Questions tagged [heat-transfer-equation]

Questions on the symbolic (DSolve) and numerical (NDSolve) solutions of the heat transfer equation in Mathematica.

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2answers
109 views

Heat balance in layer coupled via BC

I am trying to solve the heat balance in a 3 layer system in Mathematica 12.0.0. Each layer has different thermal properties. The layers are coupled via the boundary conditions (Neumann type) to ...
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0answers
49 views

2D transient heat equation resolution

I'm writing a code to get the analytical solution mentioned in Sun et al., 2013. I will copy here the passages needed to solve the equation. The 2D transient heat equation is: First of all, I set the ...
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1answer
72 views

Including point heat sources in a 2D transient PDE heat equation

I want to be sure that I included the point heat sources correctly in my PDE. The result should be: I wrote: ...
3
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0answers
85 views

Heat balance multilayer with heat flux continuity

I am solving the heat equation (diffusion only) over a 3-layer system (physical properties varying from layer to layer). ...
1
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1answer
67 views

Graphing Heat Equation

I was able to solve the heat equation, but am unsure of how to graph the partial sums of the solution. The problem: a.) Solve the heat equation subject to ...
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2answers
467 views

Solving a 2D heat equation on a square with Dirichlet boundary conditions

I am trying to solve the following heat equation problem on the square [0,1]x[0,1]. \begin{equation*} \begin{gathered} u_t = u_{xx} + u_{yy} + f(x,y,t), \qquad u(x,y,0) = 0, \qquad u=0 \text{ on ...
1
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1answer
63 views

Solving a Heat Equation

I'm new to Mathematica and am a little lost as to how to code this heat equation: u(0, t) = 0, u(L, t)=0, t>0 u(x, 0)= 1, 0<x<L/2 = 0, L/2<x<L How would I go about solving this?
2
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1answer
189 views

Discontinuity problem with 3D cylindrical heat equation (possibly due to a conversion between Cartesian to Cylindrical coordinates)

I have been working a certain type of 3D cylindrical diffusion equation for a bit now. I am trying to simulate a longitudinal diffusion process in a cylinder with a dislocation defect that will make ...
5
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2answers
187 views

Solving a heat equation on a finite interval with Neuman boundary conditions

I am new to Mathematica and need to verify my numerical result. Can anyone please show me how to solve the following heat equation problem $$ u_t = u_{xx}$$ on the interval $ x \in [0,1]$. The initial ...
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1answer
82 views

Trying to find values of two parameters that match the boundary conditions defined

I have been trying to solve for the values of two parameters that satisfy the boundary conditions set for a set of equations set. The below is the code. ...
0
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1answer
85 views

Solve complex heat transfer equation

I am trying to solve a heat transfer problem related to calculating the cooling rate after quenching a sample. The equations (and explanations) are taken from this paper: https://sci-hub.se/10.1029/...
3
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1answer
56 views

How to convert a Piecewise function in a HeavisideTheta function?

I want to convert a Piecewise function in a HeavisideTheta function in order to correctly evaluate its derivatives at the point of abrupt change and then integrate the result along the axis. How can I ...
3
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2answers
112 views

Super Slow Manipulate with Heat Equation - - Assumption, and Interpolate

Aim: solve coupled heat equation with cubic source, use DensityPlot and observe how system will react to changes in parameter(like kand ...
1
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1answer
97 views

How to solve a 3D permanent conjugate heat transfer problem?

I want to solve this conjugate heat transfer problem on NDSolve but Mathematica always quits the Kernel. Can you help me? I do not know exactly how to use the numerical methods that are NDSolve method ...
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0answers
74 views

NDSOLVE-Error (“For the method \!\(\”IDA\“\), only machine real code is available. ”

I am trying to solve mass transfer equations in an adsorption column to obtain the breakthrough curves. Previously, I asked a question on this and received great help:NDSolve for system of PDEs -Error:...
2
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1answer
221 views

Solving a PDE (2D Laplacian) coupled with an ODE

I have asked this question before, but this is my new attempt and so instead of cluttering the previous one, I am making a new post. I am trying to analytically solve a PDE ($\nabla^2 T(x,y)=0$) ...
3
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1answer
110 views

NDSolve Inconsistent equation dimensions

I am getting an error while trying to solve for a differential equation. It is saying "NDSolveValue::femper: PDE parsing error... Inconsistant equation dimensions." I was wondering if anyone ...
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0answers
98 views

Why is there no heat being transfered in my program?

