Heat equation Definition and 257 Threads

I have tried to solve the cylindrical case of the heat equation and reached the second order differential equation for the function R(r): R'' + (1/r)*R' + (alfa/k)*R = 0 (alfa, k are constants) I couldn't find material on the web for non-constant coefficients, does anyone know how to...

Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here r=\sqrt{x^{2}+y^{2}} is the cylindrical coordinate. From the three dimensional heat equation derive the equation u_{t}=k(u_{rr}+\frac{u_{r}}{r}). My...

helloo! i'm trying to find the Green's function for the modified heat equation $u_t = t * grad(u) $ in n dimensions using scaling arguments. I know how to do this for the regular heat equation, by switching to polar coordinates and noticing that the equation and initial conditions are...

regarding 1-d Head Equations on rods. I am aware of how to long a rod with length x=0 to x=L. and initial conditions of u(0,t)=0 degrees and u(L,t)=100 degrees. But how does the problem change if before t=0 the rod at x=0 was at 100 degrees and x=L was at 0 degrees. So at time=0 the rod was...

Greetings all, I have a question in regards to my initial conditions. The problem as given is: ut=uxx with u' = 0 at x=0 and u=0 at x=L I was also given u={1 0<x<L/2, 0 L/2<x<L I understand the set up of the problem and the solving of it for the most part, however I'm having...

Homework Statement Plate in the shape of the circular halo (inner radius a, outer radius b>a), the inner edge is being kept at a constant temperature T_0, and the outer at the temperature given by the function f(\phi)=T_0\cos(2\phi). Find the equilibrium distribution of the heat everywhere...

My professor asked us to use the heat equation to compare temperature change with the specific heat of three metals. I have no idea of what the heat equation is. I read a little bit about it on Wikipedia, but the equation there makes no sense to me. I'd really appreciate if anyone could help me...

solve the heat equation ut = kuxx -infinity < x < infinity and 0 < t < infinity with u(x,0)= x2 and uxxx(x,0)= 0 first i showed that uxxx(x,t) solves the equation (easy part) the next step is to conclude that u(x,t) must be of the form A(t)x2 + B(t)x + C(t). i...

Hi everyone, I am a Cybernetics student and as part of a project I need to determine the feasability of building a small ice rink (approximately 2m x 4m) The plan is to build a frame and exterior (the sides and bottom of a box) out of wood, and have a layer of internal insulation such as...

dTB/dt = -k(TB-TM). TM is held constant. (TB-TM) = Q, so: dTB/Q = -kdt. How would I change this equation so that instead of integrating wrt to TB, I can instead integrate wrt Q?

Homework Statement I'm having trouble deriving the following equation \frac {\partial^2 {\theta}}{\partial {x'^2}} = -y^2*exp(\theta) and y = x/x' my main problem is the exponent Homework Equations The Attempt at a Solution Normally i would use the equation (x')'' + k^2*x' = 0 x' = c1...

Homework Statement Let's say you have a 3m long copper pipe, 3mm in thickness with a diameter of 170mm. You fix one end at 1K and insulate it to prevent conduction or convection between the air and the pipe itself. There is still radiation. Assume that the inside of the pipe has no effect...

Homework Statement Assume that a bar is insulated at the endpoints. If it loses heat through its lateral surface at a rate per unit length proportional to the difference u(x,t) - T, where T is the temperature of the medium surrounding the bar, the equation of heat propagation is now u_{t} =...

Problem: u (sub t) = (1/2)u (sub xx) find the solution u(x,t) of the heat equation for the following initial conditions: u(x,0) = x u(x,0) = x^2 u(x,0) = sinx u(x,0) = 0 for x < 0 and 1 for x>=0 i'm really flying blind here. I've taken differential equations years ago but nothing...

Homework Statement Find the temperature distribution in the long thin bar −a ≤ x ≤ a with a given initial temperature u(x,0) = f(x). The side walls of the bar are insulated, while heat radiates from the ends into the surrounding medium whose temperature is u = 0. The radiation is taken...

Consider the heat equation in a radially symmetric sphere of radius unity: u_t = u_{rr}+{2 \over r}u_r \ for \ (r,t) \in (0,1) x (0,\infty) with boundary conditions \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0 Now, using separation of variables u=R(r)T(t) leads to the...

Hello there, hope you are having a good one. My problem is to solve the heat equtaion in cylindrical coordinates. This has been done by others for me, so a closed form solution is available, please see attached (please note the problem is 1 - D due to initial conditions depending only on...

