Coupled Nonlinear Differential Equations

In summary, the conversation discusses a set of coupled differential equations with boundary conditions and unknown variables. The speaker is having trouble solving the equations with Mathematica and asks for help. The other person questions the validity of the boundary conditions and suggests using the method of characteristics.
  • #1
Thomas_W
3
0
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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  • #2
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.
 
  • #3
Thanks for your fast response.

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
 
  • #4
Thomas_W said:
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?
 
  • #5
yus310 said:
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
 
  • #6
Can't the method of characteristics be used here?
 

1. What are coupled nonlinear differential equations?

Coupled nonlinear differential equations are a set of mathematical equations that describe the relationships between multiple variables in a system, where the equations are nonlinear (i.e. they involve terms with powers higher than 1) and the variables are interdependent (i.e. changes in one variable affect the values of the other variables).

2. How are coupled nonlinear differential equations used in science?

Coupled nonlinear differential equations are used in many fields of science, including physics, biology, engineering, and economics. They allow scientists to model complex systems and predict how changes in one variable will impact the behavior of the entire system.

3. What are some examples of systems that can be described by coupled nonlinear differential equations?

Examples include the motion of celestial bodies in space, the behavior of chemical reactions, the growth of populations, the spread of diseases, and the dynamics of weather patterns.

4. How are coupled nonlinear differential equations solved?

Unlike linear differential equations, there is no general method for solving coupled nonlinear differential equations. In most cases, numerical methods are used, such as Euler's method or Runge-Kutta methods, to approximate solutions. In some cases, analytical solutions can be found using techniques such as perturbation theory or series expansions.

5. What are the challenges of working with coupled nonlinear differential equations?

Coupled nonlinear differential equations can be very difficult to solve, and there is no guarantee that a solution can be found. They also often have multiple solutions, making it difficult to determine which solution is the most accurate. In addition, small changes in the initial conditions or parameters of the equations can lead to drastically different results, making it important to carefully validate and interpret the solutions.

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