PDE Help = Characteristic Curves / Method of Characterization

In summary, if the velocity field (V) is a fixed vector, then the characteristic curves will be a family of parallel-straight lines. This can be solved by using the parametric equations for the curves, which will result in X(t) and Y(t). These curves will be of a specific type. If you are looking for help on how to get started, it is suggested to solve the parametric equations explicitly and determine the type of curves.
  • #1
rexasaurus
14
0
1. Show that, if the velocity field (V) is a fixed (spatially constant) vector, then the characteristic curves will be a family of parallel-straight lines.



2. ut+V1ux+V2uy=f
f=S-[dell dotted with V]u

characteristic curves:
dX/dt=V1(X,Y) & dY/dt=V2(X,Y)


3. really looking for help on how to get started on this. any suggestions would be appreciated.
 
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  • #2
Since V is fixed (i.e. [itex]V_1(x,y) = C_1; V_2(x,y) = C_2[/itex], you can solve the parametric equations for the characteristic curves explicitly, and obtain X(t), and Y(t). What types of curves are these?
 

FAQ: PDE Help = Characteristic Curves / Method of Characterization

What is the Method of Characterization in PDE?

The Method of Characterization is a mathematical technique used to solve partial differential equations (PDEs) by transforming them into systems of ordinary differential equations (ODEs). This method involves identifying the characteristic curves of the PDE and using them to solve the equation.

What are Characteristic Curves in PDE?

Characteristic curves in PDE are curves that represent the paths along which the solution to the PDE is constant. These curves are determined by the coefficients and initial/boundary conditions of the PDE and are used in the Method of Characterization to solve the equation.

How does the Method of Characterization differ from other methods of solving PDEs?

The Method of Characterization differs from other methods of solving PDEs in that it transforms the PDE into a system of ODEs, which can be solved using standard techniques. This method is particularly useful for nonlinear PDEs, as it allows for a more straightforward solution.

What types of PDEs can be solved using the Method of Characterization?

The Method of Characterization can be used to solve first and second-order PDEs, both linear and nonlinear. It is commonly used in applications involving fluid dynamics, heat transfer, and wave propagation.

Are there any limitations to the Method of Characterization?

While the Method of Characterization is a powerful tool for solving PDEs, it does have some limitations. It may not be applicable to higher-order PDEs or those with irregular boundary conditions. Additionally, the method may become computationally intensive for complex PDEs.

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