unknown operator in homogenous model of heat application on plate.
$begingroup$
I got the following assingment form my employer.
Consider the following partial differential equation describing the heating of a two dimensional plate:
$rho(x,y)c(x,y)frac{partial T}{partial t}(x,y,t)=begin{bmatrix}frac{partial}{partial x} frac{partial}{partial y} end{bmatrix} K(x,y) begin{bmatrix} frac{partial T(x,y)}{partial x} \ frac{partial T(x,y)}{partial y} end{bmatrix}+u(x,y,t) qquad qquad qquad (1)$
In which:
$x$ and $y$ are positions on the plate, $rho$ is the density, $c$ is the heat capacity, $T$ is the temperature of the plate, $K=kappa (x,y)I$ in which $kappa$ is the thermal conductivity and $I$ the 2x2 identity matrix and $u$ is the input.
Next we assume that the model is homogenous and thus:
$rho(x,y)=rho (>0)$
$c(x,y)=c (>0)$
$kappa(x,y)=kappa (>0)$
Show that under these conditions, $(1)$ admits a solution of the form $T(x,y,t)=a(t)varphi^{(x)}(x)varphi^{(y)}(y)$ whenever $u(x,y,t)=0$. Here $a, varphi^{(x)}$ and $varphi^{(y)}$ are scalar valued functions on $mathbb{R}$ that satisfy the separated differential equations
$ddot{varphi}^{(x)}-lambda_x varphi^{(x)}=0, quad ddot{varphi}^{(y)}-lambda_y varphi^{(y)}=0, quaddot{a}-lambda a=0$
for suitable constants $lambda_x, lambda_y$ and $lambda$
I would love to get a complete answer but lets just start at this:
What does this mean $ddot{varphi}^{(x)}$? The second time derivative of $frac{partial varphi}{partial x}$?
dynamical-systems mathematical-modeling
asked Dec 20 '18 at 13:14
user463102user463102
7913
$endgroup$
add a comment |
$begingroup$
I got the following assingment form my employer.
Consider the following partial differential equation describing the heating of a two dimensional plate:
$rho(x,y)c(x,y)frac{partial T}{partial t}(x,y,t)=begin{bmatrix}frac{partial}{partial x} frac{partial}{partial y} end{bmatrix} K(x,y) begin{bmatrix} frac{partial T(x,y)}{partial x} \ frac{partial T(x,y)}{partial y} end{bmatrix}+u(x,y,t) qquad qquad qquad (1)$
In which:
$x$ and $y$ are positions on the plate, $rho$ is the density, $c$ is the heat capacity, $T$ is the temperature of the plate, $K=kappa (x,y)I$ in which $kappa$ is the thermal conductivity and $I$ the 2x2 identity matrix and $u$ is the input.
Next we assume that the model is homogenous and thus:
$rho(x,y)=rho (>0)$
$c(x,y)=c (>0)$
$kappa(x,y)=kappa (>0)$
Show that under these conditions, $(1)$ admits a solution of the form $T(x,y,t)=a(t)varphi^{(x)}(x)varphi^{(y)}(y)$ whenever $u(x,y,t)=0$. Here $a, varphi^{(x)}$ and $varphi^{(y)}$ are scalar valued functions on $mathbb{R}$ that satisfy the separated differential equations
$ddot{varphi}^{(x)}-lambda_x varphi^{(x)}=0, quad ddot{varphi}^{(y)}-lambda_y varphi^{(y)}=0, quaddot{a}-lambda a=0$
for suitable constants $lambda_x, lambda_y$ and $lambda$
I would love to get a complete answer but lets just start at this:
What does this mean $ddot{varphi}^{(x)}$? The second time derivative of $frac{partial varphi}{partial x}$?
dynamical-systems mathematical-modeling
asked Dec 20 '18 at 13:14
user463102user463102
7913
$endgroup$
1
$begingroup$
In this context $varphi^{(x)}$ is independent of $t$, hence I think $ddot{varphi}^{(x)} = frac{d^2varphi^{(x)}}{dx^2}$. Do you have any initial conditions, i.e. what $T(x,y,0)$ looks like? You could prescribe this and get a solution, but if this is not in the requested form, you won't achieve your goal. Another thing: Do you have boundary conditions or is your plate infinite, e.g. $mathbb{R}^2$?
$endgroup$
– humanStampedist
Dec 20 '18 at 14:26
add a comment |
$begingroup$
I got the following assingment form my employer.
Consider the following partial differential equation describing the heating of a two dimensional plate:
$rho(x,y)c(x,y)frac{partial T}{partial t}(x,y,t)=begin{bmatrix}frac{partial}{partial x} frac{partial}{partial y} end{bmatrix} K(x,y) begin{bmatrix} frac{partial T(x,y)}{partial x} \ frac{partial T(x,y)}{partial y} end{bmatrix}+u(x,y,t) qquad qquad qquad (1)$
In which:
$x$ and $y$ are positions on the plate, $rho$ is the density, $c$ is the heat capacity, $T$ is the temperature of the plate, $K=kappa (x,y)I$ in which $kappa$ is the thermal conductivity and $I$ the 2x2 identity matrix and $u$ is the input.
