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Physical meaning of PDE
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When studying PDE, I want to ask if any physical meaning for the following PDE:
$$fracpartial Wpartial u=fracpartial^2 Wpartial x^2$$ such that $t>0, 0<x<infty$ and $W(0,x)=delta_0(x)$. Standard book has physical meaning if $-infty<x<infty$ (Placing a unit impulse at $x=0$ and let it diffuse and recording the temperature). In the example given, can we use the same interpretation? What makes me confuse is that the domain defined. Usually, we do not define at $x=0$ and $t=0$ when I read the books. So I guess $W(u,x)=0$ since $delta_0(x)=0$ if $xneq 0$. But I can think of $W(u,x)neq 0$. $frac1sqrt4pi ue^-fracx^24u$ is a solution
pde heat-equation regularity-theory-of-pdes parabolic-pde
asked Jul 18 at 6:38
will_cheuk
1078
add a comment |Â
up vote
2
down vote
favorite
When studying PDE, I want to ask if any physical meaning for the following PDE:
$$fracpartial Wpartial u=fracpartial^2 Wpartial x^2$$ such that $t>0, 0<x<infty$ and $W(0,x)=delta_0(x)$. Standard book has physical meaning if $-infty<x<infty$ (Placing a unit impulse at $x=0$ and let it diffuse and recording the temperature). In the example given, can we use the same interpretation? What makes me confuse is that the domain defined. Usually, we do not define at $x=0$ and $t=0$ when I read the books. So I guess $W(u,x)=0$ since $delta_0(x)=0$ if $xneq 0$. But I can think of $W(u,x)neq 0$. $frac1sqrt4pi ue^-fracx^24u$ is a solution
pde heat-equation regularity-theory-of-pdes parabolic-pde
asked Jul 18 at 6:38
will_cheuk
1078
In your equation, should it be $partial W/partial t$ instead of $partial W/partial u$? Also, you need to specify boundary conditions at $x=0$.
– Rahul
Jul 18 at 6:50
yes, it is $fracpartial Wpartial t$. No, I do not want to specify boundary condition.
– will_cheuk
Jul 18 at 7:05
What are the boundary conditions at $x=0$ then? A physical interpretation would depend on what these are.
– Winther
Jul 22 at 19:03
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
When studying PDE, I want to ask if any physical meaning for the following PDE:
$$fracpartial Wpartial u=fracpartial^2 Wpartial x^2$$ such that $t>0, 0<x<infty$ and $W(0,x)=delta_0(x)$. Standard book has physical meaning if $-infty<x<infty$ (Placing a unit impulse at $x=0$ and let it diffuse and recording the temperature). In the example given, can we use the same interpretation? What makes me confuse is that the domain defined. Usually, we do not define at $x=0$ and $t=0$ when I read the books. So I guess $W(u,x)=0$ since $delta_0(x)=0$ if $xneq 0$. But I can think of $W(u,x)neq 0$. $frac1sqrt4pi ue^-fracx^24u$ is a solution
pde heat-equation regularity-theory-of-pdes parabolic-pde
asked Jul 18 at 6:38
will_cheuk
1078
When studying PDE, I want to ask if any physical meaning for the following PDE:
$$fracpartial Wpartial u=fracpartial^2 Wpartial x^2$$ such that $t>0, 0<x<infty$ and $W(0,x)=delta_0(x)$. Standard book has physical meaning if $-infty<x<infty$ (Placing a unit impulse at $x=0$ and let it diffuse and recording the temperature). In the example given, can we use the same interpretation? What makes me confuse is that the domain defined. Usually, we do not define at $x=0$ and $t=0$ when I read the books. So I guess $W(u,x)=0$ since $delta_0(x)=0$ if $xneq 0$. But I can think of $W(u,x)neq 0$. $frac1sqrt4pi ue^-fracx^24u$ is a solution
pde heat-equation regularity-theory-of-pdes parabolic-pde
asked Jul 18 at 6:38
will_cheuk
1078
asked Jul 18 at 6:38
will_cheuk
1078
asked Jul 18 at 6:38
will_cheuk
1078
asked Jul 18 at 6:38
will_cheuk
1078
1078
In your equation, should it be $partial W/partial t$ instead of $partial W/partial u$? Also, you need to specify boundary conditions at $x=0$.
– Rahul
Jul 18 at 6:50
yes, it is $fracpartial Wpartial t$. No, I do not want to specify boundary condition.
– will_cheuk
Jul 18 at 7:05
What are the boundary conditions at $x=0$ then? A physical interpretation would depend on what these are.
– Winther
Jul 22 at 19:03
add a comment |Â
In your equation, should it be $partial W/partial t$ instead of $partial W/partial u$? Also, you need to specify boundary conditions at $x=0$.
