Heat equation Definition and 282 Threads

Homework Statement A player bounces a basketball on the floor, compressing it to 80.5% of its original volume. The air (assume it is essentially N2 gas) inside the ball is originally at a temperature of 20.5°C and a pressure of 1.80 atm. The ball's diameter is 23.9 cm. By how much does the...

Homework Statement Solve the time dependent 1D heat equation using the Crank-Nicolson method. The conditions are a interval of length L=1, initial distribution of temperature is u(x,0) = 2-1.5x+sin(pi*x) and the temperature in the ends of the interval are u(0,t) = 2; u(1,t) = 0.5. Homework...

ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo i have this equation... i was thinking of taylor series expansion to solve it and make it easier... any ideas on how to solve?

Homework Statement Homework Equations Heat equation The Attempt at a Solution I can derive E(t) to get integral of du/dt over 0 to L, which is the same as integrating the right hand side of the original equation (d2u/dx2+sin(5t); while this allows me to take care of the d2u/dx2, I don't know...

I have not much experience in solving pde before except using the separation of variables. I am trying to solve the following equation where omega is a box. Is there a close form of the solution? How should I approach the problem? Much thanks!

Homework Statement We previously solved the heat conduction problem in a ring of radius a, and the solution is c into the sum, perform the sum first (which is just a geometric series), and obtain the general solution, which should only involve one integral in ϑHomework Equations...

Can you derive the heat conduction equation from the navier stokes equations (particularly the energy eqn)?

Hi PF! For the longest time I thought an energy balance and the heat equation were identical procedures. However, recently I saw an example of a steady state, constant property, laminar flow of fluid between two flat surfaces where the top surface moves in the ##x## direction at ##V_1## and we...

Homework Statement A ball of radius a, originally at T0, is immersed to boiling water at T1 at t=0. From t≥0, the surface (of the ball) is kept at T1 Define u(r,t)=R(r)Q(t)=T(r,t)-T1 ΔT=T0-T1<0 r,t≥0 Homework Equations ∇2u=r-2 ∂/∂r ( r2 ∂u/∂r ) =D-1∂u/∂t D>0 The Attempt at a Solution...

Homework Statement Show that if v(x,t) and w(y,t) are solutions of the 1-dimensional heat equation (v_t = k*v_xx and w_t = k*w_yy), then u(x,y,t) = v(x,t)w(y,t) satisfies the 2-dimensional heat equation. Can you generalize to 3 dimensions? Is the same result true for solutions of the wave...

Homework Statement $$u_{t}=u_{xx}+f(x) \\ u(0,t)=50 \\ u(\pi , t)=0 \\ u(x,0)=g(x)$$ $$0<x<\pi \\ t>0$$ $$f(x)=\begin{cases} 50 & 0<x<\frac{\pi}{2} \\ 0 & \frac{\pi}{2}\leq x< \pi \end{cases}$$ $$g(x)=\begin{cases} 0 & 0<x<\frac{\pi}{2} \\ 50 & \frac{\pi}{2}\leq...

Homework Statement I have the solution to the heat equation, with the BC's and everything but the IC applied. So I am just trying to solve for the coefficients, the solution without the coefficients is $$u(x,t) = \sum_{n=1}^{\infty} A_n\sin(nx)e^{-n^2t}$$ If the initial condition is ##u(x,0) =...

Hi all Do you know how to write code Alternating Direct Implicit(ADI) method in Matlab? I have given 2d heat equation for this. Thank you

I am trying to solve the steady state heat equation with a heat source. I am starting out in 1 dimension (my book gives the solution in 2, but I'm just trying to get a feel right now) and I have a heat source Q, located at 0. It radiates heat through an infinite medium. So what would the steady...

So one of my least favorite things that textbooks do is using the words "clearly", "it should be obvious", etc. In my PDEs class, we've started the Fourier Transform, and I missed the first day of it so I am trying to read through my book. Regarding the heat equation on an infinite domain, it...

Hey! I'm currently solving the heat equation using finite differences. I have a conductivity k(u) that varies greatly with temperature. It even drops to zero at u=0. I have discretized the equations the following way: \frac{\partial}{\partial x}\left( k(u) \frac{\partial u}{\partial x}\right) =...

Dear all, I am investigating a Transient Optimal Heating Problem with distributed control and Dirichlet condition. The following are the mathematical expression of the problem: Where Ω is the domain, Γ is the boundary, y is the temperature distribution, u...

Homework Statement 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}+2U_r/r)...

