This could be the solution to your confusion.

In summary, boundary conditions are necessary to specify the form of a solution to a problem, and can be either initial conditions or more general gradients.
  • #1
dorist84
7
0
Hello hello,

I cannot for the life of me wrap my head around the idea of a boundary condition. I understand the idea (at least I think I do) of solving a differential equation with given initial conditions. But is solving for a magnetic field or electric field while enforcing boundary conditions sort of the same thing? If so, why is it solved in terms of its perpendicular and parallel components?

I read at one point that the purpose of a boundary condition is so specify a "point in space" or to obtain a sense of direction/position when solving for a magnetic or electric field when given a specific geometry. Makes sense to me: it would do no good to just say a magnetic field is n-amount Gauss or Teslas. But isn't that why we use coordinate systems? I guess I just don't understand the idea of "enforcing boundary conditions in order to solve a problem." ...

The best I can make of it is perhaps, hypothetically, if a current or charge density is specified upon a surface with a known geometry, then do you use that to extrapolate information for the B-Field and E-field...(i.e. field will be 0 here...will also flow in this direction...etc ..) But if so, how does all the parallel and perpendicular stuff come in?

I apologize for the convuluted way I've asked this question. I think the problem is more that I'm confused to the extent that I don't even really know HOW to ask the question. So hopefully, if someone is patient enough with me...I can weed through this boundary conditions business.

Thanks for everything. You guys are great!

~**S
 
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  • #2
I think you're on the right track. When you're solving some sort of situation (generally a differential equation) you need initial condition or boundary conditions to completely specify the solution. Otherwise you can only come up with the form of the solution - i.e. inverse square law, or sine function etc.
The difference between an initial condition and a boundary condition is generally that boundary conditions are always. for instance, with magnetic / electric fields - generally there is a boundary condition that the field is zero at infinity, or the field is continuous at some boundary. An initial condition is used when something is changing, so perhaps if you are describing a dynamic field (changing in time) you would specify its initial form (i.e. maybe it starts as zero everywhere, then grows etc).

Initial and boundary conditions due serve the exact same purpose --> to fill in the details of a solution.
 
  • #3
Boundary conditions are indeed similar to initial conditions, in terms of coming up with a specific solution instead of a general solution containing "constants of integration".

Boundary conditions are, and this is important, made up. Sometimes the boundary condition is simply the value of something, like the magnetic field (note that vector fields require more than a single number to specify the value). Other times, the boundary conditions can be more general statements, like "the normal component of the field is continuous across a boundary".

There's two general classes of boundary conditions- the value of something, or the gradient of the something. They can be time-dependent, space-dependent, both, neither. Boundary conditions can be prescribed on a real (material) boundary, or an artificial boundary surface.

But in the end, whatever the specifics of the boundary condition, the role is to generate a specific solution from a class of solutions.
 
  • #4
If your observation is limited to a finite space, then you can't make any predictions without information what is happening on the boundaries of that space: the future is not determined, since something might come from outside and influence the events in your finite space.

Understanding of initial or boundary conditions required to solve a particular problem is easier if you can imagine a numerical method of solving that problem. For example: you could place a grid in your volume, define one variable for the solution at every point and replace all derivatives in differential equations with finite differences. But you would get more variables than equations, since evaluation of a finite difference also requires known neibour points. Therefore boundary conditions are needed to provide the missing equations.
 

1. What are boundary conditions?

Boundary conditions are a set of rules or constraints that are applied to a mathematical model or physical system in order to define the behavior of the system at its boundaries. They help determine the values or properties of the system at its boundaries, which in turn affect the overall behavior of the system.

2. Why are boundary conditions important?

Boundary conditions are important because they allow us to accurately model and understand the behavior of physical systems. They provide a framework for solving mathematical equations and predicting the behavior of a system, as well as helping us to understand how the system will react to changes in its environment or inputs.

3. What types of boundary conditions exist?

There are several types of boundary conditions, including fixed or prescribed boundary conditions, where the value of a variable at the boundary is known and does not change; free or natural boundary conditions, where the value of a variable is not constrained and can vary freely at the boundary; and periodic boundary conditions, where the values at opposite boundaries are equal to each other.

4. How are boundary conditions applied in numerical simulations?

In numerical simulations, boundary conditions are typically applied by specifying the values or properties of the system at the boundaries as initial conditions for the simulation. These conditions are then used to solve the equations that describe the behavior of the system, and the results are used to predict the behavior of the system over time.

5. What happens if boundary conditions are not properly defined?

If boundary conditions are not properly defined, it can lead to inaccurate or incorrect predictions of the behavior of a system. This can result in unrealistic simulations or models, and can also cause errors in calculations or analyses. It is important to carefully consider and define boundary conditions in order to ensure accurate and meaningful results.

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