– Rahul
Jul 18 at 6:50
yes, it is $fracpartial Wpartial t$. No, I do not want to specify boundary condition.
– will_cheuk
Jul 18 at 7:05
What are the boundary conditions at $x=0$ then? A physical interpretation would depend on what these are.
– Winther
Jul 22 at 19:03
In your equation, should it be $partial W/partial t$ instead of $partial W/partial u$? Also, you need to specify boundary conditions at $x=0$.
– Rahul
Jul 18 at 6:50
In your equation, should it be $partial W/partial t$ instead of $partial W/partial u$? Also, you need to specify boundary conditions at $x=0$.
– Rahul
Jul 18 at 6:50
yes, it is $fracpartial Wpartial t$. No, I do not want to specify boundary condition.
– will_cheuk
Jul 18 at 7:05
yes, it is $fracpartial Wpartial t$. No, I do not want to specify boundary condition.
– will_cheuk
Jul 18 at 7:05
What are the boundary conditions at $x=0$ then? A physical interpretation would depend on what these are.
– Winther
Jul 22 at 19:03
What are the boundary conditions at $x=0$ then? A physical interpretation would depend on what these are.
– Winther
Jul 22 at 19:03
add a comment |Â
2 Answers
2
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votes
up vote
0
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This equation describes e.g. heat diffusion.
Your initial condition describes a unit source at $(0, 0)$.
The complex version is e.g. the Schrödinger equation.
answered Jul 18 at 6:55
mvw
30.5k22250
Does it has a solution
– will_cheuk
Jul 18 at 7:04
This article might have it: rose-hulman.edu/~bryan/lottamath/heatkern.pdf
– mvw
Jul 18 at 7:16
Thanks. The paper focus on $-infty<x<infty$. I would like to know if we have the same meaning when we consider half plane. Someone requests that the initial function to be a continuous function in $C$ to make further discussions reasonable. (say the boundary $x=0$, we impose a continuous function in $t$ such that initial condition and boundary condition coincide at $t=0,x=0$. I wonder if we can loose this restriction, say at time 0, $W(0,x)=delta_0(x)$ and make heat $W(t,x)=frac1sqrtt$ when $t>0$)
– will_cheuk
Jul 18 at 7:31
add a comment |Â
up vote
0
down vote
The heat-type equation you are studying is the differential formulation (i.e. a formulation involving pointwise values of the quantities involved, like spatial derivatives) of the general balance equation, and the $delta(x)$ represents a source term located in $x=0$ and independent of $t$. To see this, let's analyze the $n$-dimensional case and then particularize the results for $n=1$.
Let $GsubsetmathbbR^n$ a domain for which some form of the divergence theorem holds, for example a Caccioppoli set, and suppose that this set is characterized by a time-dependent quantity $Q(t)$.
The quantity $Q$ can vary respect time only only under the action of sources lying in the interior of $G$ or by flowing across the boundary $partial G$ of $G$: this is the heuristic content of the balance equation
$$
fracmathrmdQ(t)mathrmdt=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdxtag1label1
$$
where
- $boldsymbolq(x,t)$ is the flux density vector of the quantity $Q(t)$ across $partial G$ at the point $x$
- $boldsymboln_x$ is the inner (measure theoretic, for a Caccioppoli set) normal at the point $x$.
- $s(x,t)$ is a general inner term (representing for example processes of absorption/generation of the given quantity taking place within $G$).
The further step in the deduction is to assume a particular form for $Q(t)$ and for the flux $boldsymbolq(x,t)$. This is precisely the meaning of the following equations:
$$
Q(t)=int_G c(x)rho(x) w(x,t),mathrmdx,tag2label2
$$
where
- $c(x)$ is a function which depends only on the properties of the medium which fills $G$ (if we are dealing with heat, it is called heat capacity),
- $rho(x)$ is a function which depends only on the density of the medium (it is commonly called mass density),
and
$$
boldsymbolq(x,t)=boldsymbolhatK(x)nabla w(x,t),tag3label3
$$
where $boldsymbolhatK(x)$ is tensor that again takes care of the possibly anisotropic behavior of the flux.
