Heat equation Definition and 282 Threads

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton in 1982 and culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003. Certain solutions of the heat equation known as heat kernels provide subtle information about the region on which they are defined, as exemplified through their application to the Atiyah–Singer index theorem.The heat equation, along with variants thereof, is also important in many fields of science and applied mathematics. In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time. In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the heat equation have also been given much attention in the numerical analysis literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.

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  1. J

    Heat Equation: Boundary Value Problem

    http://img821.imageshack.us/img821/7901/heatp.png Uploaded with ImageShack.us I'm having difficulty with the boundary conditions on this problem. I don't need a solution or a step by step. I've just never solved a boundary condition like this. Its the u(pi,t) = cos(t) that is giving me...
  2. M

    MHB Solve Heat Equation with Initial Conditions

    Solve $\begin{aligned} & {{u}_{tt}}={{u}_{xx}},\text{ }x\in [0,1],\text{ }t>0, \\ & u(x,0)=f(x), \\ & {{u}_{t}}(x,0)=0,\text{ }u(0,t)=u(1,t)=0 \\ \end{aligned} $ where $f(x)$ is defined by $f(x)=x$ if $0\le x\le \dfrac12$ and $f(x)=1-x$ if $\dfrac12\le x\le1.$ I'm not sure how to...
  3. M

    Looking for method to use in final step in heat equation problem

    Homework Statement The original problem is to solve u_t=u_xx+x with u(x,0)=0 and u(0,t)=0 by assuming there is a solution t^a*u(r), where r=x/t^b and a,b are constants Homework Equations The Attempt at a Solution This is a long problem, so I'm not writing everything. Following the...
  4. fluidistic

    Heat equation, Fourier cosine transform

    Homework Statement Problem 8-17 from Mathew's and Walker's book: Use a cosine transform with respect to y to find the steady-state temperature distribution in a semi-infinite solid x>0 when the temperature on the surface x=0 is unity for -a<y<a and zero outside this strip. Homework...
  5. M

    MHB Solving a Heat Equation with $\sin \pi x$

    Hi! I need to find out how to solve this type of heat equations: $$\large \frac{du}{dt} - \frac{d^2u}{dx^2} = \sin \pi x$$ $$\large u|_{t=0} = \sin 2\pi x $$ $$\large \large u|_{x=0} = u|_{x=1} = 0$$ I know what the solution to this but I can't solve it myself. The problem is that all over...
  6. N

    Steady State Heat Equation in a One-Dimensional Rod

    Homework Statement Determine the equilibrium temperature distribution for a one-dimensional rod composed of two different materials in perfect thermal contact at x=1. For 0<x<1, there is one material (cp=1, K0=1) with a constant source (Q=1), whereas for the other 1<x<2 there are no sources...
  7. R

    Formal Solution for Heat Equation using Fourier Series

    Homework Statement Find a formal solution of the heat equation u_t=u_xx subject to the following: u(0,t)=0 u_x(∏,t)=0 u(x,0)=f(x) for 0≤x≤∏ and t≥0 Homework Equations u(x,t)=X(x)T(t)The Attempt at a Solution I first began with a separation of variables. T'(t)=λT(t) T(t) =...
  8. fluidistic

    Fourier transform and the heat equation, don't understand the provided answer

    Homework Statement Using Fourier sine transform with respect to x, show that the solution to \frac{\partial u }{\partial t}=\frac{\partial ^2 u }{\partial x^2} with x and t >0 subject to the conditions u(0,t)=0 and u(x,0)=1 for 0<x<1, u(x,0)=0 otherwise, with u(x,t) bounded gives the solution u...
  9. C

    Help for Heat Equation - Questions from Chen

    1 .) i added several problems which i couldn't understand the approach for solving them. i'm kinda confused, will glad if someone would guide me. 2.) in class we've talked about and calculated . n orders(0,1,2,3) of moments, how is it to do with the heat equations? and what is the...
  10. M

    Heat Equation for Compressible Fluids: Valid or Not?

    Hi everyone, I am wondering if the heat equation is valid for compressible fluids like air. This is assuming constant 100% humidity. If it is not then how close is the appoximation. The model assumes that heat moves through an array of air only by conduction. At the moment I use a...
  11. H

    How Can Boundary Conditions Influence Solutions to the Heat Equation?

    Homework Statement Let s = x/\sqrt{t} and look for a solution to the heat equation u_{t} = u_{xx} which is of the form u(x,t) = f(s) and satisfies the IC u(x, 0) = 0 and the BC u(0, t) = 1 and u(∞, t) = 0. Homework Equations ∫e^{x^{2}} = \sqrt{\pi} The Attempt at a Solution Let f(s) = u(x...
  12. A

    Heat equation in the first quadrant.

