Long time behaviour of inhomogeneous wave equation
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I have been given the following wave equation:
$$u_{tt}(x,t) = c^2u_{xx}(x,t)-ru_t(x,t)+g(x)sin(omega t)$$
where $t >0, xin (0,l)$ and $g: [0,l] rightarrow mathbb{R}$ is in $L^2$. All the parameters are greater zero. Given initial values $u = u_t = 0 , t=0$ as well as homogeneous boundary conditions: $u(0,t) = u(l,t) = 0$ .I would like to find out the following:
For which values of $omega$ does the solution grow over time $t$ , when $r = 0$ and for which values $omega$ when $r > 0$? I am not quite sure how to tackle this problem. In the first case, i.e. $r=0$, I thought I might find an explicit solution but I do not think that this is the best way to solve this, since in the second case I would not be able to find any solution.
I do not necessarily need a complete solution, but any hints or help on how this could be approached would be greatly appreciated!
pde wave-equation
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
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show 1 more comment
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I have been given the following wave equation:
$$u_{tt}(x,t) = c^2u_{xx}(x,t)-ru_t(x,t)+g(x)sin(omega t)$$
where $t >0, xin (0,l)$ and $g: [0,l] rightarrow mathbb{R}$ is in $L^2$. All the parameters are greater zero. Given initial values $u = u_t = 0 , t=0$ as well as homogeneous boundary conditions: $u(0,t) = u(l,t) = 0$ .I would like to find out the following:
For which values of $omega$ does the solution grow over time $t$ , when $r = 0$ and for which values $omega$ when $r > 0$? I am not quite sure how to tackle this problem. In the first case, i.e. $r=0$, I thought I might find an explicit solution but I do not think that this is the best way to solve this, since in the second case I would not be able to find any solution.
I do not necessarily need a complete solution, but any hints or help on how this could be approached would be greatly appreciated!
pde wave-equation
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
$endgroup$
$begingroup$
Have you tried taking Fourier series (in x only)? In that way you should get second order inhomogeneous ODEs in t and they are solvable.
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– ydx
Dec 18 '18 at 21:31
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You probably have a Dirichlet boundary condition (or some other boundary condition) at $x=0, x=l$.
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– Giuseppe Negro
Dec 18 '18 at 21:34
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@GiuseppeNegro Yes, I forgot to mention that - I updated the question
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– GreenLogic
Dec 18 '18 at 21:36
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@ydx In the second case this might get messy but I will give this a try - thanks!
$endgroup$
– GreenLogic
Dec 18 '18 at 21:53
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In steps: 1. Use an eigenfunction on $u(x,t)$ to eventually write out a ODE for the coefficients in the fourier series as a function of $t$ (say $a_n(t)$). 2. You will then only need to discuss the behavior of this second order ODE. An explicit solution isn't completely necessary, but you do like two thirds of it.
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– DaveNine
Dec 19 '18 at 0:32
|
show 1 more comment
$begingroup$
I have been given the following wave equation:
$$u_{tt}(x,t) = c^2u_{xx}(x,t)-ru_t(x,t)+g(x)sin(omega t)$$
where $t >0, xin (0,l)$ and $g: [0,l] rightarrow mathbb{R}$ is in $L^2$. All the parameters are greater zero. Given initial values $u = u_t = 0 , t=0$ as well as homogeneous boundary conditions: $u(0,t) = u(l,t) = 0$ .I would like to find out the following:
For which values of $omega$ does the solution grow over time $t$ , when $r = 0$ and for which values $omega$ when $r > 0$? I am not quite sure how to tackle this problem. In the first case, i.e. $r=0$, I thought I might find an explicit solution but I do not think that this is the best way to solve this, since in the second case I would not be able to find any solution.
I do not necessarily need a complete solution, but any hints or help on how this could be approached would be greatly appreciated!
pde wave-equation
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
$endgroup$
I have been given the following wave equation:
$$u_{tt}(x,t) = c^2u_{xx}(x,t)-ru_t(x,t)+g(x)sin(omega t)$$
where $t >0, xin (0,l)$ and $g: [0,l] rightarrow mathbb{R}$ is in $L^2$. All the parameters are greater zero. Given initial values $u = u_t = 0 , t=0$ as well as homogeneous boundary conditions: $u(0,t) = u(l,t) = 0$ .I would like to find out the following:
For which values of $omega$ does the solution grow over time $t$ , when $r = 0$ and for which values $omega$ when $r > 0$? I am not quite sure how to tackle this problem. In the first case, i.e. $r=0$, I thought I might find an explicit solution but I do not think that this is the best way to solve this, since in the second case I would not be able to find any solution.
I do not necessarily need a complete solution, but any hints or help on how this could be approached would be greatly appreciated!
pde wave-equation
pde wave-equation
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
edited Dec 18 '18 at 21:36
GreenLogic
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
asked Dec 18 '18 at 21:09
GreenLogicGreenLogic
2951312
2951312
$begingroup$
Have you tried taking Fourier series (in x only)? In that way you should get second order inhomogeneous ODEs in t and they are solvable.
$endgroup$
– ydx
Dec 18 '18 at 21:31
$begingroup$
You probably have a Dirichlet boundary condition (or some other boundary condition) at $x=0, x=l$.