Now, by using equations eqref1, eqref2, eqref3 and the Gauss-Green-Ostrogradsky (divergence) theorem we obtain
$$
beginsplit
fracmathrmdQ(t)mathrmdt &=int_G c(x)rho(x)fracpartial w(x,t)partial t,mathrmdx\
&=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_partial GboldsymbolhatK(x)nabla w(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_Gbig[nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)+ s(x,t)big]mathrmdx
endsplit
$$
and then
$$
int_Gleft[ c(x)rho(x)fracpartial w(x,t)partial t-nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)-s(x,t)right]mathrmdx=0
$$
The above equation integral equation is satisfied if and only if
$$
c(x)rho(x)fracpartial w(x,t)partial t=nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big) + s(x,t)tag4label4
$$
Equation eqref4 is the general linear (divergence form) parabolic equation: if we assume $c(x)equivrho(x)equiv1$, $boldsymbolhatK(x)equivboldsymbol1$ and $s(x,t)=delta(x)timesdelta(t)=delta(x)delta(t)$ (the tensor products of Dirac deltas respect to the variables $xinmathbbR^n$ and $tinmathbbR$) we obtain the standard heat equation:
$$
beginsplit
fracpartial w(x,t)partial t&=Delta w(x,t)+delta(x)delta(t)quad n>1\
fracpartial w(x,t)partial t&=fracpartial^2w(x,t)partial x^2+delta(x)delta(t)quad n=1
endsplit
$$
If you need to specify only the presence of a time independent term, basically considering only $delta(x)$ or, the other way out, only a time-dependent term, you can put
$$
s(x,t)=
begincases
delta(x) & xinmathbbR^n\
delta(t) & tinmathbbR
endcases
$$
therefore deltas respect to single variables or their tensor products can be interpreted as source terms in the balance equation eqref1.
Notes
Many (if not all) evolution equations of mathematical physics can be deduced from the general balance equation eqref1 (for example the wave equation), just by properly choosing the defining equations eqref2 and eqref3, which therefore play the rôle of constitutive axioms of a theory. In particular eqref3 is the Fourier-Duhamel law of heat conduction.
A deduction of the one-dimensional heat equation (without considering the source term), however conceptually identical to the deduction shown above, is offered by Cannon (1984) (§1.1, pp. 13-15) and by Widder (1978) (§2 and §3, pp. 1-5). I prefer the one above since it don't requires the knowledge of any special (however elementary) physical concepts, being valid also for the general linear diffusion equations and other parabolic PDEs.
[1] Cannon, J. R. (1984), The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23 (1st ed.), Reading etc.: Addison-Wesley Publishing Company, pp. XXV+483, ISBN 978-0-521-30243-2, MR 0747979, Zbl 0567.35001
[2] Widder, D. V. (1978), The Heat Equation, Pure and Applied Mathematics 67, Academic Press, pp. xiv+267, ISBN 0-12-748540-6, MR0466967, Zbl 0322.35041.
answered Jul 22 at 12:34
Daniele Tampieri
7301313
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
This equation describes e.g. heat diffusion.
Your initial condition describes a unit source at $(0, 0)$.
The complex version is e.g. the Schrödinger equation.
answered Jul 18 at 6:55
mvw
30.5k22250
Does it has a solution
– will_cheuk
Jul 18 at 7:04
This article might have it: rose-hulman.edu/~bryan/lottamath/heatkern.pdf
– mvw
Jul 18 at 7:16
Thanks. The paper focus on $-infty<x<infty$. I would like to know if we have the same meaning when we consider half plane. Someone requests that the initial function to be a continuous function in $C$ to make further discussions reasonable. (say the boundary $x=0$, we impose a continuous function in $t$ such that initial condition and boundary condition coincide at $t=0,x=0$. I wonder if we can loose this restriction, say at time 0, $W(0,x)=delta_0(x)$ and make heat $W(t,x)=frac1sqrtt$ when $t>0$)
– will_cheuk
Jul 18 at 7:31
add a comment |Â
up vote
0
down vote
This equation describes e.g. heat diffusion.
Your initial condition describes a unit source at $(0, 0)$.
The complex version is e.g. the Schrödinger equation.
answered Jul 18 at 6:55
mvw
30.5k22250
Does it has a solution
– will_cheuk
Jul 18 at 7:04
This article might have it: rose-hulman.edu/~bryan/lottamath/heatkern.pdf
– mvw
Jul 18 at 7:16
Thanks. The paper focus on $-infty<x<infty$. I would like to know if we have the same meaning when we consider half plane. Someone requests that the initial function to be a continuous function in $C$ to make further discussions reasonable. (say the boundary $x=0$, we impose a continuous function in $t$ such that initial condition and boundary condition coincide at $t=0,x=0$. I wonder if we can loose this restriction, say at time 0, $W(0,x)=delta_0(x)$ and make heat $W(t,x)=frac1sqrtt$ when $t>0$)
– will_cheuk
Jul 18 at 7:31
add a comment |Â
up vote
0
down vote
up vote
0
down vote
This equation describes e.g. heat diffusion.
Your initial condition describes a unit source at $(0, 0)$.
The complex version is e.g. the Schrödinger equation.
answered Jul 18 at 6:55
mvw
30.5k22250
This equation describes e.g. heat diffusion.
Your initial condition describes a unit source at $(0, 0)$.
