Characteristic Curves: Solving PDEs

In summary, the equation $2u_{xx}-u_{tt}+u_{xt}=f(x, t)$ can be rewritten as $\left (\frac{2\partial^2}{\partial{x^2}}-\frac{\partial ^2}{\partial{t^2}}+\frac{\partial ^2}{\partial{x}\partial{t}}\right )u=f$, and in order to find the characteristics, we must solve the homogeneous equation $\frac{2\partial^2}{\partial{x^2}}-\frac{\partial ^2}{\partial{t^2}}+\frac{\partial ^2}{\partial{x}\partial{t}}=0$. These characteristics are the lines along which the
  • #1
mathmari
Gold Member
MHB
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Hey! :eek:

We have the equation $$2u_{xx}-u_{tt}+u_{xt}=f(x, t)$$

This is equal to $$\left (\frac{2\partial^2}{\partial{x^2}}-\frac{\partial ^2}{\partial{t^2}}+\frac{\partial ^2}{\partial{x}\partial{t}}\right )u=f$$

To find the characteristics do we solve the homogeneous equation $$\frac{2\partial^2}{\partial{x^2}}-\frac{\partial ^2}{\partial{t^2}}+\frac{\partial ^2}{\partial{x}\partial{t}}=0$$ ?? (Wondering)
 
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  • #2
Yes, you would solve the homogeneous equation in order to find the characteristics. The characteristics are the lines along which the solution of the equation can travel. In this case, they will be the lines that satisfy the equation $2u_{xx}-u_{tt}+u_{xt}=0$.
 

FAQ: Characteristic Curves: Solving PDEs

What are characteristic curves in the context of solving PDEs?

Characteristic curves are trajectories in the solution space of a partial differential equation (PDE) that follow the direction of the gradient of the solution. They are used to determine the behavior of the solution at different points in space and time.

How do characteristic curves help in solving PDEs?

By following the direction of the gradient, characteristic curves provide information about how the solution changes at different points. This information can then be used to construct a solution to the PDE.

What types of PDEs can be solved using characteristic curves?

Characteristic curves are most commonly used to solve hyperbolic and parabolic PDEs, which involve the transport or diffusion of a physical quantity over time and space. However, they can also be applied to elliptic PDEs in certain cases.

How are characteristic curves calculated?

The specific method of calculating characteristic curves varies depending on the type of PDE and the boundary conditions. In general, it involves solving a system of ordinary differential equations that describe the behavior of the solution along the characteristic curves.

What are the limitations of using characteristic curves to solve PDEs?

Characteristic curves can only be used for PDEs with certain properties, such as being linear or having constant coefficients. They also may not provide a complete solution to the PDE, as some boundary conditions may not be satisfied along the characteristic curves. In these cases, additional techniques may be needed to fully solve the PDE.

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