Heat equation Definition and 257 Threads

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

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...

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...

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...

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...

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...

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...

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...

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...

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} +...

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...

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)...

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...

[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)}...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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)...

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...

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...

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

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...

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 , $f(0)=0$, $f(1)=0$...

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

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...

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...

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...

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...