How would I go about finding temperature distribution in a thin square plate during the the first few milliseconds (or actually a fraction of a millisecond) after t=0s. Initial temperature distribution throughout the plate is known, there is heat flux to one side = Qinj, while heat flux from all...

Hi, Let's consider the heat equation as \frac{\partial T}{\partial t}=\alpha \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}} In order to have a second accuracy system, one can use the Crank-Nicolson method as \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}\approx \frac{1}{2}\left(...

I had these code in this forum but comes out error as below, any suggestion? Error 1 error C4430: missing type specifier - int assumed. Note: C++ does not support default-int c:\users\username\documents\visual studio 2010\projects\fdm 001\fdm 001\explicit 001.cpp 27 Error 2...

Ok, I've built a numerical model to show the cooling of hot magma sills entered into the crust over time. The results show that the volume of the "hot" zone when the emplacement of a constant volume of hot sills is all done will vary as a matter of two things: the overall rate at which the magma...

Homework Statement Find the solution ##u(x, t)## to the semi-infinite interval problem $$ u_t = u_{xx} - 4u, \hspace{2 mm} 0 < x < \infty, \hspace{2 mm} t>0\\ u_x(0,t) = -1, \hspace{2 mm} t>0\\ \lim_{x \to \infty}u(x,t) = 0, \hspace{2 mm} t>0\\ u(x,0) = e^{-x}...

Homework Statement The temperature variation at the surface is described by a Fourier series \theta(t)=\sum^\infty_{n=-\infty}\theta_n e^{2\pi i n t /T} find an expression for the complex Fourier series of the temperature at depth d below the surface Homework Equations Solution of the...

What is the best solution of the heat equation that described a transmission of heat from a source kept at certain temperature to a reservoir with an initial constant temperature (lower than the source) where its ends are not insulated from the surroundings and the surrounding is kept at a fixed...

hi pf! i'm wondering if you can help me with the heat eq for a basic cylinder wire problem. namely, we have a wire with radius ##r_i## and length ##L##and resistance is ##R## and current is ##I##. Thus heat produced $$Q = R I^2 \pi r_i^2 L$$. When using the heat eq, we assume time rate of...

Homework Statement Consider \frac{\partial u}{\partial t} = k\left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \\ 0<x<L\\ y>0 subject to the initial condition IC: u(x,y,0) = f(x,y) And solve with the following boundary conditions: BC1: \quad u(0,y,t) = 0...

Homework Statement The problem is f(x) = sin2πx - (1/πsquare)*sinπx and its given Bn sin (nπx) = f(x) Question is find Bn. Homework Equations Bn = 2/L ∫ (sin2πx - (1/πsquare)*sinπx) * sin(nπx/L) where L is 1 The Attempt at a Solution I did [/B] ∫ sin2πx * sin (nπx) - (1/πsquare)*sin...

Homework Statement Let a slab 0 \le x \le c be subject to surface heat transfer, according to Newtons's law of cooling, at its faces x = 0 and x = c , the furface conductance H being the same on each face. Show that if the medium x\le0 has temperature zero and medium x=c has the...

Homework Statement A 6.00-kg piece of solid copper metal at an initial temperature T is placed with 2.00 kg of ice that is initially at -20.0C. The ice is in an insulated container of negligible mass and no heat is exchanged with the surroundings. After thermal equilibrium is reached, there is...

Hello friends, I am new for numerical methods and programming. i have been trying to devolop a program in 2D poisson heat equation in cylinder (r,angle) by finite difference method ∂2u/∂r2 + 1/r * ∂u/∂r + 1/r2 * ∂2u/∂θ2 = Q(u,θ)discritized equation :- ui+1,j − 2uij + ui−1,j/(∆r)2 + 1/ri *...

Homework Statement I'm currently trying to follow a derivation done by Shankar in his "Basic Training in Mathematics" textbook. The derivation is on pages 343-344 and it is based on the solution to the two dimensional heat equation in polar coordinates, and I'm not sure how he gets from one...

Hi PF! Given: ##u_t = u_{xx} +1## (heat equation) with the following B.C.: ##u_x(0,t)=1, u_x(L,t)= B, u(x,0)=f(x)##. My professor then continued by stating that in equilibrium, we have ##0 = u_{xx} +1 \implies u = -x^2/2 + C_1 x + C_2##. So far I'm on board, although by "equilibrium" does he...

Homework Statement A bar of length ##L## has an initial temperature of ##0^{\circ}C## and while one end (##x=0##) is kept at ##0^{\circ}C## the other end (##x=L##) is heated with a constant rate per unit area ##H##. Find the distribution of temperature on the bar after a time ##t##. Homework...

