Specification of the boundary condition in high order PDE

In summary: However, it is not always explicitly stated as it is often implied. Specifying only one boundary condition at the origin may be due to the nature of the problem or simplification in the computation process. It is mathematically sound as long as the other boundary condition is implicitly satisfied. The physical meaning of this is that the solution approaches zero at infinity, representing a stable system.
  • #1
jollage
63
0
Hi all,

I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation
[itex]\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2}[/itex] subject to the initial condition [itex]u(x,0)=g(x)[/itex] with addition boundary condition.
I saw in some references that the boundary condition is specified as [itex]u(0,t)=f(t)[/itex]. So there is only one boundary condition at the origin, while for completeness, I think there should be another boundary condition for the viscous term. What I suppose is that the implicit unsaid is [itex]u(\infty,t)=0[/itex]. But I don't know, it's a guess.

Could you who are experienced in PDE clarify what's the physical meaning of only specifying a boundary condition at the origin even for a high-order PDE? And to what extent is this formulation mathematically sound? Thank.

Jo
 
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  • #2
jollage said:
Hi all,

I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation
[itex]\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2}[/itex] subject to the initial condition [itex]u(x,0)=g(x)[/itex] with addition boundary condition.
I saw in some references that the boundary condition is specified as [itex]u(0,t)=f(t)[/itex]. So there is only one boundary condition at the origin, while for completeness, I think there should be another boundary condition for the viscous term. What I suppose is that the implicit unsaid is [itex]u(\infty,t)=0[/itex]. But I don't know, it's a guess.

Could you who are experienced in PDE clarify what's the physical meaning of only specifying a boundary condition at the origin even for a high-order PDE? And to what extent is this formulation mathematically sound? Thank.

Jo

The usual assumption is [itex]u(\infty,t) = 0[/itex].
 

Related to Specification of the boundary condition in high order PDE

1. What is a boundary condition in a high order PDE?

A boundary condition in a high order PDE is a set of constraints or requirements that must be satisfied at the edges or boundaries of the domain in which the PDE is being solved. These boundary conditions help to define the behavior of the solution at the boundaries and are necessary for obtaining a unique solution to the PDE.

2. How do boundary conditions affect the solution of a high order PDE?

Boundary conditions play a crucial role in determining the behavior of the solution to a high order PDE. By specifying the behavior of the solution at the boundaries, they help to define the problem and allow for a unique solution to be obtained. Different boundary conditions can result in different solutions to the same PDE, highlighting the importance of carefully choosing and specifying the boundary conditions.

3. What are the types of boundary conditions used in high order PDEs?

There are several types of boundary conditions that can be used in high order PDEs, depending on the specific problem being solved. These include Dirichlet boundary conditions, which specify the value of the solution at the boundary, Neumann boundary conditions, which specify the derivative of the solution at the boundary, and Robin boundary conditions, which combine both Dirichlet and Neumann conditions. Other types of boundary conditions include periodic, mixed, and symmetry boundary conditions.

4. How are boundary conditions specified in high order PDEs?

Boundary conditions can be specified in different ways, depending on the numerical method being used to solve the PDE. In some cases, the boundary conditions may be explicitly specified as part of the problem formulation. In other cases, the boundary conditions may be implicitly defined through the use of numerical schemes or algorithms. In either case, it is important to carefully consider and properly specify the boundary conditions to ensure an accurate and meaningful solution to the PDE.

5. What are some common challenges when specifying boundary conditions in high order PDEs?

One common challenge when specifying boundary conditions in high order PDEs is ensuring that the boundary conditions are consistent with the problem being solved. This can require a deep understanding of the physical system and the mathematical formulation of the PDE. Another challenge is ensuring that the boundary conditions are properly implemented within the numerical method being used, as errors in the implementation can lead to inaccurate solutions. Finally, choosing the appropriate type and number of boundary conditions can also be a challenge, as it may require trial and error or extensive sensitivity analysis to determine the best set of boundary conditions for a particular problem.

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