We are going to give several forms of the heat equation for reference purposes, but we will only be really solving one of them. Note that with this assumption the actual shape of the cross section i. Note that the 1-D assumption is actually not all that bad of an assumption as it might seem at first glance.
If we assume that the lateral surface of the bar is perfectly insulated i. This means that heat can only flow from left to right or right to left and thus creating a 1-D temperature distribution. The assumption of the lateral surfaces being perfectly insulated is of course impossible, but it is possible to put enough insulation on the lateral surfaces that there will be very little heat flow through them and so, at least for a time, we can consider the lateral surfaces to be perfectly insulated.
As indicated we are going to assume, at least initially, that the specific heat may not be uniform throughout the bar.
Note as well that in practice the specific heat depends upon the temperature. While this is a nice form of the heat equation it is not actually something we can solve. As noted the thermal conductivity can vary with the location in the bar. Also, much like the specific heat the thermal conductivity can vary with temperature, but we will assume that the total temperature change is not so great that this will be an issue and so we will assume for the purposes here that the thermal conductivity will not vary with temperature.
First, we know that if the temperature in a region is constant, i. Next, we know that if there is a temperature difference in a region we know the heat will flow from the hot portion to the cold portion of the region. For example, if it is hotter to the right then we know that the heat should flow to the left. Finally, the greater the temperature difference in a region i. Note that we factored the minus sign out of the derivative to cancel against the minus sign that was already there.
In this case we generally say that the material in the bar is uniform. Under these assumptions the heat equation becomes. There are four of them that are fairly common boundary conditions.
The first type of boundary conditions that we can have would be the prescribed temperature boundary conditions, also called Dirichlet conditions. The prescribed temperature boundary conditions are. The next type of boundary conditions are prescribed heat fluxalso called Neumann conditions.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I am looking for a hint on how to continue from here. The result is similar to a Fourier series, but not a conventional one. Here's a similar answer where I went a bit further on orthogonality.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 1 year, 1 month ago. Active 1 year, 1 month ago. Viewed 87 times. Thank you. Active Oldest Votes. Dylan Dylan I did get the sine version at first, but then got mixed up with the notations.
I did not know this could be solved by using orthogonality, thank you for the help. Sign up or log in Sign up using Google. Sign up using Facebook.
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Heat equation/Solution to the 2-D Heat Equation
Email Required, but never shown. The Overflow Blog. Featured on Meta. Feedback on Q2 Community Roadmap. Request for input: What makes a good title? Autofilters for Hot Network Questions. Linked 0. Related 1.Let u x, t denote the solution of the heat equation subject to the initial condition:. Consider a heat transfer problem for a thin straight bar or wire of uniform cross section and homogeneous material. Suppose further that the lateral surface of the rod are perfectly insulated so that no heat transferes through them.
We also assume that the cross-sectional dimensions are so small that the temperature u can be considered constant on any cross section. Then u is a function only of the axial coordinate x and the time t. Assuming that no heat is being generated within the rod, the variation of temperature in it is goverged by a partial differential equation of parabolic type :. Since the left hand side depends only on time variable tand its right side depends on spacial variable xand these variables are independent, these funvtions could be equal only when they are a constant.
Example: We are going to find the temperature u x,t at any time t is a metal rod 9 cm long, insulated on the sides. Ali and L.
Zhang, Relativistic heat conductionInternational Journal of Heat and Mass transfer48, Chester, Second sound in solids, Physical Review 15 Lienhard V and J.
Lienhard, A Heat Transfer textbook V. Brovman, Application of isothermal surfaces for calculation of unsteady heat transfer processes, Journal of Engineering Physics and ThermophysicsVol 71, N0 5, Swenson, Robert J. Email: Prof. Vladimir Dobrushkin. Preface This section concerns about two dimensional wave equation. One dimensional Heat Transfer Equation in infinite strip.During these challenging times, we guarantee we will work tirelessly to support you.
We will continue to give you accurate and timely information throughout the crisis, and we will deliver on our mission — to help everyone in the world learn how to do anything — no matter what.
Thank you to our community and to all of our readers who are working to aid others in this time of crisis, and to all of those who are making personal sacrifices for the good of their communities. We will get through this together. The heat equation is a partial differential equation describing the distribution of heat over time.
It only takes a minute to sign up. Could someone help me out with a question? There is a PDE, which is the heat equation. There is also a solution. In the solution, it says that an odd extension of the PDE has to be computed. The wording of the solution is misleading because it implies you can tell whether an odd or even extension can be used just by looking at the initial conditions.
