Which numerical procedure to use

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Discussion Overview

The discussion revolves around the numerical procedures for solving a system of linear differential equations with known boundary conditions. Participants explore both the general solution to such systems and the implementation of these solutions in MATLAB.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant inquires about the general solution to a system of linear differential equations, suggesting it involves exponentials and eigenvalues but expresses uncertainty about the full solution.
  • Another participant responds that the general description may not allow for a systematic solution, indicating that the system might not have a solution at all.
  • A third participant proposes a specific form for the general solution involving eigenvectors and eigenvalues, questioning if the solution is primarily about fitting it to the boundary conditions.
  • One participant confirms the previous assertion about fitting the solution to boundary conditions and draws a parallel to solving systems of linear equations, noting the potential for non-unique solutions.
  • There is mention of MATLAB's symbolic differential equation solver as a tool for handling such systems.

Areas of Agreement / Disagreement

Participants express differing views on the existence and uniqueness of solutions to the system of differential equations. While there is some agreement on the form of the general solution, uncertainty remains regarding the conditions under which solutions can be found.

Contextual Notes

The discussion highlights limitations related to the generality of the problem, including the potential absence of solutions and the dependence on specific boundary conditions. There is also an acknowledgment of the need for further clarification on the analytical solution.

Who May Find This Useful

Readers interested in numerical methods for solving differential equations, MATLAB programming, and the theoretical aspects of linear differential systems may find this discussion relevant.

aaaa202
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I have a system of linear differential equations with known boundary conditions. First of all what is the general solution to such a system? I know it is exponentials with the eigenvalues, but I couldn't find any place where the exact full solution was stated.
Second of all, I want to write a program in MATLAB that solves this set of linear differential equations. In general how am I better off:
- Simulate the solutions numerically starting from scratch i.e. dy1 dy2, dt and everything are numerical quantities
- Find eigenvalues to the coefficient matrix numerically, plug into the analytical solution (which is the one I am not completely sure off, but I think you can write up a general formula) and solve for the unknown constants by applying boundary conditions.
 
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aaaa202 said:
I have a system of linear differential equations with known boundary conditions. First of all what is the general solution to such a system? I know it is exponentials with the eigenvalues, but I couldn't find any place where the exact full solution was stated.
That's because the description is too general for there to be a systematic way of finding the solution.
i.e. the system may not have a solution.

http://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx
... this sort of thing is the closest you'll get to a general approach to a solution.

To do this in matlab, you'd construct the matrix A (see link) and then use Matlab's matrix tools to analyse it. i.e. eigs() will find the eigenvalues and vectors.
 
But as far as I can see from the link you provided isn't the most general solution simply:

x(vector) = c1 * eigenvector_1 * exp(eigenvalue_1 * t) + c2 * eigenvector_2 * exp(eigenvalue_2 * t) + ...

So it is all about fitting that solution to the boundary conditions? Or am I wrong?
 
That's right.
It's a bit like solving systems of linear equations... i.e. there may not be a unique solution, and so on.

Looks like MATLAB has a symbolic differential equation solver that also does systems of equations.
http://www.mathworks.com/help/symbolic/dsolve.html
 

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