The complex version is e.g. the Schrödinger equation.
answered Jul 18 at 6:55
mvw
30.5k22250
edited Jul 18 at 7:01
answered Jul 18 at 6:55
mvw
30.5k22250
answered Jul 18 at 6:55
mvw
30.5k22250
answered Jul 18 at 6:55
mvw
30.5k22250
30.5k22250
Does it has a solution
– will_cheuk
Jul 18 at 7:04
This article might have it: rose-hulman.edu/~bryan/lottamath/heatkern.pdf
– mvw
Jul 18 at 7:16
Thanks. The paper focus on $-infty<x<infty$. I would like to know if we have the same meaning when we consider half plane. Someone requests that the initial function to be a continuous function in $C$ to make further discussions reasonable. (say the boundary $x=0$, we impose a continuous function in $t$ such that initial condition and boundary condition coincide at $t=0,x=0$. I wonder if we can loose this restriction, say at time 0, $W(0,x)=delta_0(x)$ and make heat $W(t,x)=frac1sqrtt$ when $t>0$)
– will_cheuk
Jul 18 at 7:31
add a comment |Â
Does it has a solution
– will_cheuk
Jul 18 at 7:04
This article might have it: rose-hulman.edu/~bryan/lottamath/heatkern.pdf
– mvw
Jul 18 at 7:16
Thanks. The paper focus on $-infty<x<infty$. I would like to know if we have the same meaning when we consider half plane. Someone requests that the initial function to be a continuous function in $C$ to make further discussions reasonable. (say the boundary $x=0$, we impose a continuous function in $t$ such that initial condition and boundary condition coincide at $t=0,x=0$. I wonder if we can loose this restriction, say at time 0, $W(0,x)=delta_0(x)$ and make heat $W(t,x)=frac1sqrtt$ when $t>0$)
– will_cheuk
Jul 18 at 7:31
Does it has a solution
– will_cheuk
Jul 18 at 7:04
Does it has a solution
– will_cheuk
Jul 18 at 7:04
This article might have it: rose-hulman.edu/~bryan/lottamath/heatkern.pdf
– mvw
Jul 18 at 7:16
This article might have it: rose-hulman.edu/~bryan/lottamath/heatkern.pdf
– mvw
Jul 18 at 7:16
Thanks. The paper focus on $-infty<x<infty$. I would like to know if we have the same meaning when we consider half plane. Someone requests that the initial function to be a continuous function in $C$ to make further discussions reasonable. (say the boundary $x=0$, we impose a continuous function in $t$ such that initial condition and boundary condition coincide at $t=0,x=0$. I wonder if we can loose this restriction, say at time 0, $W(0,x)=delta_0(x)$ and make heat $W(t,x)=frac1sqrtt$ when $t>0$)
– will_cheuk
Jul 18 at 7:31
Thanks. The paper focus on $-infty<x<infty$. I would like to know if we have the same meaning when we consider half plane. Someone requests that the initial function to be a continuous function in $C$ to make further discussions reasonable. (say the boundary $x=0$, we impose a continuous function in $t$ such that initial condition and boundary condition coincide at $t=0,x=0$. I wonder if we can loose this restriction, say at time 0, $W(0,x)=delta_0(x)$ and make heat $W(t,x)=frac1sqrtt$ when $t>0$)
– will_cheuk
Jul 18 at 7:31
add a comment |Â
up vote
0
down vote
The heat-type equation you are studying is the differential formulation (i.e. a formulation involving pointwise values of the quantities involved, like spatial derivatives) of the general balance equation, and the $delta(x)$ represents a source term located in $x=0$ and independent of $t$. To see this, let's analyze the $n$-dimensional case and then particularize the results for $n=1$.
Let $GsubsetmathbbR^n$ a domain for which some form of the divergence theorem holds, for example a Caccioppoli set, and suppose that this set is characterized by a time-dependent quantity $Q(t)$.
The quantity $Q$ can vary respect time only only under the action of sources lying in the interior of $G$ or by flowing across the boundary $partial G$ of $G$: this is the heuristic content of the balance equation
$$
fracmathrmdQ(t)mathrmdt=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdxtag1label1
$$
where
- $boldsymbolq(x,t)$ is the flux density vector of the quantity $Q(t)$ across $partial G$ at the point $x$
- $boldsymboln_x$ is the inner (measure theoretic, for a Caccioppoli set) normal at the point $x$.
- $s(x,t)$ is a general inner term (representing for example processes of absorption/generation of the given quantity taking place within $G$).