    Homework Statement Solve the heat equation u_t=u_{xx}+u_{yy} fot t>0 in the first quadrant of \mathbb{R}^2. The boundary conditions are u(0,y,t)=u(x,0,t)=0 and the initial temperature distribution is f(x,y)= \begin{cases} 1 \;\;\;\; \text{in the square } \; 0<x<1; \; 0<y<1 \\ 0 \;\;\;\...
  13. L

    Calculating Latent Heat of Vaporisation: 3 kW Kettle, 2.0 kg Water @ 100oC

    A 3 kW kettle contains 2.0 kg of water at a temperature close to 100oC. Latent heat of vaporisation for water: Lv=2256 (kJ kg^-1) Q= Lv x mass Ok I understand this problem because I now the answer but I don't understand the process. Like my teacher wrote 2256x10^3 why he wrote...
  14. C

    Why Does the Cylindrical Wall Heat Equation Solution Include Logarithms?

    I'm a little bit rusty with my differential equations, and can't seem to see how solving for 1/r d/dr (r dT/dr)=0 has the solution T(r)=C_1*ln⁡(r)+C_2
  15. J

    Help with similarity solutions to Heat Equation

    I'm trying to solve the Heat equation by assuming a similarity solution of the form U=f(z) where z = x / √t also subject to U=H(x) at t=0 *H(x) is heaviside function. The question want the answers to be given in terms of the error function and also checked by using the fundamental solution of...
  16. P

    Finite Element and CFL condition for the heat equation

    I am solving the heat equation in a non comercial C++ finite elements code with explicit euler stepping, and adaptive meshes (coarse in the boundaries and finer in the center). I am aware the CFL condition for the heat equation depends on dt/h**2 for the 1D, 2D, 3D case. When I solve the...
  17. R

    Using Crank-Nicolson Method to solve Heat Equation

    I'm a bit stuck with using the C-N method The question I'm trying to solve is the standard heat equation with: U=[sin(pi)*x] at \tau = 0 & U = 0 at x = 0 & x = 1 for \tau \geq 0 The intervals are 0.2 in x AND 0.02 in \tau up to \tau = 0.06 I've been asked to solve using an...
  18. M

    Solving the heat equation on the real line using fourier transforms

    Homework Statement Solve the heat equation \frac{∂}{∂t}u(x,t)=\frac{1}{α^2}\frac{∂^2}{∂t^2]}u(x,t) on the real line ℝ with the initial conditions: u(x,0)= 1 if |x| \leq1, 0 for any other value of x Homework Equations I don't even know where to start for this question, but I...
  19. T

    How to Find Temperature Distribution in a Heated Rod with Convection?

    Homework Statement A thin rod of length ∏ is heated at one end to temperature T_0. It is insulated along its length and cooled at the other end by convection in a fluid of temperature T_f . Find the transient and steady-state temperature distribution in the rod, assuming unit thermal diff...
  20. T

    Heat Equation - Trouble Finding a General Solution

    Homework Statement Solve: Ut=kUxx U(x,0)=e^3x Homework Equations The Heat Equation: The Attempt at a Solution g(y) in the heat equation for this problem is e^3y. I'm having serious trouble solving this because my professor hasn't taught us the method, and it isn't in the...
  21. B

    Heat Equation: Cooking a Turkey

    Homework Statement It's that time of the year. I'm trying to determine how long it will take to cook a 15 pound turkey at 400 degrees to reach a center temperature of 180 degrees, given that it takes 90 minutes to cook a 5 pound turkey to the desired center temperature. The roast is initially...
  22. M

    Solving Heat Equation w/ Neumann BCs Different Domain

    Hi guys! I'm to find the solution to \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} Subject to an initial condition u(x,0) = u_0(x) = a \exp(- \frac{x^2}{2c^2}) And Neumann boundary conditions \frac{\partial u}{\partial x} (-1,t) = \frac{\partial...
  23. K

    Cant solve heat equation on semi infiniate solid

    Hi All, I am having issues trying to work out a task my employer has given me and was wondering if there is someone who could help. Question: A Semi-infiniate solid (L=3M) x>0, is initially at temperature zero. At time t=0, a constant temperature Uo>0 is applied and maintained at the face...
  24. W

    Laplace Transform of Heat Equation

    Homework Statement T(x,t) is the temperature distribution for t > 0 in a semi-infinite slab occupying x > 0 T(x, 0) = T_0 e^{-ax} for x > 0 (with a positive constant) T(0, t) = T_1 for t > 0 \tau(x, s) is the Laplace transform of T(x, t) show that \tau(x, s) = \frac{T_0}{s - Ka^2}e^{-ax} +...
  25. D

