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Natural Science Forum / Physics / General Physics / October 2004One-dimensional heat equation
Hi, I'm a little confused about a one-dimensional heat equation problem. If anybody can guide me in the right direction I'd appreciate it a lot. Thanks! We start with the heat equation dU/dt = d^2U/dt^2 on a finite interval (0,L) with the following conditions: U(t,0) = U(t,L) = 0 U(0,x) = U(x) and furthermore f(x) admits a representation f(x) = sum{A_n*sin(n*pi*x/L)} for n>0 so the question is, does the integral of U(t,x) with respect to x change with time over the whole real line? I have a text that poses this is a problem, and the hint is to "think of a wire whose ends are perfectly insulated with temperature = 0." The answer is yes, this integral does change, but by conservation of heat, shouldn't it be constant? I don't understand what happens as t approaches infinity. Thank you all! KH >dU/dt = d^2U/dt^2 > [quoted text clipped - 13 lines] >integral does change, but by conservation of heat, shouldn't it be >constant? I don't understand what happens as t approaches infinity. A few remarks: 1. You have a typo in the heat equation. The correct equation is: dU/dt = d^2U/dx^2 2. Are you sure you are quoting the text correctly? Saying "think of a wire whose ends are perfectly insulated with temperature = 0" is like saying "think of a blond who is a redhead". If the ends are exposed to temperature 0, then the are _exposed_, therefore they are not insulated. The boundary conditions U(t,0) = U(t,L) = 0 say that the ends are exposed to temperature 0. If the ends were insulated, then the boundary conditions would have been dU/dx(t,0) = dU/dx(t,L) = 0. 3. You write: "does the integral of U(t,x) with respect to x change with time over the whole real line?" Do you really mean "the whole real line"? The variable x is takes values only on the interval 0 to L. It is not defined on the whole real line. The variable t takes values on the interval 0 to +infinity. It is not defined in the whole real line either. In the light of remarks above, your reference to conservation of heat is ambiguous. If the wire were insulated, then yes, heat would have been conserved and the integral of U(t,x) with respect to x over the interval (0,L) would have been independent of t. Since your wire is not insulated, it can be shown that that integral converges to 0 as t goes to infinity.  Signaturerr [snip] > Are you sure you are quoting the text correctly? Saying "think > of a wire whose ends are perfectly insulated with temperature = 0" > is like saying "think of a blond who is a redhead". Mmmmmmm! Blonde redheads! Ohhhhhhhh.... Ok, I'm back. Socks > Hi, I'm a little confused about a one-dimensional heat equation > problem. If anybody can guide me in the right direction I'd [quoted text clipped - 18 lines] > integral does change, but by conservation of heat, shouldn't it be > constant? I don't understand what happens as t approaches infinity. Indeed, if the ends of the wire are "perfectly insulated", I don't see how heat can escape the wire or enter the wire. I'd rather think of the ends of the wire as lying in an ice and water mixture, at a constant temperature of zero Celsius. The hint has me confused, so I'll leave it at that. David Bernier >Hi, I'm a little confused about a one-dimensional heat equation >problem. If anybody can guide me in the right direction I'd [quoted text clipped - 19 lines] >constant? I don't understand what happens as t approaches infinity. >Thank you all! The hint you got is what you need. Consider how would you express the condition of "ends of wire are insulated". Consider whether your wire satisfies this condition. Mati Meron | "When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same" >Hi, I'm a little confused about a one-dimensional heat equation >problem. If anybody can guide me in the right direction I'd >appreciate it a lot. Thanks! >We start with the heat equation >dU/dt = d^2U/dt^2 >on a finite interval (0,L) with the following conditions: >U(t,0) = U(t,L) = 0 >U(0,x) = U(x) >and furthermore f(x) admits a representation >f(x) = sum{A_n*sin(n*pi*x/L)} for n>0 >so the question is, does the integral of U(t,x) with respect to x >change with time over the whole real line? Yes, you seem to be confused. The integral of U(t,x) over the whole real line does not exist: U(t,x) is only defined on the interval [0,L]. > I have a text that poses >this is a problem, and the hint is to "think of a wire whose ends are >perfectly insulated with temperature = 0." This makes no sense. If the ends are insulated, they won't stay at temperature 0. An insulated boundary condition would be dU/dx = 0. > The answer is yes, this >integral does change, but by conservation of heat, shouldn't it be >constant? If the ends are kept at temperature 0, the integral (over the interval 0 to L, not over the whole real line) changes, because heat can escape out the ends. If the ends are insulated, the integral over the interval does not change. