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. N

    MHB Understanding the Heat Equation: What Does $T_j^n$ Represent?

    Please refer to the attached image, Question 1, which i have pointing an arrow at.Is this simply asking me to sub in t=0 into (5), which would leave me with $B_{l}\sin(\pi l x)$ inside the sum? would they expect anything further? Thanks!
  2. D

    MHB Change of variables heat equation

    \[ \alpha^2T_{xx} = T_t + \beta(T - T_0) \] where \(\beta\) is a constant and \(T_0\) is the temperature of the surrounding medium. The initial temperature distribution...
  3. H

    Solving Heat Equation by Fourier Transform

    When the rod is infinite or semi-infinite, I was taught to use Fourier transform. But I don't know when should the full Fourier transform or sine/cosine transform be used. how's the B.C. related to the choice of the transform ?
  4. P

    PDE:Cauchy Problem for Heat Equation

    Homework Statement Solve the Cauchy problem ut =kuxx, x ∈ R, t>0, u(x, 0) = φ(x), for the following initial conditions. (a) φ(x)=1if |x|<1 and φ(x)=0 if |x|>1. Write the solutions in terms of the erf function. Homework Equations u(x,t)=∫G(x-y,t)*φ(y)dy from -∞, to ∞ where G(x,t) is...
  5. A

    3d heat equation with constant point source

    Hi all, I'd like to solve the following problem in 3 dimensions: \partial_t u(r,t) = D\Delta u(r,t) u(r,0) = 0 u(0,t) = C_o In words, I am looking at a point 'source' that is turned on at t=0 and held at constant temperature. The ultimate goal is to then convolve this solution with...
  6. C

    What is the Best Approximation for Heat Transfer in Two-Dimensional Systems?

    Hi there. At first I tought of posting this thread on the homework category, but this is a conceptual doubt rather than anything else. While revisiting Heat Transfer I stumbled upon a simple problem, that yet got me thinking. It is as follows: Before anything else, let me show...
  7. D

    Resources for Solving Heat Equation

    Hey guys, I am just looking for some online resources for solving the heat equation. So far I have looked at Paul's Online Math Notes: http://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx But I don't feel very confident with the material yet. I would really like some more...
  8. P

    How Do You Solve the 1D Heat Equation with Trigonometric Initial Conditions?

    problem u_t=u_xx, x is in [0,1], t>0 with u(0,t)=u(1,t)=0, t>0 u(x,0)=sin(pi*x)-sin(3*pi*x), x is in (0,1) i think its solution is of the form u(x,t)=sigma(n=1 to infinity){a_n*sin(n*pi*x)*exp(-n^2*pi^2*t) where a_n=2*integral(0 to 1){ (sin(pi*x)-sin(3*pi*x)) * sin(n*pi*x) }...
  9. B

    Heat Equation with cylindrical rod

    Hello, I am looking to apply to heat equation to a cylindrical rod and solving with explicit finite difference scheme. I have never worked with cylindrical coordinates before, what would be the best way to model this? I am having a hard time understanding the advantage of using cylindrical...
  10. I

    How Do You Solve the Steady Temperature Distribution for a Semi-Infinite Plate?

    Homework Statement A flat plate lies in the region: 0<x<35, 0<y<inf The temperature is steady (not changing with time), and the boundary conditions are: T = { x if 0<x<35; y=0 70-x if 35<x<70; y=0 0 if x=0 0 if x=70 } Enter the temperature at (x = 42, y = 21) Homework...
  11. M

    Solving Fourier Heat Equation: Analytical Solutions

    Can anyone tell me if there exist analytical solution to the Fourier heat equation rhoCdt/dt= ∇.(k∇T) + S Thanks
  12. G

    Applying boundary condition on heat equation

    Homework Statement hey, i have a heat equation question which asks to solve for u(x,t) given that u(0,t)=Q_0 + ΔQsin(ωt).Homework Equations d_xx u = k d_t u u(0,t)=Q_0 + ΔQsin(ωt) The Attempt at a Solution so you can solve the equation pretty easily with separation of variables, i.e...
  13. S

    Radiative/Convective Boundary Conditions for Heat Equation

    Hi everyone, I'm attempting to create a computer program to solve the transient 3d heat equation using the Crank Nicolson method. I would like to model the boundaries of my domain as losing heat via convection and radiation due to the temperature difference between the boundary and the air in...
  14. M

    Solving the Heat Equation: Exploring Solutions to Temperature Differentials

    I've been teaching myself some thermodynamics, and I've been thinking about solving the heat equation. \frac{\partial T}{\partial t} = K\frac{\partial ^2 T}{\partial x^2} I haven't taken a course in PDEs. I have noticed that if I assume an exponential solution, there are not non-decaying...
  15. O

    Backward euler method for heat equation with neumann b.c.