    Heat equation solving quadratic equation with complex numbers

    Homework Statement given that kλ2-ρcpuλ-ρcpωi=0 plug into the quadratic formula and get out an equation that looks like this λ=α+iβ±γ√(1+iδ) where α,β,γ,and δ are in terms of ρ,cp,u,k, and ω Homework Equations (-b±√b2-4ac)/2a kλ2-ρcpuλ-ρcpωi=0 λ=α+iβ±γ√(1+iδ) The Attempt at a...
  26. Telemachus

    Heat equation with boundary conditions

    Hi. I'm trying to solve the heat equation with the initial boundary conditions: u(0,t)=f_1(t) u(x_1,t)=f_2(t) u(x,0)=f(x) 0<x<x_1 t>0 And the heat equation: \frac{\partial u}{\partial t}-k\frac{\partial^2 u}{\partial x^2}=0 So when I make separation of variables I get: \nu=X(x)T(t)...
  27. U

    How can I graph a heat equation with multiple variables in Maple?

    Homework Statement Graph T(x,t) = T1e^(lambda*x)sin(wt-lambda*x) in Maple lambda = -.2 T1 = 10 omega = constant Homework Equations Maybe heat equation ut(x,t) = uxx(x,t) The Attempt at a Solution I'm really unsure on how to graph a multiple variable function/equation in Maple. I...
  28. R

    Verification of solution to Heat Equation

    [b]1. verify that u(t,x,y)=e-λtsin(αt)cos(βt) (for arbitrary α, β and with λ=α2+β2) satisfies the 2-D Heat Equation. [b]2. ut=Δu [b]3. I began with: Δu=uxx+uyy. note the equation does not contain variable "x" so uxx=0 i.e. Δu=uyy uy=e-λtsin(αt){-βsin(βt)}...
  29. S

    Solve Heat Equation PDE with Boundary Conditions

    Homework Statement u_{t}=3u_{xx} x=[0,pi] u(0,t)=u(pi,t)=0 u(x,0)=sinx*cos4x Homework Equations The Attempt at a Solution with separation of variables and boundry conditions I get: u(x,t)= \sumB_{n}e^-3n^{2)}}*sinnx u(x,0)=sinx*cos4x f(x)=sinx*cos4x=\sumB_{n}*sinnx...
  30. C

    Steady state heat equation in concentric spherical shells

    Homework Statement Homework Equations The Attempt at a Solution I'm trying to find the steady state solution to the heat equation for a system of spherical shells (looks like http://correlatingcancer.com/wp-content/uploads/2009/01/nanoshell-thumb.jpg" ) where heat generation Q occurs in...
  31. M

    Can the heat equation apply to gases?

    I've never known this but the equation only seems to contain a conduction term so I assume it can only apply to solids. Is there a similar equation for the time-evolution of temperature fields in gases, where convection is also considered? (how about radiation? although that sounds like it will...
  32. D

    Modified heat equation steady state

    Homework Statement determine the steady state equation for the given heat equation and boundary conditions Homework Equations Ut=Uxx-4(U-T) U(0,T)=T U(4,T)=0 U(x,0)=f(x) The Attempt at a Solution I put Ut=0 so 0=UInf''-4(Uinf-T) then once I tried to integrate I ended up with a...
  33. B

    Heat Equation with movable point source

    1. I would like to find and plot the temperature for all points in a 1 dimensional rod of length L, due to a heat source of q placed at point xo where 0<xo<L. The ends of the rod are kept at a constant of 300 Kelvin. The thermal diffusivity constant is a. I'm also looking for the steady state...
  34. T

    What is the Relationship Between Heat Energy and Temperature?

    Homework Statement Show that the heat energy per unit mass necessary to raise the temperature of a thin slice of thickness \Deltax from 0^o{} to u(x,t) is not c(x)u(x,t). but instead \int_0^uc(x,\overline{u})d\overline{u}. Homework Equations According to the text, the relationship...
  35. U

    Analytical solution for heat equation with simple boundary conditions

    I am trying to solve the following heat equation ODE: d^2T/dr^2+1/r*dT/dr=0 (steady state) or dT/dt=d^2T/dr^2+1/r*dT/dr (transient state) The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2. I have searched the analytical solution for this kind of ODEs in polar coordinate...
  36. Y

    Find an equilibrium solution of heat equation Please help

    Hello I need help with this heat equation I need to find steady state solution please help me I ve tried but I could not get it. Homework Statement http://www.alm5zn.com/upfiles/m0w00932.jpg Homework Equations The Attempt at a Solution I ve got C= Cx/2 +ax+b...
  37. T

    Derivation of Heat Equation for frustum-shaped rod

    Homework Statement Derive the Heat Equation for a rod in the shape of a frustum. Assume the specific heat c and density p are all constant. Use the "exact" method (through an integral) to derive the heat equation. Also, there is no heat source in the rod. Homework Equations The...
  38. J

    Is this an acceptable method to solve a convectihe heat equation?