The further step in the deduction is to assume a particular form for $Q(t)$ and for the flux $boldsymbolq(x,t)$. This is precisely the meaning of the following equations:
$$
Q(t)=int_G c(x)rho(x) w(x,t),mathrmdx,tag2label2
$$
where
- $c(x)$ is a function which depends only on the properties of the medium which fills $G$ (if we are dealing with heat, it is called heat capacity),
- $rho(x)$ is a function which depends only on the density of the medium (it is commonly called mass density),
and
$$
boldsymbolq(x,t)=boldsymbolhatK(x)nabla w(x,t),tag3label3
$$
where $boldsymbolhatK(x)$ is tensor that again takes care of the possibly anisotropic behavior of the flux.
Now, by using equations eqref1, eqref2, eqref3 and the Gauss-Green-Ostrogradsky (divergence) theorem we obtain
$$
beginsplit
fracmathrmdQ(t)mathrmdt &=int_G c(x)rho(x)fracpartial w(x,t)partial t,mathrmdx\
&=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_partial GboldsymbolhatK(x)nabla w(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_Gbig[nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)+ s(x,t)big]mathrmdx
endsplit
$$
and then
$$
int_Gleft[ c(x)rho(x)fracpartial w(x,t)partial t-nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)-s(x,t)right]mathrmdx=0
$$
The above equation integral equation is satisfied if and only if
$$
c(x)rho(x)fracpartial w(x,t)partial t=nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big) + s(x,t)tag4label4
$$
Equation eqref4 is the general linear (divergence form) parabolic equation: if we assume $c(x)equivrho(x)equiv1$, $boldsymbolhatK(x)equivboldsymbol1$ and $s(x,t)=delta(x)timesdelta(t)=delta(x)delta(t)$ (the tensor products of Dirac deltas respect to the variables $xinmathbbR^n$ and $tinmathbbR$) we obtain the standard heat equation:
$$
beginsplit
fracpartial w(x,t)partial t&=Delta w(x,t)+delta(x)delta(t)quad n>1\
fracpartial w(x,t)partial t&=fracpartial^2w(x,t)partial x^2+delta(x)delta(t)quad n=1
endsplit
$$
If you need to specify only the presence of a time independent term, basically considering only $delta(x)$ or, the other way out, only a time-dependent term, you can put
$$
s(x,t)=
begincases
delta(x) & xinmathbbR^n\
delta(t) & tinmathbbR
endcases
$$
therefore deltas respect to single variables or their tensor products can be interpreted as source terms in the balance equation eqref1.
Notes
Many (if not all) evolution equations of mathematical physics can be deduced from the general balance equation eqref1 (for example the wave equation), just by properly choosing the defining equations eqref2 and eqref3, which therefore play the rôle of constitutive axioms of a theory. In particular eqref3 is the Fourier-Duhamel law of heat conduction.
A deduction of the one-dimensional heat equation (without considering the source term), however conceptually identical to the deduction shown above, is offered by Cannon (1984) (§1.1, pp. 13-15) and by Widder (1978) (§2 and §3, pp. 1-5). I prefer the one above since it don't requires the knowledge of any special (however elementary) physical concepts, being valid also for the general linear diffusion equations and other parabolic PDEs.
[1] Cannon, J. R. (1984), The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23 (1st ed.), Reading etc.: Addison-Wesley Publishing Company, pp. XXV+483, ISBN 978-0-521-30243-2, MR 0747979, Zbl 0567.35001
[2] Widder, D. V. (1978), The Heat Equation, Pure and Applied Mathematics 67, Academic Press, pp. xiv+267, ISBN 0-12-748540-6, MR0466967, Zbl 0322.35041.
answered Jul 22 at 12:34
Daniele Tampieri
7301313
add a comment |Â
up vote
0
down vote
The heat-type equation you are studying is the differential formulation (i.e. a formulation involving pointwise values of the quantities involved, like spatial derivatives) of the general balance equation, and the $delta(x)$ represents a source term located in $x=0$ and independent of $t$. To see this, let's analyze the $n$-dimensional case and then particularize the results for $n=1$.
Let $GsubsetmathbbR^n$ a domain for which some form of the divergence theorem holds, for example a Caccioppoli set, and suppose that this set is characterized by a time-dependent quantity $Q(t)$.
The quantity $Q$ can vary respect time only only under the action of sources lying in the interior of $G$ or by flowing across the boundary $partial G$ of $G$: this is the heuristic content of the balance equation
$$
fracmathrmdQ(t)mathrmdt=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdxtag1label1
$$
where
- $boldsymbolq(x,t)$ is the flux density vector of the quantity $Q(t)$ across $partial G$ at the point $x$
- $boldsymboln_x$ is the inner (measure theoretic, for a Caccioppoli set) normal at the point $x$.
- $s(x,t)$ is a general inner term (representing for example processes of absorption/generation of the given quantity taking place within $G$).
