Solving the PDE u_(xy) = ku with some initial conditions

• quasar987
In summary, the conversation discusses solving two different partial differential equations for the function u with given initial conditions. The first equation involves a constant k, while the second equation involves a constant K. The solution for the second equation, with boundary conditions, is found using separation of variables and is a superposition of normal modes with frequencies that differ from the usual wave equation. The conversation also considers whether the solution for the first equation can be obtained from the solution for the second equation through a change of variables.
quasar987
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Homework Statement

Does anyone know how to solve this PDE for u:R-->R and some initial conditions?

$$u_{xy}=ku$$

where k is a positive constant.

Or this one, also for u:R-->R and some initial conditions:

$$u_{tt}=u_{xx}-Ku$$

where K is a positive constant.

The Attempt at a Solution

I can solve the second one for u:[0,L]-->R and the boundary conditions of the fixed string u(0,t)=u(L,t)=0 by separation of variable. The solution, a superposition of normal modes, differs only from the solution of the usual wave equation u_tt=u_xx in that the frequencies are $$\sqrt{\lambda_n^2+K}$$, where lambda_n=npi/L is the usual nth eigenfrequency for the usual wave equation.

In the usual wave equation solution, the normal modes are superpositions of traveling waves. And traveling waves are the general solution to the "free" wave equation u_tt=u_xx for u:R-->R, obtained by d'Alembert's method. Can I conclude that the solution for u:R-->R of u_{tt}=u_{xx}-Ku are also traveling waves?

The first one reduces to the second one by the change of variables
$$x=\frac{1}{2}\,(v+u),\,y=\frac{1}{2}\,(v-u)$$

1. What is a PDE?

A PDE (partial differential equation) is a type of mathematical equation that involves multiple variables and their partial derivatives. It is used to describe the relationship between a function and its derivatives.

2. What does u_(xy) = ku represent?

This represents a second-order partial derivative, where u is a function of both x and y, and k is a constant.

3. How do you solve a PDE?

To solve a PDE, you typically use a combination of analytical and numerical methods. This involves breaking down the equation into simpler parts, solving them individually, and then combining the solutions to get the overall solution.

4. What are initial conditions?

Initial conditions refer to the values of the function and its derivatives at a specific point in the domain. These values are used as starting points for solving the PDE.

5. How does the constant k affect the solution?

The constant k can affect the solution in different ways, depending on the specific PDE and initial conditions. In general, it can impact the behavior of the function and its derivatives, leading to different solutions or patterns in the solution.

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