    Is this an acceptable method to solve a convectihe heat equation? I am trying to solve the following PDE: \frac{\partial f(x,t)}{\partial t}=a(x,t)\frac{\partial^2 f(x,t)}{\partial x^2}+b(x,t)\frac{\partial f(x,t)}{\partial x}+c(x,t)f(x,t) where a, b and c are known functions of x and t...
  39. B

    Neumann Boundary Conditions for Heat Equation

    I wrote a program that uses the FEM to approximate a solution to the heat conduction equation. I was lazy and wanted to test it, so I only allowed Neumann boundary conditions (I will program in the Dirichlet conditions and the source terms later). When I input low values for the heat flux, I...
  40. S

    Fourier Transform to solve heat equation in infinite domain

    I'm having trouble following a step in my notes: first off the heat equation is given by: \frac{\partial u}{\partial t}=k^{2}\frac{\partial^{2}u}{\partial x^{2}} then take the Fourier transform of this w.r.t.x, where in this notation the Ftransform of u(x,t) is denoted by U(alpha,t)...
  41. O

    Heat equation and taylor's approximation

    Homework Statement storage of heat, T at time, t (measured in days) at a depth x (measured in metres) T(x,t)=T0 + T1 e^{-\lambda} x sin (wt - \lambdax) where w = 2pi/365 and \lambda is a positive constant show that \deltaT/\deltat = k \delta^2 T / \deltax^2Derive the second order Taylor...
  42. T

    Heat equation with nonhomogeneous boundary conditions

    Homework Statement Consider \frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2} subject to u(0,t) = A(t),\ u(L,t) = 0,\ u(x,0) = g(x). Assume that u(x,t) has a Fourier sine series. Determine a differential equation for the Fourier coefficients (assume appropriate continuity)...
  43. M

    What is the Inequality for the Heat Equation?

    So I multiplied the heat equation by 2u, and put the substitution into the heat equation, and get 2uut-2uuxx=(u2)t=2(uux)x+2(ux)2. I`m not sure where to go from there, I can integrate with respect to t, then I would have a u2 under the integral on the left side, but them I`m not sure where to...
  44. D

    About solving heat equation in half plane

    Hi guys, I have trouble when solving the following heat transport equation in half plane in frequency domain. (\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\theta(x,y)=i\beta\theta(x,y),-\infty<x<+\infty,y\geq 0...
  45. W

    Is there an ODE with only time dependence and constant position?

    Hi all I am new to this forum. I need a little help on a project I am doin. The heat equation is a pde with dependence on time and position, what i want to know is is there a ode which the dependence is only on time, constant position? Any help would be great thank
  46. J

    Solving PDE Heat Equation for Temperature Distribution

    Homework Statement Find the distribution of temperatures in the rod of length L with the follow BC and NC Homework Equations u_{t}=\alpha u_{xx}\,\,\,x\in]\frac{-L}{2},\frac{L}{2} u(\frac{-L}{2},t)=u(\frac{L}{2},t)=700 u(x,0)=300\,\,\,x\in]\frac{-L}{2},\frac{L}{2} The Attempt at...
  47. W

    Fourier decomposition and heat equation

    Homework Statement In the heat equation, we have $T(t,x)=sum of a_k(t)b_k(x)$. Now I want to find a formula for computing the initial coefficients $a_k(0)$ given the initial temperature distribution $f(x)$. Homework Equations We know that in a heat equation , $f(0)=0$, $f(1)=0$...
  48. S

    Understanding the One-Dimensional Heat Equation

    why does the one-dimensional heat equation for temperature distribution contain a second derivative of the spatial variable?
  49. M

    Initial Condition Problems with Heat Equation in Mathematica

    I've been trying to work through the heat equation given in this Sous Vide cooking primer: http://amath.colorado.edu/~baldwind/sous-vide.html It gives a modified version of the heat equation with a shape parameter for simplification. The equations are shown below...
  50. N

    Heat equation with a Fourier Series on an infinitely long rod

    Homework Statement The heat equation for an infinitely long rod is shown as: \alpha^2 \frac{\partial^2}{\partial x^2}u(x,t) = \frac{\partial}{\partial t}u(x,t) u(0,t) = u(L,t) = 0,\ \forall \ t > 0 u(x,0) = sin(\pi x) \ \forall \ 1 < x...
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