Superficially simple vector differential equation problem

[cfp]

Hi,

I have the following vector differential equation (numerator layout derivatives):

$$\frac{\partial e(v)}{\partial v}=\frac{1}{\beta} \frac{\partial w(v)}{\partial v} \Gamma^{-1}$$

where both $e(v)$ and $w(v)$ are scalar functions of the vector $v$, and where $\Gamma$ is a symmetric invertible matrix with all columns (and rows) summing to 1.

The naive solution would be $e(v)=\frac{1}{\beta} w(v) \Gamma^{-1}$, but this is incorrect since $e(v)$ is a scalar.

Clearly, when $\Gamma$ is the identity matrix, $e(v)=\frac{1}{\beta} w(v)$ is a valid solution. My question is, does a solution exist for any other value of $\Gamma$?

Tom

 

[jvicens]

,

Thank you for your post. Your vector differential equation is certainly interesting and presents a challenging problem to solve. To answer your question, yes, a solution exists for any other value of $\Gamma$ as long as it is a symmetric invertible matrix with all columns and rows summing to 1. However, finding this solution may not be as straightforward as in the case of the identity matrix.

One approach to solving this problem is to use the method of separation of variables. This involves assuming that $e(v)$ can be written as a product of two functions, one of which depends only on $v$ and the other which depends only on $\Gamma$. By substituting this into the original equation and solving for each function separately, you can find a general solution for $e(v)$.

Another approach is to use numerical methods, such as finite difference or finite element methods, to approximate the solution. These methods involve discretizing the domain of $v$ and solving for the values of $e(v)$ at each point. This can provide a more accurate solution, but may require more computational resources.

I hope this helps answer your question and provides some insight into possible ways to solve this vector differential equation. Good luck with your research!