# [Numerical] System of first order ordinary diff equations with given asymptotic

1. Jul 29, 2009

### ala

I have system of first order ordinary diff equations, indipendent variable is x cordinate. I know asymptotic solution in left and right region (i.e. when x->-infinity or x->infinity, e.g. when abs(x)>1000), it's const plus exponentially falling function. I want to find numerical solution in middle region, witch will have given asymptotic in left and right region.

If I give initial value at left (i.e. at x=-1000) numerical solution blow up at right (I also have exponentially growing functions on right).

How to do this?

2. Jul 29, 2009

### John Creighto

Okay, bellow is my attempt:

Let:

$$y(x)=c(1-w(x)exp(-kx))$$

Then

$$w(x)=(1-y/c)exp(kt)$$
$$\dot{w}=-\frac{\dot{y}}{c}exp(-kx)+k \ (1-y/c)exp(-kx)$$

Now time to make some substitutions
$$\dot{w}=-\frac{f(y)}{c}exp(kt)+k \ (1-\left(c(1-w \ exp(-kx)) \right)/c)exp(kt)$$
$$=-\frac{f(y)}{c}exp(kx)-w \ \left(1-\frac{k}{c}\right)exp(kx)+k \ w \$$
where $$y$$ is given above as:

$$y=c (1- w \ exp(-kx))$$

and $$f(y)$$ is the original differential equation.

edit: The above only seems useful if $$\frac{1}{x}$$ is much bigger then $$k$$.

Last edited: Jul 29, 2009
3. Jul 30, 2009

### ala

For that x, I have asymptotic solution. I want to find numerical solution in the middle, but don't know how. (I don't have 1 ODE, I have system of ODE)