The further step in the deduction is to assume a particular form for $Q(t)$ and for the flux $boldsymbolq(x,t)$. This is precisely the meaning of the following equations:
$$
Q(t)=int_G c(x)rho(x) w(x,t),mathrmdx,tag2label2
$$
where
- $c(x)$ is a function which depends only on the properties of the medium which fills $G$ (if we are dealing with heat, it is called heat capacity),
- $rho(x)$ is a function which depends only on the density of the medium (it is commonly called mass density),
and
$$
boldsymbolq(x,t)=boldsymbolhatK(x)nabla w(x,t),tag3label3
$$
where $boldsymbolhatK(x)$ is tensor that again takes care of the possibly anisotropic behavior of the flux.
Now, by using equations eqref1, eqref2, eqref3 and the Gauss-Green-Ostrogradsky (divergence) theorem we obtain
$$
beginsplit
fracmathrmdQ(t)mathrmdt &=int_G c(x)rho(x)fracpartial w(x,t)partial t,mathrmdx\
&=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_partial GboldsymbolhatK(x)nabla w(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_Gbig[nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)+ s(x,t)big]mathrmdx
endsplit
$$
and then
$$
int_Gleft[ c(x)rho(x)fracpartial w(x,t)partial t-nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)-s(x,t)right]mathrmdx=0
$$
The above equation integral equation is satisfied if and only if
$$
c(x)rho(x)fracpartial w(x,t)partial t=nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big) + s(x,t)tag4label4
$$
Equation eqref4 is the general linear (divergence form) parabolic equation: if we assume $c(x)equivrho(x)equiv1$, $boldsymbolhatK(x)equivboldsymbol1$ and $s(x,t)=delta(x)timesdelta(t)=delta(x)delta(t)$ (the tensor products of Dirac deltas respect to the variables $xinmathbbR^n$ and $tinmathbbR$) we obtain the standard heat equation:
$$
beginsplit
fracpartial w(x,t)partial t&=Delta w(x,t)+delta(x)delta(t)quad n>1\
fracpartial w(x,t)partial t&=fracpartial^2w(x,t)partial x^2+delta(x)delta(t)quad n=1
endsplit
$$
If you need to specify only the presence of a time independent term, basically considering only $delta(x)$ or, the other way out, only a time-dependent term, you can put
$$
s(x,t)=
begincases
delta(x) & xinmathbbR^n\
delta(t) & tinmathbbR
endcases
$$
therefore deltas respect to single variables or their tensor products can be interpreted as source terms in the balance equation eqref1.
Notes
Many (if not all) evolution equations of mathematical physics can be deduced from the general balance equation eqref1 (for example the wave equation), just by properly choosing the defining equations eqref2 and eqref3, which therefore play the rôle of constitutive axioms of a theory. In particular eqref3 is the Fourier-Duhamel law of heat conduction.
A deduction of the one-dimensional heat equation (without considering the source term), however conceptually identical to the deduction shown above, is offered by Cannon (1984) (§1.1, pp. 13-15) and by Widder (1978) (§2 and §3, pp. 1-5). I prefer the one above since it don't requires the knowledge of any special (however elementary) physical concepts, being valid also for the general linear diffusion equations and other parabolic PDEs.
[1] Cannon, J. R. (1984), The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23 (1st ed.), Reading etc.: Addison-Wesley Publishing Company, pp. XXV+483, ISBN 978-0-521-30243-2, MR 0747979, Zbl 0567.35001
[2] Widder, D. V. (1978), The Heat Equation, Pure and Applied Mathematics 67, Academic Press, pp. xiv+267, ISBN 0-12-748540-6, MR0466967, Zbl 0322.35041.
answered Jul 22 at 12:34
Daniele Tampieri
7301313
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The heat-type equation you are studying is the differential formulation (i.e. a formulation involving pointwise values of the quantities involved, like spatial derivatives) of the general balance equation, and the $delta(x)$ represents a source term located in $x=0$ and independent of $t$. To see this, let's analyze the $n$-dimensional case and then particularize the results for $n=1$.
Let $GsubsetmathbbR^n$ a domain for which some form of the divergence theorem holds, for example a Caccioppoli set, and suppose that this set is characterized by a time-dependent quantity $Q(t)$.
The quantity $Q$ can vary respect time only only under the action of sources lying in the interior of $G$ or by flowing across the boundary $partial G$ of $G$: this is the heuristic content of the balance equation
$$
fracmathrmdQ(t)mathrmdt=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdxtag1label1
$$
where
- $boldsymbolq(x,t)$ is the flux density vector of the quantity $Q(t)$ across $partial G$ at the point $x$
- $boldsymboln_x$ is the inner (measure theoretic, for a Caccioppoli set) normal at the point $x$.
- $s(x,t)$ is a general inner term (representing for example processes of absorption/generation of the given quantity taking place within $G$).
