Numerical Solution to ODE System - IVP or BVP?

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Discussion Overview

The discussion revolves around the classification of a system of spatial ordinary differential equations (ODEs) as either an initial value problem (IVP) or a boundary value problem (BVP). Participants explore the implications of having boundary conditions defined at different spatial points and the requirements for solving such systems.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant describes a system of differential-algebraic equations (DAEs) with boundary conditions defined at two points, questioning whether it constitutes a BVP or an IVP.
  • Another participant asserts that having values for functions at two different points indicates a boundary value problem, emphasizing the role of boundary conditions in determining existence and uniqueness.
  • A participant seeks clarification on the use of derivatives as boundary conditions, indicating confusion about the requirements for a BVP.
  • Another participant states that for a system of first-order equations, the number of boundary conditions must match the number of equations, suggesting that the participant may not have enough conditions for a BVP if only one condition per function is provided.
  • There is a discussion about the implications of having first-order equations and the necessary conditions for solving them, with some participants expressing differing views on the number of required boundary conditions.

Areas of Agreement / Disagreement

Participants express differing opinions on the classification of the problem as an IVP or BVP, with some asserting it is a BVP due to the boundary conditions provided, while others question the sufficiency of those conditions. The discussion remains unresolved regarding the correct classification and the requirements for boundary conditions.

Contextual Notes

Participants mention the need for clarity on the order of equations and the implications of having only one boundary condition per function, indicating potential limitations in understanding the problem's requirements.

mpowers
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I have a system of spatial ODEs to solve... Actually a DAE system, but here's the issue:

The equations are vaild over a specific domain, x = 0..L

The equations are only bound at one point (their "initial point") but at either 0 or L
f1(0)=0
f2(0)=100
f3(L)=0
f4(L)=100
(also an algebraic expression that links all of the functions)

Essentially, those functions bound at L are moving "backwards" with respect to those bound at 0.

My question is, is this an initial value problem, or a boundary value problem? I started in MAPLE, and it decided (automatically) that it was a BVP. In MATLAB I am attempting to use bvp4c to come up with a solution, but is it really even a BVP if it only has one boundary condition, essentially an initial conditition, but defined at different spatial coordinates for different functions?

Your insight is appreciated.
 
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Since you are given the values, of the function and its derivative, at two different points, that's a "boundary value" problem. The mathematical difference is that "existence and uniqueness", for an initial value problem, depend only on the equation. For a boundary value problem it also depends on the boundary conditions.
 
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Just to make sure I understand...

I only have one value for each function at *one* point... (let's say L, but could be 0 for the other functions) but for the boundary at the other point (let's say 0, but other functions are not bound at 0) I can use the derivative of the function?

These are First order, btw.
 
I don't know what you mean by "use the derivative of the function".
 
Sorry... I need two boundary conditions for each function for a BVP, right? I only have one boundary condition per function... I thought you were saying that I could use the derivative of the function as a boundary condition.
 
What order are the equations? Typically, systems of equations are first order (any set of m nth order equations can be reduced to a system of mn first order equations). If you have have a system of 4 first order equations in 4 functions, then you need 4 conditions- exactly what you have. If you have 4 second or higher order equations then you don't have enough information. You can't "use" the derivative because you are given the derivative.
 
All first order. I though that if I had 4 first order equations, then I would need 8 boundary conditions for both ends of the interval for each equation. I guess you're saying that is not right?
 

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