Questions tagged [heat-equation]
The heat-equation tag has no usage guidance.
142
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Problem with pdetoode for two coupled PDEs
I tried to adapt a code for a single equation to solve the following system using 'pdetoode'
Updated answer
...
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2
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1
answer
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Need help solving cylindrical Laplacian
I'm trying to solve the cylindrical Laplacian for a heated disk in a large cylinder. The cylinder and disk have constant temperatures and I only care about the temperature field between the disk and ...
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2
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Spherical Heat Equation and Convection Boundary Conditions
I'm trying to solve transient heat conduction equation in spherical domain in Mathematica. I made the simplifying hypothesis that temperature varies only with time and radial coordinate.
The code is:
<...
3
votes
1
answer
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Simulating the Conduction of Heat on a Metal Rod
Consider the one-dimensional heat equation
$$
\frac{\partial u(x,t)}{\partial t}=\alpha\frac{\partial^2 u(x,t)}{\partial x^2}, \quad
(x,t)\in (0,L)\times (0,\infty),
$$
subject to the ...
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1
vote
1
answer
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Fitting Parameters to the Heat Equation [closed]
As part of my research, I have been trying to use a model of heat conduction through a 2D layer given an input steady-state Gaussian power profile and a heat loss term to the environment. All but one ...
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1
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1
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Plot of non-homogeneous diffusion equation
Let the initial and boundary value problem for the diffusion heat equation
\begin{align*}
u_t(x,t)&=u_{xx}(x,t)-\alpha u_x(x,t), \quad 0<x<+\infty,t>0\\
u(x,0)&=f(x), \quad x\...
2
votes
2
answers
113
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Solving 3D heat equation with an off-center boundary condition
So i have this code (albeit a simplified version but it'll do for this question)which solves a time dependant 3D heat equation on a cylinder.
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1
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1
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262
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2D transient heat equation solution
I want to calculate the time- and space- dependent temperature of a 2D system where there are 3 materials, with different thermal properties.
The system can be described by the schematics:
...
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2
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1
answer
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Setting up a strategy delivering a smooth heat transfer solution
I have a polymeric film that undergoes a two-step cooling process. First, it spends time1 in air. Then it comes in contact with a piece of steel at a temperature <...
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3
votes
2
answers
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Stefan problem with mixed bc
I am trying to solve through Mathematica the classical Stefan problem
$$
\left\{
\begin{array}{lll}
\dot{v}(x,t)=v_{xx}(x,t) & x\in(0,s(t))\\
\dot{s}(t)=-v_x(s(t),t) & x = s(t)\\
v(0,t) = 0 &...
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2
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Analytical Solution in Generalized Heat Equation
Given the following problem
$$u_{t}-\frac{v}{2}\cdot u_{xx}+\frac{v}{2}\cdot x^2 \cdot u(x)=0 $$
$$u(x,0)=f(x)$$
Where $ f(x)=y_1(x)+0.2y_4(x)+0.01y_6 (x) $
$y_n$ are the Eigenfunctions that defined ...
5
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2
answers
272
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Inaccuracy for FEM for 3D Heat Equation
I'm simulating the heat transfer within a cylindrical rod, with external heating from its sides.
...
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0
answers
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NDSolve stops solving past a certain time
This is a continuation of the post I've made Unable to solve Delay PDEs Error in Boussinesq Approximation. I apologise if I shouldn't have posted a seperate question for this but I think that the ...
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Unable to solve Delay PDEs Error in Boussinesq Approximation
I'm trying to solve the set of equations below describing the flow of a pot of water being heated slightly. The equations are 2D axisymmetric in nature.
...
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2
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279
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Meshing an irregular domain using quads to solve conjugate heat transfer problem
I am trying to mesh the following domain to solve a heat transfer + fluid flow problem:
The continuity+momentum equations are to be solved in $ABGH$, while the energy equation is to be solved across ...
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1
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Implementation of FEM to steady-state coupled fluid flow and heat transfer
This is a steady-state conjugate heat transfer problem (the time-independent version of this problem). The problem is conjugate as the energy equation is being solved in thermally connected solid and ...
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1
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Modelling heat transfer in periodically reversing flow
This is a heat transfer problem, which involves reciprocating (fully-reversing) fluid flow over a heated block of solid. The objective is to determine the temperature field in the solid and the fluid ...
