Heat equation Definition and 282 Threads

Homework Statement du/dt = d2u/dx2 + u Bc: u'(0) = u'(1) = 0 Ic: u(x,0) = 1 Homework Equations Using sturm-liouville to solve for eigenvalues. The Attempt at a Solution After first separating variables in the equation we get G'/G - 1 = F'' = λ after using Sturm-Liouville we...

Homework Statement I am trying to solve the Laplacian Equation with mixed boundary conditions on a rectangular square that is 1m x 1m. Homework Equations \nabla2T=0 .....T=500K ....________ ....|@@@@| T=500K...|@@@@|...T=500K ....|@@@@| ....|______.| ....Convection ....dT...

Homework Statement heat equation of for 2 dimensional body(stationary)...heat is supplied to a body per unit volume and per unit time and by using conservation of energy principle the following equation is derived.. \intQt dA= \intqt d\ell the intergral on the right is a line integral on a...

OK, so I was trying to solve the Heat Equation with Inhomogeneous boundary conditions for a rod through Fourier Series when I got stuck at the solution for the coefficient c_n, the part where I'm stuck is highlighted in red. The following is just a step-by-step solution of how I got to c_n...

thanks allot they worked out fine, just another quick question if could help. A semi-infinite bar 0<x<infinity is subject to periodic heating at x=0; the temperature at x=0 is T_0cos\omegat and is zero at x=infinity. By solving the heat equation show that T(x,t)=...

Consider the following non-homogenous heat equation on 0 \leq x \leq \pi u_t = k u_{xx} - 1 with u(x,0) = 0, u(0,t) = 0, u(\pi, t) = 0 Find a solution of the form \displaystyle \sum_1^{\infty} b_n(t) \phi_n (x) where \phi_n(x) are the eigenfunctions of an appropriate homogenous...

Homework Statement The Heat equation in two space is \alpha ^2 \left[\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2} \right]=\frac{\partial u}{\partial t} Assuming separation solution of the form u(x,y,t)=F(x)G(y)H(t) find ordinary differential equations satisfied by...

Hi, I have the following problem. I am tried to numerically solve the 1D heat equation for a metal bar of length L. Using the forward time, centered space equation a(t+1) = a(t)+(alpha*deltaA/(deltaX)^2)*(a(x+1,t)-2*a(x,t)+a(x-1,t)) The problem is that I only have ONE heat source at...

du/dt=d2u/dx2 Show that u(x,t)=(t^a) * theta(xi) where xi=x/sqrt(t) and a is a constant, then theta(xi) satisfys the ODE a*theta - 0.5 * xi * dtheta/dxi = d2theta/dxi2 Not sure how to start this. Any help most appreciated (sorry if question isn't easy to ready)

Homework Statement du/dt=d2u/dx2, u(0,t)=0, u(pi,t)=0 u(x,0) = sin^2(x) 0<x<pi Find the solution Also find the solution to the initial condition: du/dt u(x,0) = sin^2(x) 0<x<pi The Attempt at a Solution From separation of variables I obtain u(x,t) = B.e^(-L^2t).sin(Lx)...

I'm trying to solve the homogeneous heat equation of a semi-infinite cylinder in cylindrical coordinates for a semi-infinite cable (no theta dependence): \frac{\partial U}{\partial t}=D\left(\frac{\partial^{2} U}{\partial r^{2}}+\frac{1}{r}\frac{\partial U}{\partial r}+\frac{\partial^{2}...

Homework Statement Find U(x,t) dU/dt = d2U/dx2 + sin x Boundary Conditions: dU/dx (0,t) = 0 and U(1,t) = 0 Initial Condition: U(x,0) = cos 7*π*x 2. The attempt at a solution I start off with: d2(Un)/dx2 = λnUn (as an initial value problem) [d(Un)/dx](0) = 0...

Hi, i'm just going through my lecture notes reading a proof of the maximum principle for the heat equation. It goes roughly like this: Maximum Principle: Heat equation: u_{t}=ku_{xx} u(x,0)=\Psi(x) x\in[0,L], t\in[0,T]. Given a C2 solution of the HE then the maximum u(x,t) is attained at...

I got a solution to the heat equation using Fourier transforms with the special case g(x) = GH(x-a) u(x,t) = G/2[1+erf(x-a/(2\sqrt{t}))]. But I just wanted to know what this special case represents physically. I should probably ask what does any special case to the heat equation represent...

1. I have a rod of length 4,cross section 1 and thermal conductivity 1.Nothing is mentioned about the end at the origin x=0, but at the opposite end x=4, the rod is radiating heat energy at twice the difference between the temperature of that end and the air temperature of 23 celcius. Find the...

The question :Two rods L1 and L2 of different materials( hence different thermal conductivities) and different cross-sectional areas,are joined at x=a. The temperature is continuous, And NO HEAT ENERGY IS LOST AT a, so all heat energy that flows from L1 flows into L2. ? What equation...

Homework Statement I have a rod of density \rho and length l. It's located at 0\leq x\leq l . The density of internal energy per mass is E = c(T-T_0) + E_0 where T is the tempertature in Kelvin, E_0 is a constant and c is the specific heat capacity. We assume that the temperature is not...

I have the following PDE PDE: ut=A2(uxx+uyy)+f(x,y,t) such that A is a constant BC: ux(1,y,t)=0 ux(0,y,t)=0 uy(x,1,t)=0 u(x,0,t)=0 IC: u(x,y,0)=0 I've set up my ODEs using separation of variables to get X''/X=k1 Y''/Y=k2 T'/(A2)T=k1+k2 where k1 and k2 are constants. How do I account for my...

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...

help solving the heat equation! 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...

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...

I got the following PDE. {\frac {\partial }{\partial t}}u \left( y,t \right) = \nu \cdot {\frac {\partial ^{2}}{\partial {y}^{2}}}u \left( y,t \right) With boundary conditions: y=0: u(0,t)=0 y=h: u(h,t)=U_0 \cdot cos(\omega \cdot t) Now I need to show by using substitution that...

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...