frac{dy}{dx}y=B(x)$$
Here, let us regard the operator L as applied to a sum of two functions y(x) and z(x).

$$L(y+z)=(\frac{d}{dx}(y+z))(y+z)=(\frac{dy}{dx}+\frac{dz}{dx})(y+z)=\frac{dy}{dx}y+\frac{dz}{dx}z+\frac{dy}{dx}z+\frac{dz}{dx}y\neq{L}(y)+L(z)= \frac{dy}{dx}y+\frac{dz}{dx}z$$
Thus, L in this case is non-linear.

Thus, your idea that "non-linearities" sort of mix together solutions or at least functions is spot on; the associated differential operator is unable to keep the two component functions separate as a linear combination, cross-terms of mixture appears.

 

[fisico30]

My compliments, nice answer arildno!

regarding your comment:

For example, one nasty feature with non-linear equations is that the may exhibit singularities dependent upon initial conditions. This is not the case with linear diff.eqs, whose singularities (if any!) will always coincide with the singularities in the coefficient functions

where can I find some references about "singularities dependent upon initial conditions" in nonlinear equations? Any suggestion on a clear textbook, or examples where this is discussed?

thanks!

 

[Mapes]

1) For linear PDEs, there is a distinction between parabolic, hyperbolic, and elliptic and depending on B^2-4AC. Is there a similar distinction for nonlinear equations?

You can surely classify them by various means (the power of the coefficient, for example). Also, many nonlinear differential equations are named, which helps when searching the literature:

Clairaut's equation: $y=y'x+f(y')$
Laguerre's equation: $xy''+(1-x)y'+ny=0$
Hermite's equation: $y''-2xy'+2ny=0$
Legendre's equation: $(1-x^2)y''-2xy'+n(n+1)y=0$

Some of these are no doubt like old friends to the math folks here.

 

[fisico30]

Hi Mapes,

sorry but to me those ODEs are all linear... the y and its derivative are only multiplied by functions of the independent variable... which makes them linear... right?

 

[Mapes]

You're right, I goofed. Only the first is possibly nonlinear, depending on the function f.

 

[Haelfix]

Generally speaking there are very few nonlinear differential equations which admit exact solutions. Of the top of my head, things like the Ricatti equation and the Bernouilli equation spring to mind. They are interesting too b/c you can change variables for instance and get a linear but second order equation in some cases.

 

[arildno]

My compliments, nice answer arildno!

regarding your comment:


where can I find some references about "singularities dependent upon initial conditions" in nonlinear equations? Any suggestion on a clear textbook, or examples where this is discussed?

thanks!

A trivial case is the following separable diff.eq:
$$\frac{dy}{dx}=1+y^{2}$$
Dividing both sides with (1+y^2), and separating, yields:
$$arctan(y)=x+C$$
Thus, for x=0, we get C=arctan(y_{0}), y_{0}=y(0).
And the solution is:
$$y(x)=tan(x+arctan(y_{0}))$$
This funmction blows up to infinity at $$x=\frac{\pi}{2}-arctan(y_{0})$$,
that is, its singularity depends upon the initial condition.