- 297
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1
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142
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Directly calculating the cyclic steady state of a time-periodic conjugate heat transfer problem
Context
The following transient problem is the reciprocating (i.e., fully reversing) flow of a fluid $0<x<L, 0<y<d$ over a thick heated block $0<x<L, -e<y<0$ until the system ...
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3
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1
answer
137
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Conjugate heat transfer modelling of reciprocating flow crashes for long flow times
The following transient problem is essentially the reciprocating (i.e., fully reversing) flow of a fluid over a thick heated block until the system reaches a cyclic steady-state (i.e., the system ...
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1
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Solving a Caputo fractional diffusion equation in cylindrical coordinates
I would like to solve this following equation with d, the order of the factional derivative (in the sense of Caputo) :
So I tried the following code :
...
2
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0
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Coupled PDEs with different dimensions and boundary conditions
The geometry of the problem is described by a cylinder inside a rectangular domain. The equations describes the region in between them.
I have 2 PDEs for U(x,y,z) in the bulk region and V(z) for -0.5&...
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1
answer
162
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Steady state heat equation/Laplace's equation special geometry
I would like to solve the Laplace's equation in between the square domain and a disk. However, using the code below, I was able to generate mesh but not able to obtain results with correct boundary ...
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1
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Plotting the solution of the following equation with integral boundary conditions
This is my equation:
E1 = D[u[x, t], t] + D[u[x, t], x] + u[x, t] == 0;
ic = {u[x, 0] == Sin[\[Pi] x]};
bcc = {u[0, t] == Integrate[u[x, t], {x, 0, Infinity}]};
...
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1
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Power Series Method on Heat Equation
I’m interested in solving the heat equation: d[K[z] d[T[z],z],z]=0, equipped with T(-h/2)=Tm and T(h/2)=Tc.
Is it possible to get a symbolic solution in Mathematica using the power series method?
The ...
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votes
1
answer
116
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How to solve PDEs for function $\Psi(z,\bar{z})$ dependent on independent variables $z,\bar{z}$? (Wirtinger derivatives)
For the past few days, I have been struggling to convey to mathematica to solve a PDE that is in terms of the independent variables $(z,\bar{z})$. I know mathematica supports solving PDEs with respect ...
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1
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How to solve this initial boundary value problem for heat conduction equation?
I want to solve the following PDE:
heqn=D[u[x,t],t]==D[u[x,t],{x,2}]-D[u[x,t],{x,1}];
with the initial conditions:
...
- 1
3
votes
1
answer
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Symbolic solution for steady-state heat equation i.e. Laplace equation inside cylinder
I want to symbolically solve the following boundary value problem in my textbook. It's steady-state heat conduction equation i.e. Laplace equation inside a cylinder
$$
\left\{\begin{array}{l}
\Delta u=...
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1
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Heat Equation with Mobile Boundary
I am trying to use NDSolve to find the solution to a set of coupled diffEQs.
They represent 1d (radial) heat balance in a spherical shell.
The goal is to specify a heat flux into the base of the shell,...
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1
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Is there any alternative for Manipulate or Animate?
I have a project to animate heat equation in 3D. I have the following problem
$$u_t=u_{xx},\qquad 0\leq x\leq 1 \land t>0$$
$$u(0,t)=u(1,t)=0$$
$$u(x,0)=\begin{cases}
2x, & 0\leq x<\frac{1}{...
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0
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Neumann Boundary Condition for thermal radiation between two bodies
I'm trying to formulate and solve a pde which models thermal radiation between two bodies using a Neumann boundary condition.
I've created a mesh and want to capture the thermal radiation heat ...
4
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1
answer
170
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Speeding up Boussinesq equations solving?
I am working on Boussinesq equation. The notebook can run perfectly for only 0.6 steps and then the calculation starts running slowly after 0.7. All boundary conditions seemed fine. I am unsure if I ...
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1
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Why do I see Part::partw error when using NDSolveValue?
Bug introduced in version 12.0 (or earlier) and fixed in version 13.0 (or earlier)
I am using Mathematica 12.0. The present calculation is a simple "test case" to prove efficacy, but I am ...
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votes
1
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Coupled heat transfer equations using collocation method
Using the collocation method proposed here, recently this problem has been solved. I am trying to solve a similar problem described by the equations given below. My attempt in Mathematica is ...