Hi, I need help to solve this problem, about 1-D heat equation \partialu / \partialt = k (\partial2u / \partialx2)-2u (0< x <1) u(x,0)=e-x u(0,t)=e-2t u(1,t)=0 I need to solve it with separation variable

Hi, it's been a while since I touched mathematics and I'm a little rusty... I'm looking at a problem right now that I find difficult to understand, conceptually. Any insight would be greatly appreciated. (A direct solution would help immensely as well, not only because that's what I need to...

One critic of the Fourier heat equation \frac{\partial T}{\partial t}=k\nabla^2 T that I recently came across is that it gives rise to infinite speed of heat propagation. I understand that the speed cannot be infinite because it contradict special relativity that no speed should be...

I don't be able to convert the following code(HEAT EQUATION BACKWARD-DIFFERENCE ALGORITHM in the Burden-faires numerical analysis book).I need heat EQUATION FORWARD-DIFFERENCE ALGORITHM C like following code.I don't be able to convert FORWARD-DIFFERENCE the following code .Please help me. if...

I don't be able to convert the following code(HEAT EQUATION BACKWARD-DIFFERENCE ALGORITHM in the Burden-faires numerical analysis book).I need HEAT EQUATION FORWARD-DIFFERENCE ALGORITHM C like following code.I don't be able to convert FORWARD-DIFFERENCE the following code .Please help me. /*...

Homework Statement http://img444.imageshack.us/img444/7641/20240456gw8.png Homework Equations http://img14.imageshack.us/img14/5879/63445047rj2.png Note that the rightside of the rod is insulated. The Attempt at a Solution I get this model: \frac{ \partial{u} }{ \partial{t} } = \kappa...

Homework Statement http://img3.imageshack.us/img3/5020/84513876dm0.png The Attempt at a Solution I found that f(t) =exp \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) Is this correct?

Homework Statement If there is heat radiation in a rod of length L, then the 1-D heat equation might take the form: u_t = ku_xx + F(x,t) exercise deals with the steady state condition => temperature u and F are independent of time t and that u_t = 0. u_t = partial derivative with...

Problem:IF there is heat radiation within the rod of length L , then the 1 dimensional heat equation might take the form u_t = ku_xx + F(x,t) Find u(x) if F = x , k = 1 , , u(0)=0 , u(L) = 0 the problem is that i am not sure what this is asking me , how can i find u(x) if i have...

Hello, I believe this is my first post. I would like to solve the heat equation PDE with some special (but not complicated) initial conditions, my scenario is as follows: A perfectly spherical mass of water, where the outer surface is at some particular temperature at t=0 (but not held at...

Hi all, I need to solve the heat equation (Ut=C*Uzz) with the following boundary conditions: U(max z,t)=0 and Uz(0,t)=-B. where B is a constant. My initial condition is U(z,0)=Uo where Uo is a constant. I know how to solve the equation for simple 0 boundary conditions. The Neumann BC...

Homework Statement Solve the initial-value problem for the heat equation ut = K\nabla2u in the column 0< x < L1, 0< y < L2 with the boundary conditions u(0,y;t)=0, ux(L1,y,t)=0, u(x,0;t)=0, uy(x,L2;t)=0 and the initial condition u(x,y;0)=1. Find the relaxation time. Can anyone please explain...

Homework Statement So I'm trying to solve Evans - PDE 2.5 # 12... "Write down an explicit formula for a solution of u_t - \Delta u + cu=f with (x,t) \in R^n \times (0,\infty) u(x,0)=g(x)" Homework Equations The Attempt at a Solution I figure if I can a fundamental solution...

Homework Statement Solve the heat equation: u_{t} = u_{xx} - u - x*e^{-t} BC: u(0,t) = 0, u(1,t) = 0 IC: u(x,0) = x Homework Equations The Attempt at a Solution The only progress I've made so far is figuring out that the steady state solution is zero. Other than that I don't...

Hi, I have the following problem and would appreciate any help you might be able to give. In reality, I have a 2D/3D problem (self heating of a semiconductor device), but I'm absolutely happy if I would understand the 1D simplification: I have a rod of length L (semi infinite would be...

1. Solve the one dimensional heat equation for a rod of length 1 with the following boundary and initial conditions: BC: \partialu(0,t)/\partialt = 0 \partialu(1,t)/\partialt = 0 these are the wrong boundary conditions (see below) Actual BC: \partialu(0,t)/\partialx = 0...

inhomogeneuos heat equation! Homework Statement ∂θ/∂t= D∇2θ + K, the mensioned equation is the heat equation for a cylindrical rod , and the requaired is to find the ordinary differential equation for θ(r) .where the radius of the rod is R , and K is constant ( correspond to a constant rate...

