A set of nonlinear coupled ODE's

In summary, the author is looking for an analytical solution to a set of coupled equations and is unsure if it exists. However, if there is a specific function or vectors that are known, it may still be possible to find a general solution.
  • #1
gouranja
11
0
Hi,

I'm looking for an analytical solution to this set of coupled equations:

[tex]U^{'}_{i}(x)=c (f_i(x)-(\vec{U}\cdot\vec{f})U_i(x))[/tex]

Where the vectors are 3 dimentional, c is constant and f is a vector of given functions:

[tex]f_i(x) , i\in 1..3[/tex]

There probably isn't one but I thought I'd try anyway.

thanks
 
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  • #2
right now it looks like you have 1 equation in two or more unknowns, which will be impossible to solve.
 
  • #3
Not really

There are 3 equations for each component of [tex]\vec{U}(x)[/tex] since the index i goes between 1..3. For example the first equation can be explicitly written as:
[tex]U_1^{'}(x)=c(f_1(x)-U_1(f_1 U_1+f_2 U_2+f_3 U_3))[/tex]
As for the unknows, that doesn't mean you can't formulate a general solution. consider the equation:
[tex]U_i^{'}(x)=f_iU_i(x)[/tex] (no summation)
whose solution is:
[tex]U_i(x)=\exp{\left(\int_0^x f_i(t)dt\right)}[/tex]
 
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  • #4
I think if we knew something more about the U's or the f's it may be possible to get somewhere, otherwise I wouldn't know where to begin.

Is there anything more to know?
 
  • #5
I wasn't sure what kind of summation notation you were using, but after playing with it for a few minutes I was able to move the u1's and f1's to one side, so if you really need that analytical solution I would guess that its there, as you could wirte

G(f1,u1,u1')=-F2u2-F3u3
G(f2,u2,u2')=-F1u1-F3u3
G(f3,u3,u3')=-F1u1-F2u2
(note I used the same G as it would be the same separation procedure)
which can then be written as the matrix equation

G=FU
then you might be able to play with it some more to get the G's and u's together in order to make 3 first orderdifferential equations, but I don't know, but you would have to assume that det(F) does not equal 0.

hmm one thing I do notice there though is that if G=(FU)^t than it would become three ordinary differential equations in 1 variable. does anybody know of an identity that will let you do that?

if you do come up with a general solution post it, i'd love to see it. also out of curiosity what's it for?

EDIT:edited for reasons of gross inaccuracy
 
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  • #6
You are right Matthew the f's and u's have some very neat properties which I left out because a lot of mathematicians are not familiar with the Minkovsky product which is the appropriate product for unified space time. Let's denote it by star, such that:
[tex]\vec{f}(x)\star\vec{g}(x)=f_1g_1-f_2g_2-f_3g_3[/tex]
Note the diffrence in sign because this is Minkovsky space not Euclidian space. So in fact in my original equation you should replace Euclidiean product [tex]\vec{U}(x)\cdot\vec{f}(x)[/tex] with [tex]\vec{U}(x)\star\vec{f}(x)[/tex]. This of course doesn't really shed more light on the problem, however The U's are unit vectors under the Minkovsky product and the f's are null vectors. Namely:
[tex]\vec{U}(x)\star\vec{U}(x)=1[/tex]
[tex]\vec{f}(x)\star\vec{f}(x)=0[/tex]
Note that for Euclidian vectors null vectors are always trivial, because a zero norm implies that the vector is the zero vector. This is not true for Minkovsky space vectors, that can be non trivial if they have a zero norm under the Minkovsky product.

Luke: These equations are related to the dynamics of confined quarks within mesons or Baryons.
 
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  • #7
hmm well i you ha the other function, and the initial conditions you could use a lplace or Fourier transform to get the answer, but /i do't think you'll be able to form a general solution
 

Related to A set of nonlinear coupled ODE's

1. What are nonlinear coupled ODE's?

A set of nonlinear coupled ordinary differential equations (ODE's) is a system of equations in which the dependent variables are functions of the independent variable and each other. This means that the equations cannot be easily solved by traditional methods and often require numerical techniques.

2. What makes nonlinear coupled ODE's difficult to solve?

Nonlinear coupled ODE's are difficult to solve because they do not follow a linear relationship between the dependent and independent variables. This means that the equations cannot be manipulated into a standard form and require more complex techniques to find a solution.

3. How are nonlinear coupled ODE's used in science?

Nonlinear coupled ODE's are used in many scientific fields, such as physics, biology, and engineering, to model complex systems and phenomena. They allow for a more accurate representation of real-world situations and can help predict future behaviors or outcomes.

4. What are some techniques for solving nonlinear coupled ODE's?

There are several techniques for solving nonlinear coupled ODE's, including numerical methods such as Euler's method, Runge-Kutta methods, and finite difference methods. Other approaches include perturbation methods, series solutions, and phase plane analysis.

5. Are there any limitations to using nonlinear coupled ODE's?

While nonlinear coupled ODE's can be powerful tools for modeling complex systems, they also have limitations. These equations can become very complicated and difficult to solve, especially for larger systems. Additionally, small errors in the initial conditions or parameters can greatly affect the accuracy of the solution.

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