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2
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Reciprocating flow in a channel over a heated surface
The following is a coupled heat transfer and fluid flow problem.
A thick plane channel is being heated with a constant flux from the bottom (at $y=-e$) with a constant heat flux $q$ as shown in the ...
- 297
5
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2
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question on documentation convention for heat PDE used by Finite Elements methods in Mathematica
Why FEM documentation says heat PDE is second order in time? This makes it looking same as the wave PDE. Is this meant to be that $m=0$ for the heat pde? But this looks confusing. Could this be just ...
- 137k
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1
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FEM nonlinear anisotropic heat transfer with element markers
I'm trying to model a 2D nonlinear anisotropic heat transfer using FEM with element markers. I got stuck with errors and have no idea what to do to make it work...
I have simple toy-model: two ...
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0
answers
139
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Boundary Value of Function Disagrees with Neumann Boundary Heat Equation
I'm trying to solve a time-dependent 2D heat equation with a source, initial specified heat distribution and heat flux loss at the boundary which goes with the temperature difference. Thus far, I've ...
1
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1
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how we can extract the value of the solution of a PDE in a point x? (NDSolve) [closed]
Please I need your help, I calculate the solution of heat equation using methode of line
This is my code:
...
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votes
2
answers
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Modified Heat Transfer in Fluid Flow
I am trying to simulate Modified Heat Transfer in Fluid Flow (based on Buoyancy-Driven Flow in a Square Cavity ).
The modified heat transfer takes the form:
with the solid volume fraction:
The ...
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1
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Different answer between HeatTransferPDEComponent and Matlab
I've been working on a 2-layer HeatTransferPDE Model.
The diagram of the model is
The two materials have an initial temperature of 37℃, and the outside is 75℃. The model is for heat insulation so it ...
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2
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NDSolve coupled PDE Grad function in source term error
I am working on a multiphysics problem involving heat transfer and electrostatics. I have been messing around with the Joule Heating Tutorial Case and got stuck with the heat source term not working. ...
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1
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Adding maximum value for function within diffusion equation
I have a function that I want to max out at a certain value, say 1 for simplicity. There is a pump that will heat in a certain area, but once its reaches the cap, it no longer heats past this (as if ...
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1
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Mathematica heat transfer tutorial problem
I want to solve some heat transfer problems with Mathematica. I am trying to run the heat transfer tutorial cases but I am always getting the error "NDSolveValue: Equation or list of equations ...
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1
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264
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1D Nonlinear Diffusion Equation with NDSolve Graphs
I am trying to model/solve a specific instance of a 1D diffusion equation in which I have a nonlinear Neumann boundary condition at x=1 (length of unit 1). My equations that I have are:
D[u[x,t],t] == ...
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0
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72
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Setting Up to Solve the Diffusion Equation with Nonlinear Neumann [duplicate]
I am really new to Mathematica, so please bear with me if I ask any relatively easy questions. I am trying to model/solve a specific instance of a 1D diffusion equation in which I have a nonlinear ...
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1
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How we can solve heat equation with this particular boundary conditions (error NDsolve)?
Please I need help to solve this heat equation with this particular boundary conditions:
This the Code I tried in Mathematica:
...
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1
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How can I solve this system please?
I need to solve the following impulsive heat equation:
$$
\left\{\begin{array}{ll}
\partial_{t} \psi(x,t)-\partial_{xx} \psi(x,t)=0, & (x,t)\in (0,1) \times((0, 2) \backslash\{1\}) \\
\...
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votes
1
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Please I need help, how can I solve this heat impulsive system? [closed]
Please I need your help!!
I need to solve the following system:
$$
\left\{\begin{array}{ll}
\partial_{t} \psi(x,t)-\Delta \psi(x,t)=0, & (x,t)\in (0,1) \times((0, 2) \backslash\{1\}) \\
...
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2
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Total flux of the gradient of the numerical solution of a PDE through a surface
Trying to solve the following PDE with BC T==1 on a spherical cap of a unit sphere and T==0 at infinity (approximated as ...
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votes
1
answer
437
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Heat transfer with functions defined on different domains
I'm attempting to model a situation in which a polymer initially at a higher temperature is sandwiched between two cooler metallic mold pieces with conductive boundary conditions in between the ...
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