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada > Hi, I'm a little confused about a one-dimensional heat equation > problem. Yes, it is confusing the way you stated the problem. > If anybody can guide me in the right direction I'd > appreciate it a lot. Thanks! A familiar statement and solution of heat equation possibly closer to your textbook question may be found at http://www.math.ubc.ca/~feldman/demos/demo7.html --- rest snipped ================== Mohan Pawar MIO Instruments LLC Oh my, how embarassing! This is what I get for posting late at night on very little sleep. I transcribe the problem incorrectly, and thankyou very much to all for pointing this out! We start with the heat equation dU/dt = d^2U/dx^2 on a finite interval (0,L) with the following conditions: U(t,0) = U(t,L) = 0 U(0,x) = U(x) and furthermore f(x) admits a representation f(x) = sum{A_n*sin(n*pi*x/L)} for n>0 and we want to know if the integral of u(t,x) with respect to x from -infinity to positive infinity changes over time. (I think that the above is a typo and that we are only supposed to evaluate the integral of u(t,x) with respect to x from 0 to L). The next part of the question reads "What is the behavior of this integral as t approaches infinity? Give a physical explanation; think of a finite heated wire, both of whose ends are embedded in an ice cube at constant temperature 0." Sorry for the confusion, and thanks very much for all of your replies. I don't understand how admitting this representation for f(x) allows one to show that the heat of the system converges; is this really dumb of me? Is it something obvious that I've missed? Thanks very much! KH >Oh my, how embarassing! This is what I get for posting late at night >on very little sleep. I transcribe the problem incorrectly, and [quoted text clipped - 25 lines] >one to show that the heat of the system converges; is this really dumb >of me? Is it something obvious that I've missed? First let's clarify: what is f(x) and what does it have to do with this problem? SignatureRouben Rostamian > First let's clarify: what is f(x) and what does it have to do > with this problem? I apologize again. I must seem incompetent! f(x) is the initial condition of the differential equation: U(0,x) = f(x) (so, before, when I had U(0,x) = U(x), that was a mistake) and it admits a representation as I described. Thank you very much and I apologize for taking your time. KH >We start with the heat equation >dU/dt = d^2U/dx^2 >on a finite interval (0,L) with the following conditions: >U(t,0) = U(t,L) = 0 >U(0,x) = U(x) I think you mean f(x). >and furthermore f(x) admits a representation >f(x) = sum{A_n*sin(n*pi*x/L)} for n>0 >and we want to know if the integral of u(t,x) with respect to x from >-infinity to positive infinity changes over time. (I think that the [quoted text clipped - 4 lines] >both of whose ends are embedded in an ice cube at constant temperature >0." > I don't understand how admitting this representation for f(x) allows >one to show that the heat of the system converges; is this really dumb >of me? Is it something obvious that I've missed? Hints: The solution U(x,t) can be written in terms of a very similar series with some exponential factors involving t ... What do you get when you integrate sin(n*pi*x/L) for x from 0 to L? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada Hmmm...so, we find that the integral of sin(n*pi*x/L) from 0 to L is 0 for n even, and 2*L/(n*pi) for n odd. So the integral of f(x) from 0 to L is ... + A_{-3}*-2*L/(3*pi) + A_{-1}*-2*L/(pi) + A_{1}*2*L/(pi) + A_{3}*2*L/(3*pi) + ... I assume the similar series is the fourier series of U(x,t) for fixed x, which is sum{c_n*e^(i*n*t)} where the c_n are defined in the usual way. How does integral of sin(n*pi*x/L) help me here, can I use it to calculate the fourier coefficients in the series above? Thanks so much! KH > Hints: > The solution U(x,t) can be written in terms of a very similar series [quoted text clipped - 4 lines] > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada >I assume the similar series is the fourier series of U(x,t) for fixed >x, which is >sum{c_n*e^(i*n*t)} No, I mean the Fourier series of U(x,t) as a function of x. The usual series solution of the heat equation with these boundary conditions. Look in your textbook, it must be there. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |
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