In fact, there is no way to tell whether an odd or even extension would be used just by looking at the initial condition the time condition. Just by changing the boundary conditions, this problem might as well give you an even extension. The solution said odd extension because it had already done the calculations for this particular problem in Assignment 4 and found that the solution in space involves only sine functions.
This pde problem has been solved by the method separation of variables. To understand more about this method, you'd need to review what you went over in your class. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 2 years, 3 months ago.
Active 2 years, 3 months ago. Viewed times. My questions are: Why an odd extension is used and not an even one? Thank you in advance for your replies! The Phenotype 4, 9 9 gold badges 18 18 silver badges 33 33 bronze badges. Active Oldest Votes.
The Phenotype The Phenotype 4, 9 9 gold badges 18 18 silver badges 33 33 bronze badges. However, if we do an odd extension, isn't the function undefined due to a discontinuous jump from 1 to -1? Also, if we do an odd extension, are all the cn terms 0?
In the odd extension the terms are non-zero. As I can see, they have already been calculated in Assignment 4. The reason for that is that some of the questions are recycled from the previous years, but that doesn't matter. Anyways, thank you very much for your time!
With your help, I have now understood about odd and even extensions. I am new to math.Okay, it is finally time to completely solve a partial differential equation.
In the previous section we applied separation of variables to several partial differential equations and reduced the problem down to needing to solve two ordinary differential equations. In this section we will now solve those ordinary differential equations and use the results to get a solution to the partial differential equation. We are going to do the work in a couple of steps so we can take our time and see how everything works.
The first thing that we need to do is find a solution that will satisfy the partial differential equation and the boundary conditions. At this point we will not worry about the initial condition. Okay the first thing we technically need to do here is apply separation of variables.
This leaves us with two ordinary differential equations. We did all of this in Example 1 of the previous section and the two ordinary differential equations are. The positive eigenvalues and their corresponding eigenfunctions of this boundary value problem are then. Note however that we have in fact found infinitely many solutions since there are infinitely many solutions i.
So, there we have it. The function above will satisfy the heat equation and the boundary condition of zero temperature on the ends of the bar. The problem with this solution is that it simply will not satisfy almost every possible initial condition we could possibly want to use.
This is actually easier than it looks like. This is almost as simple as the first part. Doing this gives. The Principle of Superposition is, of course, not restricted to only two solutions.
For instance, the following is also a solution to the partial differential equation. Doing this our solution now becomes. This may still seem to be very restrictive, but the series on the right should look awful familiar to you after the previous chapter.
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The series on the left is exactly the Fourier sine series we looked at in that chapter. That almost seems anti-climactic. This was a very short problem.
Of course, some of that came about because we had a really simple constant initial condition and so the integral was very simple. While the example itself was very simple, it was only simple because of all the work that we had to put into developing the ideas that even allowed us to do this.In physics and mathematicsthe heat equation is a partial differential equation that describes how the distribution of some quantity such as heat evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower.
It is a special case of the diffusion equation. This equation was first developed and solved by Joseph Fourier in to describe heat flow. However, it is of fundamental importance in diverse scientific fields.Transformation of Black Scholes PDE to Heat Equation
In probability theorythe heat equation is connected with the study of random walks and Brownian motionvia the Fokker—Planck equation. In financial mathematicsit is used to solve the Black—Scholes partial differential equation. In quantum mechanicsit is used for finding spread of wave function in potential free region.
Using Newton's notation for derivatives, and the notation of vector calculusthe heat equation can be written in compact form as. However, it also describes many other physical phenomena as well. With this simplification, the heat equation is the prototypical parabolic partial differential equation.
By the second law of thermodynamicsheat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the thermal conductivity of the material between them. When heat flows into or out of a material, its temperature increases respectively, decreasesin proportion to the amount of heat divided by the amount mass of material, with a proportionality factor called the specific heat capacity of the material.
The value at some point will remain stable only as long as it is equal to the average value in its immediate surroundings. This is a property of parabolic partial differential equations and is not difficult to prove mathematically see below.
If a certain amount of heat is suddenly applied to a point the medium, it will spread out in all directions in the form of a diffusion wave. Unlike the elastic and electromagnetic wavesthe speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.
For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy Cannon By Fourier's law for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it:.
The equation becomes. That is. This derivation assumes that the material has constant mass density and heat capacity through space as well as time. This quantity is called the thermal diffusivity of the medium. An additional term may be introduced into the equation to account for radiative loss of heat.