Hi!, I'm working on a personal project: Solve the heat equation with the semi discretization method, using my own Mathematica's code, (W. Mathematica 9). The code: I'm having problems with the variable M (the number of steps). It works with M=1-5, but no further, I do not know what's going...

Hey! :o I have to solve the following problem: $$u_t=u_{xx}, x \in \mathbb{R}, t>0$$ $$u(x,0)=f(x)=H(x)=\left\{\begin{matrix} 1, x>0\\ 0, x<0 \end{matrix}\right.$$ I have done the following: We use the method separation of variables, $u(x,t)=X(x)T(t)$. I have found that the eigenfunctions...

Hello! I have written the code in Maple for Heat equation with Neumann B.C. Could anyone check it? I will be very grateful! Heat equation: diff(u(x,t),t)=diff(u(x,t),x,x); Initial condition: U(x,0)=2*x; Boundary conditions: Ux(0,t)=0; Ux(L,t)=0; I use centered difference approximation for...

Hey! :o I haven't really understood the following proof that the solution of the heat equation is unique. Could you explain it to me? Heat equation with Dirichlet boundary conditions: $$\left.\begin{matrix}u_t=u_{xx}, 0<x<L, t>0\\ u(0,t)=u(L,t)=0, t>0\\ u(x,0)=f(x)=0...

Homework Statement A nuclear fuel of thickness ##2L## has a steel slab to the left and right, each slab of thickness ##b##. Heat generates within the rod at a rate ##\dot{q}## and is removed by a fluid at ##T_{\infty}## (the question doesn't say, but I believe ##T_{\infty}## is temperature of...

Homework Statement I have been given a complex function I have been given a complex function \widetilde{U}(x,t)=X(x)e(i\omega t) Where X(x) may be complex I have also been told that it obeys the heat equation...

Homework Statement Use the Fourier transform directly to solve the heat equation with a convection term u_t =ku_{xx} +\mu u_x,\quad −infty<x<\infty,\: u(x,0)=\phi(x), assuming that u is bounded and k > 0. Homework Equations fourier transform inverse Fourier transform convolution thm The...

Homework Statement Solve the Dirichlet problem for the heat equation u_y=u_{xx}\quad 0<x<2\pi, \: t>0u(x,0)=\cos xu(0,t)=u(2\pi,t)=e^{-t} Homework Equations The Attempt at a Solution I have no idea what to do here. It seems to me like it's a mix of the solutions we learned. I...

Hi, it is easy solving these PDEs with the idealized homogeneous BCs they throw out in class, but I am having some difficulty solving the transient problem posed in the images below. I have tried working through it, but I don't have confidence in the result. I overlook the solution when the...

If I have a hot wire, the distribution of its temperature with respect to radius (from the center of the wire) and time follows the heat/diffusion equation. However, now consider two wires, or even an array of many such wires. Say we can ignore the z coordinate and treat them as a point...

Hello to everyone, I urgently need to solve the following pde: ∂u/∂t +∂²u/∂x² = So*δ(x-xo)*sin(wo*t) It's the heat equation with a cyclic source. The lentgh of the cable is L. I have no clue how to do this with such a source, all i have learned was to do a separation of variables, but it...

\frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{r}^2}+ \frac {2}{r} \frac {\partial{u}}{\partial{r}}+\frac{1}{r^2}\left[\frac{\partial^2{u}}{\partial{\theta^2}}+\cot\theta \frac{\partial{u}}{\partial {\theta}} +\csc\theta\frac{\partial^2{u}}{\partial{\phi}^2}\right]+q(r,\theta,t)...

I have two questions: (1)As the tittle, if u(a,\theta,t)=0, is \frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2} and \frac{\partial^2{u}}{\partial...

I want to solve the one-dimensional heat PDE backward in time ∂u/∂t = -∇2u = -∂2u/∂x2 , x element of [0,L] Basically, I want to find what the initial temperature profile u(x,t=0) should be such that after some time t1 of diffusion, I am left with the bar at a uniform temperature u(x,t1)=c...

Homework Statement solve the heat equation over the interval [0,1] with the following initial data and mixed boundary conditions.Homework Equations \partial _{t}u=2\partial _{x}^{2}u u(0,t)=0, \frac{\partial u}{\partial x}(1,t)=0 with B.C u(x,0)=f(x) where f is piecewise with values: 0...

Heat equation on a half line! Hi, I am now dealing with the heat equation on a half line, i.e., the heat equation is subject to one time-dependent boundary condition only at x=0 (the other boundary condition is zero at the infinity) and an initial condition. I searched online, it seems...