    I am trying to solve the following pde numerically using backward f.d. for time and central di fference approximation for x, in MATLAB but i can't get correct results. \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}},\qquad u(x,0)=f(x),\qquad u_{x}(0,t)=0,\qquad...
  16. stripes

    How Can the Heat Equation on a Ring Be Demonstrated in a Classroom Seminar?

    Homework Statement I have to give a seminar to my math class about the heat equation on the ring. I will be introducing the heat kernel on the circle to the class, which is as follows: H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} . I basically will...
  17. M

    MATLAB Solving Heat Equation in Cylindrical Coordinates with MATLAB's pdepe

    hello i am solving heat equation in cylindrical coordinator. i am using MATLAB "pdepe" solver to solve the partial differential equation. can anyone suggest me how to choose the initial condition?
  18. E

    Heat Equation in cylindrical coordinates

    Large, cylindrical bales of hay used to feed livestock in the winter months are D = 2 m in diameter and are stored end-to-end in long rows. Microbial energy generation occurs in the hay and can be excessive if the farmer bales the hay in a too-wet condition. Assuming the thermal conductivity of...
  19. J

    Solving heat equation for heat-pulse in a point on the surface

    Hi everybody, I'm trying to find a solution for the 3D heat equation for pulsed surface heating of a semi-infinte solid with insulated surface. I know the method of reflection is required, and that a point source in an infinite solid gives the following solution: U(x,y,z,t)=...
  20. M

    Understanding the Heat Equation and its Practical Applications

    given the heat equation \frac{\partial u}{\partial x}=\frac{\partial^2 u}{\partial x^2} what does \frac{\partial^2 u}{\partial x^2} represent on a practical, physical level? I am confused because this is not time-space acceleration, but rather a temperature-spacial derivative. thanks all!
  21. stripes

    What Linear Transformations Satisfy the 2D Heat Equation?

    Homework Statement For the heat equation in two space variables find all the linear transformations of the form (x,y) = a(x',y') for real number a such that \frac{\partial u}{\partial t} = \frac{k}{\sigma}\Delta u \Leftrightarrow \frac{\partial v}{\partial t} = \Delta'v where u(x,y,t) =...
  22. S

    MHB Solving the 1D Heat Equation with Separations of Variables

    Have been trying for hours but simply no results. Hope that someone can help me out. \[\frac{\partial u}{\partial t}=4\frac{\partial^2 u}{\partial x^2}\] for \(t>0\) and \(0\leq x\leq 2\) subject to the boundary conditions \[u_x (0,t) = 0\mbox{ and }u(2,t) = 0\] and the initial condition...
  23. fluidistic

    Steady state heat equation in a cylinder

    Homework Statement I'm unable to solve a problem of heat equation in a cylinder in steady state. The problem is a cylinder of radius a and a height L. The boundary condition are ##T(\rho , \theta , 0)=\alpha \sin \theta##, ##T(\rho, \theta , L)=0## and ##\frac{\partial T}{\partial \rho} (a...
  24. fluidistic

    Heat equation with Laplace transform

    Homework Statement Problem 8-19 in Matthews and Walker's book on mathematical physics. A straight wire of radius a is immersed in an infinite volume of liquid. Initially the wire and the liquid have temperature T=0. At time t=0, the wire is suddenly raised to temperature ##T_0## and...
  25. J

    1D Heat Equation / Greenhouse Simulation

    I am trying to model a thermal mass for a greenhouse design. A thermal mass is basically a box of mass with high Cp buried in the ground with the top exposed to the inside of the greenhouse and all the other sides insulated. My model is based on the fact that during the day, there is a net...
  26. D

    Heat equation and Wave equation problems

    1. Solve the Heat equation u_t = ku_xx for 0 < x < ∏, t > 0 with the initial condition u(x, 0) = 1 + 2sinx and the boundary conditions u(0, t) = u(∏, t) = 1 (Notice that the boundary condition is not homogeneous) 3. Find the solution of the Wave equation u_tt = 4 u_xx with u(0...
  27. M

    Heat Equation with moving source

    Hello there, I want to solve the heat PDE in a 1D domain for a source moving at constant speed. The problem has been solved already, the solution being stationary in a reference frame moving with the source. This is highly un-intuitive, and I suppose the result originate from the fact the...
  28. L

    How does the heat equation change in the real world?