I have been working on this code and am new to this coding language. I have been trying to model the transfer of heat through a hot plastic being cooled in a water bath. For some reason I have not ...
1
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0answers
153 views

Can this boundary value problem leading to a partio-integral DE be solved using finite Fourier or any integral transform?

I asked the three-dimensional version of this problem here which lead to a trivial solution. I have now tried it in 2-D $$\frac{\partial \theta_h}{\partial x} + b_h (\theta_h - \theta_w) = 0, \tag 1\\\...
5
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2answers
267 views

Mesh Generation help

I am having trouble generating a mesh of a rectangle inside another rectangle. I have been following the steps on this article: https://reference.wolfram.com/language/PDEModels/tutorial/HeatTransfer/...
8
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2answers
1k views

Three dimensional Laplacian insulated on lateral faces and convectively exposed on transverse faces (updated)

I have the three dimensional Laplacian $\nabla^2 T(x,y,z)=0$ representing temperature distribution in a cuboid shaped wall which is exposed to two fluids flowing perpendicular to each other on either ...
8
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2answers
285 views

Two-dimensional Laplacian coupled with another equation leading to a BVP with integral bc(s)

I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation. The Laplacian is defined over $x\in[0,L], y\in[0,l]$. On manipulating the second equation (which I have ...
3
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1answer
232 views

Problem with NDSolveValue : “The function value {$Failed} is not a list of numbers with \ dimensions…”

I was having fun modifying a code given to me as an answer to a previous problem here, courtesy of user Alex Trounev (Thank you again), when I encountered a certain error which I had never seen before....
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0answers
82 views

Need help with IVP for Heat Equation [closed]

I'm trying to solve the heat equation on a torus and am running into a problem... Mathematica keeps running, but never finishes/ends. Even after leaving it running overnight, I still got nada. ...
1
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1answer
166 views

Plot time dependent 3D heat equation solution with functions like Plot3D +Manipulate (or Graph3D) but using cylindrical coordinates

I have a time dependant heat diffusion equation here and I would like to plot the result of NDSolveValue. Here is the code I am using : ...
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0answers
49 views
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0answers
100 views

Heat transfer FEM with nonlinear Cp

I would like to solve a 3D heat transfer problem with nonlinear heat capacity Cp (Cp=1*T[t,x,y,z]). My problem is very simple: a slab with the bottom surface insulated and the other five surfaces ...
0
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1answer
59 views

Blank plot is returned when trying to use Manipulate[Plot3D…] on the output function of NDSolve

I am currently trying to solve a transient cylindrical diffusion problem with space variable (r,z). Using NDSolve, I get a certain Interpolation function but I can't plot it no matter what I try. ...
6
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3answers
230 views

Incorrect Result from DSolve for Diffusion IBVP?

The result of DSolve for the following IBVP doesn't seem to satisfy the IBVP: \begin{cases} u_{t} - ku_{xx} = 0 &\mbox{} k>0,x>0,t>0 \\ u(0,t)=p(t) &...
3
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3answers
175 views

Inhomogeneous Heat Equation Formula Not Satisfying IVP

From Wikipedia, However, when I plug the formula of $u(x,t)$ into Mathematica 12.1, it doesn't seem to satisfy the PDE (does not give $0$): Here is the code: ...
4
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1answer
153 views

Solve free boundary problem for heat equation

How can I use Mathematica to compute/approximate and plot the solution of the following problem? $\min\{u_t - u_{xx} -1, u \} = 0 \text{ in } (0,T)\times (-1,1)$ $u(0,\cdot) = 0 \text{ in } (-1,1)$...
3
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0answers
225 views

Problem computing a cylindrical Heat equation with a parameter alpha

i have been struggling to compute a particular instance of cylindrical 3D heat equation. Here is my code : ...
7
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3answers
383 views

