Check Homework on Partial Differential Wave Equation

In summary: I don't understand what you mean by "when you integrate the second equation your not going to get a x time a function.".
  • #1
roldy
237
2

Homework Statement


Consider the partial differential equation

[tex] u_{xx}-3u_{xt}-4u_{tt}=0 [/tex]

(a) Find the general solution of the partial differential equation in the xt-plane, if possible.
(b) Find the solution of the partial differential equation that satisfies
[tex]u(x,0)=x^3[/tex] and [tex]u_t(x,0)=-3x^2[/tex] for [tex]-\infty<x<\infty[/tex]

Homework Equations





The Attempt at a Solution



(a)

[tex]
\left(\frac{\partial^2}{\partial x^2}-\frac{3\partial^2}{\partial x\partial t}-\frac{4\partial^2}{\partial t^2}\right)u=0
[/tex]

*
[tex]
\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial t}\right)\left(\frac{\partial}{\partial x}-\frac{4\partial}{\partial t}\right)u=0
[/tex]

Set the coefficients equal to a variable...

[tex]\alpha=1, \beta=1, \gamma=1, \delta=-4[/tex]

[tex]
\zeta=\beta{x}-\alpha{t}=x-t
[/tex]
[tex]
\eta=\delta{x}-\gamma{t}=-4x-t
[/tex]

[tex]
\frac{\partial v}{\partial x}=\frac{\partial v}{\partial \zeta}\frac{\partial \zeta}{\partial x}+\frac{\partial v}{\partial \eta}\frac{\partial \eta}{\partial x}=\frac{\partial v}{\partial \zeta}-\frac{4\partial v}{\partial \eta}
[/tex]

[tex]
\frac{\partial v}{\partial t}=\frac{\partial v}{\partial \zeta}\frac{\partial \zeta}{\partial t}+\frac{\partial v}{\partial \eta}\frac{\partial \eta}{\partial t}=\frac{-\partial v}{\partial \zeta}-\frac{\partial v}{\partial \eta}
[/tex]

So now substitute these two differentials into *
[tex]
\left[\left(\frac{\partial}{\partial \zeta}-\frac{4\partial}{\partial \eta}\right)+\left(\frac{-\partial}{\partial \zeta}-\frac{\partial}{\partial \eta}\right)\right]\left[\left(\frac{\partial}{\partial \zeta}-\frac{4\partial}{\partial \eta}\right)-\left(\frac{-4\partial}{\partial \zeta}-\frac{4\partial}{\partial \eta}\right)\right]u=0
[/tex]

[tex]
\left(\frac{-4\partial}{\partial \eta}-\frac{\partial}{\partial \eta}\right)\left(\frac{\partial}{\partial \zeta}+\frac{4\partial}{\partial \zeta}\right)u=0
[/tex]

[tex]
\left(\frac{-5\partial}{\partial \eta}\right)\left(\frac{5\partial}{\partial \zeta}\right)u=0
[/tex]

[tex]
\frac{\partial}{\partial \eta}\left(\frac{\partial u}{\partial \zeta}\right)=0
[/tex]

Let [tex]h(\zeta)=\frac{\partial u}{\partial \zeta}[/tex]

So now integrate with respect to [tex]\zeta[/tex] holding [tex]\eta[/tex] fixed.

[tex]
u=\int h(\zeta)d\zeta + g(\eta)=f(\zeta)+g(\eta)
[/tex]

Substitute back in for [tex]\zeta[/tex] and [tex]\eta[/tex]...

[tex]
u(x,t)=f(x-t)+g(-4x-t)
[/tex]

Now this should be the general solution. Is this right, did I do my differentiation and arithmetic correctly?

(b)

So for the functions f and g I just picked some arbitrary functions.