    The heat equation predicts that heat spreads infinitely far over arbitrarily small time intervals. What happens in real life? How does the heat equation get modified?
  29. V

    Solving the Heat Equation with R Code

    Hi! I have some trouble understanding this question. Could someone help me with it? Thanks! Solve the following with the explicit method from t=0 to t=0.5 with h=1/10 and with μ(=k/h2)=0.5 ut = uxx, -1 ≤ x ≤ 1, t>0 u(0,x) = cos(x), -1 ≤ x ≤ 1 u(t,-1) = u(t,1) = e-tcos1, t>0 Compute...
  30. F

    Help with heat equation dirac delta function?

    Homework Statement The question was way too long so i took a snap shot of it http://sphotos-h.ak.fbcdn.net/hphotos-ak-snc7/397320_358155177605479_1440801198_n.jpg Homework Equations The equations are all included in the snapshotThe Attempt at a Solution So for question A I've done what the...
  31. D

    Finding Solution of Inhomogeneous Heat Equation

    Homework Statement Show that if u(x,t) and v(x,t) are solutions to the Dirichlet problems for the Heat equation u_t (x,t) - ku_xx (x,t) = f(x,t), u(x,0) = Φ₁(x), u(0,t) = u(1,t) = g₁(t) v_t (x,t) - kv_xx (x,t) = f(x,t), v(x,0) = Φ₂(x), v(0,t) = v(1,t) = g₂(t) and if Φ₂(x) ≤ Φ₁(x)...
  32. D

    MHB Solving the Heat Equation with Initial Conditions

    I have already solved the main portions. I have $$ T(x,t) = \sum_{n = 1}^{\infty}A_n\cos\lambda_n x\exp(-\lambda_n^2t) $$ The eigenvalues are determined by $$ \tan\lambda_n = \frac{1}{\lambda_n} $$ The initial condition is $T(x,0) =1$. For the particular case of $f(x) = 1$, numerically...
  33. D

    MHB Solving the Heat Equation with B.C. and I.C.

    Is this correct? $$ \text{B.C.}=\begin{cases} T_x(0,t) = 0\\ T(\pi,t) = 1 \end{cases} $$ The I.C. is $T(x,0) = 0$. The equation is $\frac{1}{\alpha}T_t = T_{xx}$.$$ \varphi(x) = A\cos\lambda x + B\frac{\sin\lambda x}{\lambda} $$ and $$ \psi(t) = C\exp\left(-\alpha\lambda^2t\right). $$ First...
  34. F

    Solution to a PDE (heat equation) with one initial condition

    Homework Statement By trial and error, find a solution of the diffusion equation du/dt = d^2u / dx^2 with the initial condition u(x, 0) = x^2. Homework Equations The Attempt at a Solution Given the initial condition, I tried finding a solution at the steady state (du/dt=0)...
  35. T

    Neumann Boundary Conditions using FTCS on the Heat Equation

    I am really confused with the concept of Neumann Boundary conditions. For the simple PDE ut=uxx for the domain from 0<=x<=1 I'm trying to use a ghost point (maintain a second order scheme) for the Neumann Boundary condition ux(0,t) = 0. I understand that I can setup a scheme to...
  36. A

    FDM Heat Equation FTCS Scheme with NBC

    Homework Statement I have a quiz question that I'm struggling with. We've been working on using a FTCS scheme with two Essential Boundary Conditions, and now I have a problem with one EBC (ie static) and a Natural Boundary Condition (ie a derivative). The condensed problem statement: u_{t} =...
  37. R

    PDE - Solve heat equation with convection

    Homework Statement Solve u_t -k u_xx +V u_x=0 With the initial condition, u(x,0)=f(x) Use the transformation y=x-Vt Homework Equations The solution to the equation u_t - k u_xx=0 with the initial condition is u(x,t)=1/Sqrt[4\pi kt] \int e^(-(x-y)^2 /4kt)f(y) dy The Attempt at a...
  38. A

    Solving one dimension steady state heat equation with finite differences

    I have a project where I need to solve T''(x) = bT^4 ; 0<=x<=1 T(0) = 1 T'(1) = 0 using finite differences to generate a system of equations in Matlab and solve the system to find the solution So far I have: (using centred 2nd degree finite difference) T''(x) = (T(x+h) - 2T(x) +...
  39. G