Wrong solution from multi materials FEM NDSolve

I'm quite new in Mathematica. I have a problem with obtaining the right solution for multi-layer 1D heat transfer problem. It seems that boundary condition not working. Could you advise something? I ...
3
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1answer
128 views

NDSolve 1D Heat equation with NeumannValue poorly satisfies boundary condition

Context I am trying to measure the slope of the stellar cuspide around our MilkyWay's central black hole using the observed distribution of S stars orbiting around it. Attempt Close to the ...
4
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1answer
284 views

A complicated boundary value problem leading to third order Eigen sytem [Help in continuing forward] [EDITED]

I have the following elliptic PDE (describing temperature in a plate, w in thermal contact with two fluids h and c): $$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 ...
4
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1answer
331 views

Approach to analytically solve a Coupled system of PDE in Mathematica [Heat transfer in cylindrical coordinates]

I have the following two PDEs, which describe steady-state coupled heat transport between a externally heated axi-symmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it $...
16
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4answers
736 views

How to model diffusion through a membrane?

This is a follow-up on How to handle discontinuity in diffusion coefficient? Consider diffusion of $u(t,x)$ on the domain $x \in [0,2]$ with some simple boundary conditions such as $u(0) = 2, u(2) = ...
9
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1answer
289 views

How to handle discontinuity in diffusion coefficient?

I am looking to solve the diffusion equation with a discontinuous jump in the diffusion coefficient. In 1D, the diffusion equation for $u(t,x)$ is: $$ \partial_t u = \partial_x (D \partial_x u), $$ ...
6
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3answers
286 views

How to define the boundary condition in 1D Heat transfer

I am trying to calculate the heat transfer among a 1-D rod, with one end insulated while the right end is immersed in constant temperature surface T=0. Assume that the initial temperature of the rod ...
0
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0answers
75 views

Heat Equation with insulated ends

I have a problem with Heat Equation.Imagine this.We have this equation ----> ut=uxx with boundary conditions u(0,t)=0 u(25,t)=0 and initial condition u(x,0)=x. Find u(x,t) and make the Plot ( doesnt ...
6
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1answer
370 views

Laplace equation with robin boundary conditions

I want to solve the following steady state heat transfer problem with robin boundary condition at the bottom: The following is the code for the transient solution, but how should I change the code ...
2
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3answers
168 views

Inconsistent Boundary Conditions on Transient Heat Equation Partial Differential Equation

I am attempting to solve the equation with the Wolfram one-liner ...
9
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2answers
248 views

FEM Mesh related errors

Bug introduced in 12.0. Fixed in 12.1.0. I try to solve the simple heat transfer equation but there are some errors at the stage of the pre-processing. ...
4
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2answers
332 views

Solve a one dimensional heat transfer problem with NDSolve

The following problem is a 1-D heat transfer conduction problem: where, I am trying to solve with NDSolve like this way: ...
2
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3answers
363 views

Problem with boundary condition 2D heat transfer

I want solve this 2D Heat transfer: So at Mathematica: ...
7
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4answers
510 views

What is wrong with my approach to solving a heat transfer PDE?

I wanna solve the following heat transfer PDE using Mathematica. $\qquad u_{xx}=u_{t}$ with following conditions: $\qquad \begin{cases}u(x,0)=sin(x) &0<x<\pi &,t>0\\u_{x}(0,t)=1\\...
8
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1answer
396 views

NDSolveValue -difference between Version 11.2 and Version 11.3

I am trying to learn something about simulation of heat transfer by looking at this problem which is an answer by user21. If I try this in Version 11.3 the following happens: ...
2
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1answer
1k views

Solving the 2D heat equation

I am trying to solve the 2D heat equation (or diffusion equation) in a disk: ...
1
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1answer
267 views

Errors in Solving Heat Equation

I am have trouble with what should be a fairly simple NDSolve operation for the unsteady state heat transfer in a rod. The rod starts off at 20oC, and its surface temperature is fixed at 37oC. I am ...
7
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1answer
782 views

Free Convective Heat Transfer of Non-Newtonian Power Law Fluids from a Vertical Plate

I am trying to solve a set of PDEs mentioned in this paper with NDSolve but facing quite a few issues. The PDE system is: ...