$endgroup$
– Giuseppe Negro
Dec 18 '18 at 21:34
$begingroup$
@GiuseppeNegro Yes, I forgot to mention that - I updated the question
$endgroup$
– GreenLogic
Dec 18 '18 at 21:36
$begingroup$
@ydx In the second case this might get messy but I will give this a try - thanks!
$endgroup$
– GreenLogic
Dec 18 '18 at 21:53
$begingroup$
In steps: 1. Use an eigenfunction on $u(x,t)$ to eventually write out a ODE for the coefficients in the fourier series as a function of $t$ (say $a_n(t)$). 2. You will then only need to discuss the behavior of this second order ODE. An explicit solution isn't completely necessary, but you do like two thirds of it.
$endgroup$
– DaveNine
Dec 19 '18 at 0:32
|
show 1 more comment
$begingroup$
Have you tried taking Fourier series (in x only)? In that way you should get second order inhomogeneous ODEs in t and they are solvable.
$endgroup$
– ydx
Dec 18 '18 at 21:31
$begingroup$
You probably have a Dirichlet boundary condition (or some other boundary condition) at $x=0, x=l$.
$endgroup$
– Giuseppe Negro
Dec 18 '18 at 21:34
$begingroup$
@GiuseppeNegro Yes, I forgot to mention that - I updated the question
$endgroup$
– GreenLogic
Dec 18 '18 at 21:36
$begingroup$
@ydx In the second case this might get messy but I will give this a try - thanks!
$endgroup$
– GreenLogic
Dec 18 '18 at 21:53
$begingroup$
In steps: 1. Use an eigenfunction on $u(x,t)$ to eventually write out a ODE for the coefficients in the fourier series as a function of $t$ (say $a_n(t)$). 2. You will then only need to discuss the behavior of this second order ODE. An explicit solution isn't completely necessary, but you do like two thirds of it.
$endgroup$
– DaveNine
Dec 19 '18 at 0:32
$begingroup$
Have you tried taking Fourier series (in x only)? In that way you should get second order inhomogeneous ODEs in t and they are solvable.
$endgroup$
– ydx
Dec 18 '18 at 21:31
$begingroup$
Have you tried taking Fourier series (in x only)? In that way you should get second order inhomogeneous ODEs in t and they are solvable.
$endgroup$
– ydx
Dec 18 '18 at 21:31
$begingroup$
You probably have a Dirichlet boundary condition (or some other boundary condition) at $x=0, x=l$.
$endgroup$
– Giuseppe Negro
Dec 18 '18 at 21:34
$begingroup$
You probably have a Dirichlet boundary condition (or some other boundary condition) at $x=0, x=l$.
$endgroup$
– Giuseppe Negro
Dec 18 '18 at 21:34
$begingroup$
@GiuseppeNegro Yes, I forgot to mention that - I updated the question
$endgroup$
– GreenLogic
Dec 18 '18 at 21:36
$begingroup$
@GiuseppeNegro Yes, I forgot to mention that - I updated the question
$endgroup$
– GreenLogic
Dec 18 '18 at 21:36
$begingroup$
@ydx In the second case this might get messy but I will give this a try - thanks!
$endgroup$
– GreenLogic
Dec 18 '18 at 21:53
$begingroup$
@ydx In the second case this might get messy but I will give this a try - thanks!
$endgroup$
– GreenLogic
Dec 18 '18 at 21:53
$begingroup$
In steps: 1. Use an eigenfunction on $u(x,t)$ to eventually write out a ODE for the coefficients in the fourier series as a function of $t$ (say $a_n(t)$). 2. You will then only need to discuss the behavior of this second order ODE. An explicit solution isn't completely necessary, but you do like two thirds of it.
$endgroup$
– DaveNine
Dec 19 '18 at 0:32
$begingroup$
In steps: 1. Use an eigenfunction on $u(x,t)$ to eventually write out a ODE for the coefficients in the fourier series as a function of $t$ (say $a_n(t)$). 2. You will then only need to discuss the behavior of this second order ODE. An explicit solution isn't completely necessary, but you do like two thirds of it.
$endgroup$
– DaveNine
Dec 19 '18 at 0:32
|
show 1 more comment
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$begingroup$
Have you tried taking Fourier series (in x only)? In that way you should get second order inhomogeneous ODEs in t and they are solvable.
$endgroup$
– ydx
Dec 18 '18 at 21:31
$begingroup$
You probably have a Dirichlet boundary condition (or some other boundary condition) at $x=0, x=l$.
$endgroup$
– Giuseppe Negro
Dec 18 '18 at 21:34
$begingroup$
@GiuseppeNegro Yes, I forgot to mention that - I updated the question
$endgroup$
– GreenLogic
Dec 18 '18 at 21:36
$begingroup$
@ydx In the second case this might get messy but I will give this a try - thanks!
$endgroup$
– GreenLogic
Dec 18 '18 at 21:53
$begingroup$
In steps: 1. Use an eigenfunction on $u(x,t)$ to eventually write out a ODE for the coefficients in the fourier series as a function of $t$ (say $a_n(t)$). 2. You will then only need to discuss the behavior of this second order ODE. An explicit solution isn't completely necessary, but you do like two thirds of it.
$endgroup$
– DaveNine
Dec 19 '18 at 0:32