    Mathematica [Mathematica] Solving Heat Equation in Spherical Coordinates

    Hello Folks, I have this equation to solve (expressed in LaTeX): \frac{\partial{h}}{\partial t} = \frac{1}{n} \left[ \frac{1}{r^2 \sin^2{\phi}} \frac{\partial}{\partial \theta} \left( K \frac{\partial h}{\partial \theta} \right) + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi}...
  40. J

    Heat equation and maximum principle

    Homework Statement Suppose that u(x,t) satisfies the heat equation u_{t}=u_{x x} for 0<x<L and t>0 with initial condition u(x,0)=θ(x) and boundary conditions u(0,t)=u(L,t)=0. Suppose that θ(x)>0 for 0<x<L. Explain why u(x,t)>0 for all 0<x<L and t>0 Homework Equations Strong Maximum principle...
  41. G

    2D heat equation bounday conditions for different intervals

    Homework Statement I have boundary conditions on my heat equation ## \dot{T}(x,t) = T''(x,t) ## ## T(0,t) = T(L,t), ## ## \frac{\partial T(0,t)}{\partial x} = \frac{\partial T(L,t)}{\partial x} ## Then at ## T= 0## ## T(x,0) = 1 ## for ## 0<x<L/4 ## ## T(x,0) = 0 ## for ## L/4<x<L ##...
  42. J

    How Do I Solve a Heat Equation with Unknown Forcing Term p(x, t)?

    Homework Statement vt(x,t)=vxx(x,t) + p(x,t), Neumann boundary conditions, v(x,0)=cos(∏x) Homework Equations Assume v(x,t)=X(x)T(t) The Attempt at a Solution I'm stuck. We aren't given a p(x,t) and I'm not sure what to do. Where do I go from here? Attempt so far:
  43. S

    Solving the 2D Heat Equation with FDM and Gauss-Seidel Method

    Homework Statement Given a steady-state heat transfer for a 100mx100m plate, to be discretized to 6 nodes, governed by a heat tranfer equation: U_xx + U_yy = 0 Given dirichlet boundary conditions: U(0,y)=50, U(100,y) = 100,, neumann boundary: U_y(x,0)=0, U_y(x,100) = 0. Find the...
  44. S

    2D heat equation using gauss-seidel method

    Hi guys. Badly need some help. We were given this 2D heat equation BV problem. On the square plate, values on all four edges are given (2 are Neumann, 2 are Dirichlet). And we are to solve this problem using FDM, on a 5point stencil. So I used FDM approximations to derive the formula for the...
  45. W

    Why is T''(x) Zero in the Steady State Solution of the Heat Equation?

    Hi, So if I start with the boundary conditions U(0,t) = T1 and U(L,t) = T2 and T1 does not equal T2, it seems that you are supposed to look at the 'steady state solution' (solution as t goes to infinity)? which satisfies T''(x) = 0 so the solutions are T(x) = Ax + B and then you...
  46. C

    Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform help ;)

    Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform... help! ;) Hi I'm trying to model the temperature profile of an inertia friction welding during and after welding. I have the welding outputs and have come up with a net heat flow wrt time during the process. I now want to...
  47. C

    Conduction - Heat Equation - Units Don't Add Up

    Conduction - Heat Equation - Units Don't Add Up! Hi there I have what I think/hope is a simple question: I've been working on heat inputs and outputs in inertia friction welds and have managed to produce a net power term (W) as a function of time. I now want to use that in the heat...
  48. G

    Heat Equation Homework: Find F(x,t)

    Homework Statement F(x,t) satisfies \frac{\partial^2 F(x,t)}{\partial x^2}=\frac{\partial F(x,t)}{\partial t} With the following boundary conditions \frac{\partial F(0,t)}{\partial x}=\frac{\partial F(1,t)}{\partial x} F(0,t) = F(1,t) F(x,0) = x^2 \text{ for } x \in (0,1) The...
  49. N

    Can the 3D Heat Equation Be Solved Analytically?

    Hi, I've been trying to solve \frac{\partial \vec{u}(x,y,z,t)}{\partial t} = \nu \nabla^2 \vec{u}(x,y,z,t) Where the Laplacian is 3-d. My initial conditions are \vec{u}(x,y,z,0)=\vec{u_o}(x,y,z). And my BC is that (u_z,v_z,w)= 0 at z=0. where \vec{u} = (u,v,w) I want to...
  50. S

    Heat equation with peculiar boundary conditions

    Homework Statement Find the solution to the heat equation for the following conditions: Homework Equations The Attempt at a Solution Not sure. I've only encountered the following scenarios: temperatures of both ends are arbitrary values both ends are insulated (so the first...
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