Numerical Solution to ODE System - IVP or BVP?

In summary, the conversation discusses a system of spatial ODEs (actually a DAE system) with equations that are valid over a specific domain and only have one boundary condition at either 0 or L. The question is whether this is an initial value problem or a boundary value problem, and the expert clarifies that it is a boundary value problem since there are values for the function and its derivative at two different points. The expert also mentions that for a system of 4 first order equations, 4 conditions are needed, which is what is provided in the given information. The conversation ends with some confusion about the number of boundary conditions needed for a system of first order equations.
  • #1
mpowers
4
0
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 (thier "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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  • #2
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 "existance 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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  • #3
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 derivitive of the function?

These are First order, btw.
 
  • #4
I don't know what you mean by "use the derivative of the function".
 
  • #5
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 derivitive of the function as a boundary condition.
 
  • #6
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.
 
  • #7
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?
 

1. What is the difference between an IVP and BVP in numerical solution to ODE systems?

An IVP (Initial Value Problem) involves finding a solution to an ODE system at a particular initial condition. A BVP (Boundary Value Problem) involves finding a solution to an ODE system at multiple boundary conditions. In numerical solutions, IVPs are typically solved using initial value methods such as Euler's method, while BVPs are solved using boundary value methods such as shooting or finite difference methods.

2. How do you choose the appropriate numerical method for solving an ODE system?

The choice of numerical method depends on the type of ODE system, the accuracy and stability requirements, and the available computational resources. Some common methods include Euler's method, Runge-Kutta methods, and finite difference methods. It is important to consider the strengths and limitations of each method before choosing the most suitable one for a particular problem.

3. Can you explain the concept of convergence in numerical solutions to ODE systems?

Convergence refers to the property of a numerical method where the approximate solution approaches the exact solution as the step size or grid size decreases. In other words, as we increase the number of steps or decrease the interval size, the numerical solution becomes closer to the true solution. This is an important aspect to consider when choosing a numerical method, as a higher convergence rate leads to more accurate solutions.

4. How do you handle stiffness in numerical solutions to ODE systems?

Stiffness refers to ODE systems with solutions that vary rapidly over a small range of independent variable values. This can make it difficult for numerical methods to accurately capture the behavior of the solution. To handle stiffness, specialized methods such as implicit methods or adaptive step size methods can be used. These methods are designed to handle stiffness and can provide more accurate solutions compared to standard methods.

5. What are the limitations of numerical solutions to ODE systems?

While numerical methods are often used to solve ODE systems, they do have some limitations. One limitation is that they can only provide an approximate solution, which may deviate from the true solution depending on the accuracy of the method and the chosen step size. Additionally, some numerical methods may struggle with certain types of ODE systems, such as stiff systems, leading to less accurate solutions. It is important to consider these limitations when choosing a numerical method for a particular problem.

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