The further step in the deduction is to assume a particular form for $Q(t)$ and for the flux $boldsymbolq(x,t)$. This is precisely the meaning of the following equations:
$$
Q(t)=int_G c(x)rho(x) w(x,t),mathrmdx,tag2label2
$$
where
- $c(x)$ is a function which depends only on the properties of the medium which fills $G$ (if we are dealing with heat, it is called heat capacity),
- $rho(x)$ is a function which depends only on the density of the medium (it is commonly called mass density),
and
$$
boldsymbolq(x,t)=boldsymbolhatK(x)nabla w(x,t),tag3label3
$$
where $boldsymbolhatK(x)$ is tensor that again takes care of the possibly anisotropic behavior of the flux.
Now, by using equations eqref1, eqref2, eqref3 and the Gauss-Green-Ostrogradsky (divergence) theorem we obtain
$$
beginsplit
fracmathrmdQ(t)mathrmdt &=int_G c(x)rho(x)fracpartial w(x,t)partial t,mathrmdx\
&=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_partial GboldsymbolhatK(x)nabla w(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_Gbig[nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)+ s(x,t)big]mathrmdx
endsplit
$$
and then
$$
int_Gleft[ c(x)rho(x)fracpartial w(x,t)partial t-nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)-s(x,t)right]mathrmdx=0
$$
The above equation integral equation is satisfied if and only if
$$
c(x)rho(x)fracpartial w(x,t)partial t=nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big) + s(x,t)tag4label4
$$
Equation eqref4 is the general linear (divergence form) parabolic equation: if we assume $c(x)equivrho(x)equiv1$, $boldsymbolhatK(x)equivboldsymbol1$ and $s(x,t)=delta(x)timesdelta(t)=delta(x)delta(t)$ (the tensor products of Dirac deltas respect to the variables $xinmathbbR^n$ and $tinmathbbR$) we obtain the standard heat equation:
$$
beginsplit
fracpartial w(x,t)partial t&=Delta w(x,t)+delta(x)delta(t)quad n>1\
fracpartial w(x,t)partial t&=fracpartial^2w(x,t)partial x^2+delta(x)delta(t)quad n=1
endsplit
$$
If you need to specify only the presence of a time independent term, basically considering only $delta(x)$ or, the other way out, only a time-dependent term, you can put
$$
s(x,t)=
begincases
delta(x) & xinmathbbR^n\
delta(t) & tinmathbbR
endcases
$$
therefore deltas respect to single variables or their tensor products can be interpreted as source terms in the balance equation eqref1.
Notes
Many (if not all) evolution equations of mathematical physics can be deduced from the general balance equation eqref1 (for example the wave equation), just by properly choosing the defining equations eqref2 and eqref3, which therefore play the rôle of constitutive axioms of a theory. In particular eqref3 is the Fourier-Duhamel law of heat conduction.
A deduction of the one-dimensional heat equation (without considering the source term), however conceptually identical to the deduction shown above, is offered by Cannon (1984) (§1.1, pp. 13-15) and by Widder (1978) (§2 and §3, pp. 1-5). I prefer the one above since it don't requires the knowledge of any special (however elementary) physical concepts, being valid also for the general linear diffusion equations and other parabolic PDEs.
[1] Cannon, J. R. (1984), The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23 (1st ed.), Reading etc.: Addison-Wesley Publishing Company, pp. XXV+483, ISBN 978-0-521-30243-2, MR 0747979, Zbl 0567.35001
[2] Widder, D. V. (1978), The Heat Equation, Pure and Applied Mathematics 67, Academic Press, pp. xiv+267, ISBN 0-12-748540-6, MR0466967, Zbl 0322.35041.
answered Jul 22 at 12:34
Daniele Tampieri
7301313
The heat-type equation you are studying is the differential formulation (i.e. a formulation involving pointwise values of the quantities involved, like spatial derivatives) of the general balance equation, and the $delta(x)$ represents a source term located in $x=0$ and independent of $t$. To see this, let's analyze the $n$-dimensional case and then particularize the results for $n=1$.
Let $GsubsetmathbbR^n$ a domain for which some form of the divergence theorem holds, for example a Caccioppoli set, and suppose that this set is characterized by a time-dependent quantity $Q(t)$.
The quantity $Q$ can vary respect time only only under the action of sources lying in the interior of $G$ or by flowing across the boundary $partial G$ of $G$: this is the heuristic content of the balance equation
$$
fracmathrmdQ(t)mathrmdt=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdxtag1label1
$$
where
- $boldsymbolq(x,t)$ is the flux density vector of the quantity $Q(t)$ across $partial G$ at the point $x$
- $boldsymboln_x$ is the inner (measure theoretic, for a Caccioppoli set) normal at the point $x$.
