# Coupled Nonlinear Differential Equations

#### Thomas_W

Hey,

I need your help to solve the following set of coupled differential equations numerically.

dn(t,z)/dt=I^5(t,z)+I(t,z)*n(t,z)

dI(t,z)/dz=I^5(t,z)-α(n(t,z))*I(t,z)

where I(t,0)=I0*exp(-4ln2(t/Δt)^2) and n(t,0)=0 and n(-certrain time,z)=0. Some constant parameters I did not show here.
α(n(t,z)) is just a parameter which depends linear on n(t,z)

I tried to solve it with Mathematica and NDsolve.. since it works fine if I just solve the first equation without the z dependence. But Mathematica seems to be unhappy with the boundary conditions.

Thanks!

Thomas

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#### haruspex

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There doesn't seem to be enough information. There's nothing to set how n varies with z or I with t. You said it was ok if you just took the first equation and treated z as constant, but then you have two unknown dependent variables, n and I, and only 1 equation.

#### Thomas_W

Yeah I know, but what I solved was
dn(t)/dt=I^5(t)+I(t)*n(t)
with a time dependent I(t).

n depends on z only due to the fact that I is depending on z, there is no direct dependence. dI(t,z)/dt I only can give for the case I(t,0).. so I just adapt these coupled equations from a paper.

Thank you

Thomas

#### yus310

Hey,

I need your help to solve the following set of coupled differential equations numerically.

dn(t,z)/dt=I^5(t,z)+I(t,z)*n(t,z)

dI(t,z)/dz=I^5(t,z)-α(n(t,z))*I(t,z)

where I(t,0)=I0*exp(-4ln2(t/Δt)^2) and n(t,0)=0 and n(-certrain time,z)=0. Some constant parameters I did not show here.
α(n(t,z)) is just a parameter which depends linear on n(t,z)

I tried to solve it with Mathematica and NDsolve.. since it works fine if I just solve the first equation without the z dependence. But Mathematica seems to be unhappy with the boundary conditions.

Thanks!

Thomas
This is why mathematica is not solving thsi.

"where I(t,0)=I0*exp(-4ln2(t/Δt)^2)"

are you sure this is so, what if Δt=0? then what?

#### Thomas_W

This is why mathematica is not solving thsi.

"where I(t,0)=I0*exp(-4ln2(t/Δt)^2)"

are you sure this is so, what if Δt=0? then what?
Δt is not zero. so I is just a gaussian distribution around Δt

#### hunt_mat

Homework Helper
Can't the method of characteristics be used here?