Hi. Having problems with this tricky Heat Equation Question. Managed to do part (a) and would appreciate verification that it's right. But I can't manage to finish off the second part. I've started it off so please do advice me. Thanks a lot! QUESTION...

I'm trying to solve a basic heat equation \frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2} I manage to get T=X(x)\tau(t) Then \tau(t)=A*e^{-\alpha*\lambda^2*t} and X(x)=C*sin(\lambda x) where \lambda=\pi/Ln n=1,2,3,... From here I don't know how or why I get to a...

I don't understand where to even start with this problem. This book has ZERO examples. I would appreciate some help. Show that by a suitable scaling of the space coordinates, the heat equation u_{t}=\kappa\left(u_{xx}+u_{yy}+u_{zz}\right) can be reduced to the standard form v_{t} = \Delta...

How would one obtain a Fourier Transform solution of a non homogeneous heat equation? I've arrived at a form that has \frac{\partial }{ \partial t }\hat u_c (\omega,t) + (\omega^2 + 1)\hat u_c (\omega,t) = -f(t) My professor gave us the hint to use an integrating factor, but I don't see...

Hello, I'm currently doing some research comparing efficiency of various programming languages. Being a user of Matlab, Mathematica, and Excel, c++ is definitely not my forte. I was wondering if anyone might know where I could find a simple, standalone code for solving the 1-dimensional heat...

Hi all, I have tried to solve the heat equation with a Fourier-Bessel approach but I fail to implement the boundary condition, which is a Neumann condition. Every textbook that I have available treats the corresponding Dirichlet problem but not the Neumann one. Below I have tried to summarize...

Hello to all! Homework Statement for testing my program i need a heat equation with numerical initial and boundary conditions: Derivative[2, 0][f][x, t] == Derivative[0, 1][f][x, t] f[x, 0] == numerical f[0, t] == numerical, f[numerical, t] == numerical PS. to moders: please, if...

Hello Trying to calculate and simulate with Matlab the Steady State Temperature in the circular cylinder I came to the book of Dennis G. Zill Differential Equations with Boundary-Value Problems 4th edition pages 521 and 522 The temperature in the cylinder is given in cylindrical...

Hi folks, Given the following heat equation u_t = u_{xx} + t - x^2, I'd like to find all solutions u(x,t)\in C^2(\mathbb{R}^2) such that the quotient |u(x,t)| / (|x|^5 + |t|^5) goes to zero as the sum |x| + |t| goes to infinity. I know how to do the same problem with the usual...

Homework Statement n_1 moles of a monatomic gas and n_2 moles of a diatomic gas are mixed together in a container. Derive an expression for the molar specific heat at constant volume of the mixture. My answer can only use the variables n_1 and n_2, and I'm assuming constants. Homework...

Homework Statement I am solving the heat equation 1/a^2\theta_t = \theta_x boundary conditions are \theta(0,t) = \theta(L,t) = 0 t > 0 initial conditions are \theta(x,0) = T_0sin(x\pi/L) now I have derived the steady solution to be 0 and I have derived that the general...

can anyone help me interpret what exactly this question is asking as i am quite unawares By direct substitution into the heat equation and calculation of boundary values, verify that the solution u(x, t) for a metal rod of length L which satisfies the initial temperature u(x, 0) = f(x) and...

I am taking the liberty of collecting mathwonk's "short course" for some followup comments/questions, since this topic is IMHO more interesting than the context in which it first appeared. (Hope this is OK under PF rules!). Part I: Part II: Part III: How annoying, Part IV won't...

Im trying to solve the heat equation in 2dim on a plate. 0=<x=<L, 0=<y=<L. With homogenous dirichlet conditions on the boundary and the initial condition: T(x,y)=T0sin(pi*x/L)sin(pi*y/L) With separation of variables i get the solution T(x,y,t)=\sum_{m=0}^\infty\sum_{n=0}^\infty...

I'm considering a wall, using this equation: Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right) Where QP is added effect (an oven), \dot V is air-flow, and the rest should be self-explanatory. I'm just not sure what it tells me. The...

Hey all, I've been working on learning to solve some PDE's. To do this I've been reading other people's tutorials. Here's one on the heat equation: http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node21.html This is pretty much the same as the others I've read on the heat equation...