Getting rid on internal boundaries
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Suppose I have a volume, $V$ split up into two regions $V=V_{1}cup V_{2}$ say with and interface $Sigma$ between them and I am considering the heat equation in each region:
$$frac{partial T_{1,2}}{partial t}=nablacdotleft(k_{1,2}(x)nabla T_{1,2}right),$$
Is there a way of defining a "global" $T$, such that:
$$T=leftlbrace
begin{array}{cc}
T_{1} & xin V_{1} \
T_{2} & xin V_{2}
end{array}right.$$
Which eliminates the use the usual interfacial conditions:
$$T_{1}(t,mathbf{x})=T_{2}(t,mathbf{x})quad xinSigma$$
$$frac{partial T_{1}}{partial n}(t,mathbf{x})=frac{partial T_{2}}{partial n}(t,mathbf{x})quad xinSigma$$
So basically, can I derive an equation for $T$ which just uses the boundary conditions on $V$ rather than $Sigma$?
pde boundary-value-problem heat-equation
edited Nov 27 at 20:10
Dylan
12.1k31026
asked Nov 12 at 19:35
Matthew Hunt
36916
add a comment |
up vote
0
down vote
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Suppose I have a volume, $V$ split up into two regions $V=V_{1}cup V_{2}$ say with and interface $Sigma$ between them and I am considering the heat equation in each region:
$$frac{partial T_{1,2}}{partial t}=nablacdotleft(k_{1,2}(x)nabla T_{1,2}right),$$
Is there a way of defining a "global" $T$, such that:
$$T=leftlbrace
begin{array}{cc}
T_{1} & xin V_{1} \
T_{2} & xin V_{2}
end{array}right.$$
Which eliminates the use the usual interfacial conditions:
$$T_{1}(t,mathbf{x})=T_{2}(t,mathbf{x})quad xinSigma$$
$$frac{partial T_{1}}{partial n}(t,mathbf{x})=frac{partial T_{2}}{partial n}(t,mathbf{x})quad xinSigma$$
So basically, can I derive an equation for $T$ which just uses the boundary conditions on $V$ rather than $Sigma$?
pde boundary-value-problem heat-equation
edited Nov 27 at 20:10
Dylan
12.1k31026
asked Nov 12 at 19:35
Matthew Hunt
36916
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose I have a volume, $V$ split up into two regions $V=V_{1}cup V_{2}$ say with and interface $Sigma$ between them and I am considering the heat equation in each region:
$$frac{partial T_{1,2}}{partial t}=nablacdotleft(k_{1,2}(x)nabla T_{1,2}right),$$
Is there a way of defining a "global" $T$, such that:
$$T=leftlbrace
begin{array}{cc}
T_{1} & xin V_{1} \
T_{2} & xin V_{2}
end{array}right.$$
Which eliminates the use the usual interfacial conditions:
$$T_{1}(t,mathbf{x})=T_{2}(t,mathbf{x})quad xinSigma$$
$$frac{partial T_{1}}{partial n}(t,mathbf{x})=frac{partial T_{2}}{partial n}(t,mathbf{x})quad xinSigma$$
So basically, can I derive an equation for $T$ which just uses the boundary conditions on $V$ rather than $Sigma$?
pde boundary-value-problem heat-equation
edited Nov 27 at 20:10
Dylan
12.1k31026
asked Nov 12 at 19:35
Matthew Hunt
36916
Suppose I have a volume, $V$ split up into two regions $V=V_{1}cup V_{2}$ say with and interface $Sigma$ between them and I am considering the heat equation in each region:
$$frac{partial T_{1,2}}{partial t}=nablacdotleft(k_{1,2}(x)nabla T_{1,2}right),$$
Is there a way of defining a "global" $T$, such that:
$$T=leftlbrace
begin{array}{cc}
T_{1} & xin V_{1} \
T_{2} & xin V_{2}
end{array}right.$$
Which eliminates the use the usual interfacial conditions:
$$T_{1}(t,mathbf{x})=T_{2}(t,mathbf{x})quad xinSigma$$
$$frac{partial T_{1}}{partial n}(t,mathbf{x})=frac{partial T_{2}}{partial n}(t,mathbf{x})quad xinSigma$$
So basically, can I derive an equation for $T$ which just uses the boundary conditions on $V$ rather than $Sigma$?
pde boundary-value-problem heat-equation
pde boundary-value-problem heat-equation
edited Nov 27 at 20:10
Dylan
12.1k31026
asked Nov 12 at 19:35
Matthew Hunt
36916
edited Nov 27 at 20:10
Dylan
12.1k31026
asked Nov 12 at 19:35
Matthew Hunt
36916
edited Nov 27 at 20:10
Dylan
12.1k31026
edited Nov 27 at 20:10
Dylan
12.1k31026
edited Nov 27 at 20:10
Dylan
12.1k31026
12.1k31026
asked Nov 12 at 19:35
Matthew Hunt
36916
asked Nov 12 at 19:35
Matthew Hunt
36916
asked Nov 12 at 19:35
Matthew Hunt
36916
36916
add a comment |
add a comment |
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