Next we assume that the model is homogenous and thus:
$rho(x,y)=rho (>0)$
$c(x,y)=c (>0)$
$kappa(x,y)=kappa (>0)$
Show that under these conditions, $(1)$ admits a solution of the form $T(x,y,t)=a(t)varphi^{(x)}(x)varphi^{(y)}(y)$ whenever $u(x,y,t)=0$. Here $a, varphi^{(x)}$ and $varphi^{(y)}$ are scalar valued functions on $mathbb{R}$ that satisfy the separated differential equations
$ddot{varphi}^{(x)}-lambda_x varphi^{(x)}=0, quad ddot{varphi}^{(y)}-lambda_y varphi^{(y)}=0, quaddot{a}-lambda a=0$
for suitable constants $lambda_x, lambda_y$ and $lambda$
I would love to get a complete answer but lets just start at this:
What does this mean $ddot{varphi}^{(x)}$? The second time derivative of $frac{partial varphi}{partial x}$?
dynamical-systems mathematical-modeling
asked Dec 20 '18 at 13:14
user463102user463102
7913
$endgroup$
I got the following assingment form my employer.
Consider the following partial differential equation describing the heating of a two dimensional plate:
$rho(x,y)c(x,y)frac{partial T}{partial t}(x,y,t)=begin{bmatrix}frac{partial}{partial x} frac{partial}{partial y} end{bmatrix} K(x,y) begin{bmatrix} frac{partial T(x,y)}{partial x} \ frac{partial T(x,y)}{partial y} end{bmatrix}+u(x,y,t) qquad qquad qquad (1)$
In which:
$x$ and $y$ are positions on the plate, $rho$ is the density, $c$ is the heat capacity, $T$ is the temperature of the plate, $K=kappa (x,y)I$ in which $kappa$ is the thermal conductivity and $I$ the 2x2 identity matrix and $u$ is the input.
Next we assume that the model is homogenous and thus:
$rho(x,y)=rho (>0)$
$c(x,y)=c (>0)$
$kappa(x,y)=kappa (>0)$
Show that under these conditions, $(1)$ admits a solution of the form $T(x,y,t)=a(t)varphi^{(x)}(x)varphi^{(y)}(y)$ whenever $u(x,y,t)=0$. Here $a, varphi^{(x)}$ and $varphi^{(y)}$ are scalar valued functions on $mathbb{R}$ that satisfy the separated differential equations
$ddot{varphi}^{(x)}-lambda_x varphi^{(x)}=0, quad ddot{varphi}^{(y)}-lambda_y varphi^{(y)}=0, quaddot{a}-lambda a=0$
for suitable constants $lambda_x, lambda_y$ and $lambda$
I would love to get a complete answer but lets just start at this:
What does this mean $ddot{varphi}^{(x)}$? The second time derivative of $frac{partial varphi}{partial x}$?
dynamical-systems mathematical-modeling
dynamical-systems mathematical-modeling
asked Dec 20 '18 at 13:14
user463102user463102
7913
asked Dec 20 '18 at 13:14
user463102user463102
7913
asked Dec 20 '18 at 13:14
user463102user463102
7913
asked Dec 20 '18 at 13:14
user463102user463102
7913
asked Dec 20 '18 at 13:14
user463102user463102
7913
7913
1
$begingroup$
In this context $varphi^{(x)}$ is independent of $t$, hence I think $ddot{varphi}^{(x)} = frac{d^2varphi^{(x)}}{dx^2}$. Do you have any initial conditions, i.e. what $T(x,y,0)$ looks like? You could prescribe this and get a solution, but if this is not in the requested form, you won't achieve your goal. Another thing: Do you have boundary conditions or is your plate infinite, e.g. $mathbb{R}^2$?
$endgroup$
– humanStampedist
Dec 20 '18 at 14:26
add a comment |
1
$begingroup$
In this context $varphi^{(x)}$ is independent of $t$, hence I think $ddot{varphi}^{(x)} = frac{d^2varphi^{(x)}}{dx^2}$. Do you have any initial conditions, i.e. what $T(x,y,0)$ looks like? You could prescribe this and get a solution, but if this is not in the requested form, you won't achieve your goal. Another thing: Do you have boundary conditions or is your plate infinite, e.g. $mathbb{R}^2$?
$endgroup$
– humanStampedist
Dec 20 '18 at 14:26
1
1
$begingroup$
In this context $varphi^{(x)}$ is independent of $t$, hence I think $ddot{varphi}^{(x)} = frac{d^2varphi^{(x)}}{dx^2}$. Do you have any initial conditions, i.e. what $T(x,y,0)$ looks like? You could prescribe this and get a solution, but if this is not in the requested form, you won't achieve your goal. Another thing: Do you have boundary conditions or is your plate infinite, e.g. $mathbb{R}^2$?
$endgroup$
– humanStampedist
Dec 20 '18 at 14:26
$begingroup$
In this context $varphi^{(x)}$ is independent of $t$, hence I think $ddot{varphi}^{(x)} = frac{d^2varphi^{(x)}}{dx^2}$. Do you have any initial conditions, i.e. what $T(x,y,0)$ looks like? You could prescribe this and get a solution, but if this is not in the requested form, you won't achieve your goal. Another thing: Do you have boundary conditions or is your plate infinite, e.g. $mathbb{R}^2$?
$endgroup$
– humanStampedist
Dec 20 '18 at 14:26
add a comment |
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1
$begingroup$
In this context $varphi^{(x)}$ is independent of $t$, hence I think $ddot{varphi}^{(x)} = frac{d^2varphi^{(x)}}{dx^2}$. Do you have any initial conditions, i.e. what $T(x,y,0)$ looks like? You could prescribe this and get a solution, but if this is not in the requested form, you won't achieve your goal. Another thing: Do you have boundary conditions or is your plate infinite, e.g. $mathbb{R}^2$?
$endgroup$
– humanStampedist
Dec 20 '18 at 14:26