- $s(x,t)$ is a general inner term (representing for example processes of absorption/generation of the given quantity taking place within $G$).
The further step in the deduction is to assume a particular form for $Q(t)$ and for the flux $boldsymbolq(x,t)$. This is precisely the meaning of the following equations:
$$
Q(t)=int_G c(x)rho(x) w(x,t),mathrmdx,tag2label2
$$
where
- $c(x)$ is a function which depends only on the properties of the medium which fills $G$ (if we are dealing with heat, it is called heat capacity),
- $rho(x)$ is a function which depends only on the density of the medium (it is commonly called mass density),
and
$$
boldsymbolq(x,t)=boldsymbolhatK(x)nabla w(x,t),tag3label3
$$
where $boldsymbolhatK(x)$ is tensor that again takes care of the possibly anisotropic behavior of the flux.
Now, by using equations eqref1, eqref2, eqref3 and the Gauss-Green-Ostrogradsky (divergence) theorem we obtain
$$
beginsplit
fracmathrmdQ(t)mathrmdt &=int_G c(x)rho(x)fracpartial w(x,t)partial t,mathrmdx\
&=int_partial Gboldsymbolq(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_partial GboldsymbolhatK(x)nabla w(x,t)cdotboldsymboln_xmathrmdx+int_G s(x,t),mathrmdx\
&=int_Gbig[nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)+ s(x,t)big]mathrmdx
endsplit
$$
and then
$$
int_Gleft[ c(x)rho(x)fracpartial w(x,t)partial t-nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big)-s(x,t)right]mathrmdx=0
$$
The above equation integral equation is satisfied if and only if
$$
c(x)rho(x)fracpartial w(x,t)partial t=nablacdotbig(boldsymbolhatK(x)nabla w(x,t)big) + s(x,t)tag4label4
$$
Equation eqref4 is the general linear (divergence form) parabolic equation: if we assume $c(x)equivrho(x)equiv1$, $boldsymbolhatK(x)equivboldsymbol1$ and $s(x,t)=delta(x)timesdelta(t)=delta(x)delta(t)$ (the tensor products of Dirac deltas respect to the variables $xinmathbbR^n$ and $tinmathbbR$) we obtain the standard heat equation:
$$
beginsplit
fracpartial w(x,t)partial t&=Delta w(x,t)+delta(x)delta(t)quad n>1\
fracpartial w(x,t)partial t&=fracpartial^2w(x,t)partial x^2+delta(x)delta(t)quad n=1
endsplit
$$
If you need to specify only the presence of a time independent term, basically considering only $delta(x)$ or, the other way out, only a time-dependent term, you can put
$$
s(x,t)=
begincases
delta(x) & xinmathbbR^n\
delta(t) & tinmathbbR
endcases
$$
therefore deltas respect to single variables or their tensor products can be interpreted as source terms in the balance equation eqref1.
Notes
Many (if not all) evolution equations of mathematical physics can be deduced from the general balance equation eqref1 (for example the wave equation), just by properly choosing the defining equations eqref2 and eqref3, which therefore play the rôle of constitutive axioms of a theory. In particular eqref3 is the Fourier-Duhamel law of heat conduction.
A deduction of the one-dimensional heat equation (without considering the source term), however conceptually identical to the deduction shown above, is offered by Cannon (1984) (§1.1, pp. 13-15) and by Widder (1978) (§2 and §3, pp. 1-5). I prefer the one above since it don't requires the knowledge of any special (however elementary) physical concepts, being valid also for the general linear diffusion equations and other parabolic PDEs.
[1] Cannon, J. R. (1984), The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23 (1st ed.), Reading etc.: Addison-Wesley Publishing Company, pp. XXV+483, ISBN 978-0-521-30243-2, MR 0747979, Zbl 0567.35001
[2] Widder, D. V. (1978), The Heat Equation, Pure and Applied Mathematics 67, Academic Press, pp. xiv+267, ISBN 0-12-748540-6, MR0466967, Zbl 0322.35041.
answered Jul 22 at 12:34
Daniele Tampieri
7301313
edited Jul 22 at 16:02
answered Jul 22 at 12:34
Daniele Tampieri
7301313
answered Jul 22 at 12:34
Daniele Tampieri
7301313
answered Jul 22 at 12:34
Daniele Tampieri
7301313
7301313
add a comment |Â
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In your equation, should it be $partial W/partial t$ instead of $partial W/partial u$? Also, you need to specify boundary conditions at $x=0$.
– Rahul
Jul 18 at 6:50
yes, it is $fracpartial Wpartial t$. No, I do not want to specify boundary condition.
– will_cheuk
Jul 18 at 7:05
What are the boundary conditions at $x=0$ then? A physical interpretation would depend on what these are.
– Winther
Jul 22 at 19:03