Heat equation Definition and 282 Threads

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...

Sorry, it does not seem that Latex is not compiling my code right so I will try my best to be clear. Homework Statement The curved surface of a cylinder of radius a is grounded while the end caps at z = ± L/2 are maintained at opposite potentials ψ (r,θ, ± L/2)= ± V(r,θ). Suppose that...

Hi all, I'm trying to analytically solve the heat equation with a heat source and Neumann B.C. The source term is creating some problems for me as I cannot determine the coefficients in the series that builds up the solution. If someone could could help me or at...

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...

Heat equation! Urgent Is anyone who are able to solve heat equation: {\frac{{\delta u}}{\delta t} = c^2\frac{{\delta^2 u}}{{\delta x^2}} where,u(x,t) I have looked for so many textbooks but most of them using the same method to solve which is "Assume u(x,t) = F(x)G(t) and sub...

The equation given is \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( K(x) \frac{\partial T}{\partial x} \right), 0<x<L where T is temperature, t is time, and x is the 1-d spatial coordinate, also K(x) = \kappa e^{-x}, \kappa > 0 and the boundary conditions are...

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...

Vectorize - C-->Matlab / Heat equation I want to 'translate' some programs I had in C for MATLAB but by sigtly optimizing the code for MATLAB use. I am new in matlab. So, I had the following code: (1) for K = 1:dT:M (2) for I = 1:N+1 dU(I) = (U(I+1)-2*U(I)+U(I-1))/dX^2...

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...

\frac{\partial u}{\partial t} - k \frac{\partial^2 u}{\partial x^2} = 0 for 0 <x < pi, t> 0 u(0,t) = u(\pi,t) = 0 u(x,0) = x (\pi - x) OK i know the boring part of getting u(x,t) = X(x) T(t) the infinite series part is hard part the coefficient c_{n} = \frac{2}{\pi}...

Is there a straightforward proof for the existence of the one-dimensional linear heat equation f=u_t_-a^2*u_xx_=0. Is so, how? Note: _t_ represents the subscript, i.e., the derivative t, and _xx_ represents the subscript xx. Is the heat equation well posed? Can this proven? How?

Hi! Can someone please help? I'm trying to solve the heat equation in polar coordinates. Forgive my way of typing it in, I'm battling to make it look right. The d for derivative should be partial, alpha is the Greek alpha symbol and theta is the Greek theta symbol. du/dt =...

Hi, I'm not sure how to solve problems of this form: Uxx - Ut = h(x,t) where Uxx is second derivative of U(x,t) wrt x and Ut is first derivative of U(x,t) wrt t. Boundary conditions are as follows: U(0,t)=U(a,t)=U(x,0)=0 and h(x,t) is a fairly simple function, or even constant, say h=1...

What is the dimension of soln space of the heat equation: \frac{\partial U }{\partial t}=a^2\frac{\partial^2 U}{\partial x^2} U(0,t) = U(L,t) = 0 U(x,0)= f(x) Is it infinite , if so why?

Hello friends, does anybody have a soft copy of the following paper. if yes, then please mail it to my email address: aditya_tatu@yahoo.com aditya_tatu@da-iict.org I am not sure whether it is freely available online or not? the details of the paper are: Title : The heat equation...

I was wondering what happens if I want to solve the heat equation in a sphere surface, neglecting its thickness. I have one initial condition for T(t=0), in particular this initial profile can depend on azimuth and zenith angles, it is not uniform. Perhaps I have saying something stupid but I...