Let [tex]f(x)=e^x[/tex] and [tex]g(x)=e^{-x}[/tex]

So now,

[tex]
x^3=e^x+e^{4x}
[/tex]

And also,

[tex]
-3x^2=-e^x-e^{4x}
[/tex]

And this is where I stall. I'm not sure what to do after this point. My book doesn't have an example of solving these types of problems and the notes our professor gave us doesn't show an example either.
 
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  • #2
Your new equation is fine along with your general solution. however you did not apply the boundary conditions correctly, but you need to work on your boundary conditions. You have [tex]u=f(x-t)+g(-4x-t)[/tex] Applying you boundary conditions:
[tex]
u(x,0)=f(x)+g(-4x)=x^{3}\quad\partial_{t}u(x,0)=-f'(x)-g(-4x)=-3x^{2}
[/tex]
Now solve those equations for f and g to get the solution.
 
  • #3
So I'm not suppose to pick arbitrary functions for f and g then right?. I guess this makes sense now, two equations two unknowns.
 
  • #4
Your general solution will give you the functions as you've done now you have to fit those functions to your initial conditions. I have done it so I can say if you have the correct equation or not.
 
  • #5
I'm not that I entirely understand what you mean in your last post.
 
  • #6
Now you have a set of ODE's to solve:
[tex]
\begin{array}{ccc}
f(x)+g(-4x) & = & x^{3} \\
f'(x)+g'(-4x) & = & 3x^{2}
\end{array}
[/tex]
You should be able to find what f and g are by solving the above system of ODE.
 
  • #7
Yeah, that's what I figured I needed to do but what method involved is what I'm not sure about.
 
  • #8
In the first equation set x=0 to find that f(0)+g(0)=0, integrate the second equation to obtain:
[tex]
f(x)-\frac{1}{4}g(x)=x^{3}+C
[/tex]
Then take the first equation away from the second one to obtain:
[tex]
-\frac{5}{4}g(x)=C
[/tex]
This shows that g is constant. which in turn shows that:
[tex]
f(x)=x^{3}+D
[/tex]
So the full solution becomes...

Simple integration is all that is required, what can you say about D in light of the initial conditions?
 
  • #9
I don't understand why you are setting x=0. There's no initial condition for x=0. Also, how to you know that when you integrate the second equation your not going to get a x time a function. Meaning, how do you for example that f(x) is not (1+x)?
 
  • #10
The setting x=0 was unimportant, so ignore it.

I get a constant because f is a function of one variable only.
 
  • #11
Could you maybe explain a little bit more on how that 1/4 got out there?
 
  • #12
Still confused on this one.
 
  • #13
it comes into the integral in the following way:
[tex]
\int g'(-4x)dx =-\frac{1}{4}\int g'(u)du=-\frac{1}{4}g(u)=-\frac{1}{4}g(-4x)
[/tex]
using the substitution u=-4x
 

1. What is a partial differential wave equation?

A partial differential wave equation is a mathematical equation that describes the behavior of a wave over time and space. It takes into account the partial derivatives of the wave with respect to time and space variables.

2. How is a partial differential wave equation used in science?

A partial differential wave equation is used in various fields of science, such as physics, engineering, and finance, to model and analyze wave phenomena. It is also used in solving many real-world problems, such as predicting the behavior of sound waves, electromagnetic waves, and heat diffusion.

3. What are some common methods for solving a partial differential wave equation?

Some common methods for solving a partial differential wave equation include separation of variables, Fourier series, and numerical methods such as finite difference and finite element methods. The choice of method depends on the specific characteristics and complexity of the equation.

4. Can a partial differential wave equation be applied to any type of wave?

Yes, a partial differential wave equation can be applied to various types of waves, including mechanical waves, electromagnetic waves, and quantum waves. However, the specific form of the equation may vary depending on the type of wave being studied.

5. How can I check my homework on partial differential wave equations?

The best way to check your homework on partial differential wave equations is to ensure that your solution satisfies the equation and any given boundary conditions. You can also compare your solution to known solutions or use software programs designed for solving partial differential equations to verify your work.

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