Skip to main content

Questions tagged [heat-equation]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
91 views

Solving the steady-state problem of a multilayer hollow sphere

I would like to reproduce the analytical solution for the steady state problem as described in the paper Singh2016(https://doi.org/10.1115/1.4033536). The additional conditions can be found in the ...
FMW's user avatar
  • 1
2 votes
0 answers
164 views

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 ...
S. Maths's user avatar
  • 203
2 votes
1 answer
96 views

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 ...
icebox207's user avatar
  • 131
2 votes
2 answers
336 views

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: <...
Julio Araujo Dos Santos's user avatar
3 votes
1 answer
354 views

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 ...
Bell's user avatar
  • 373
1 vote
1 answer
90 views

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 ...
Jack LeGrow's user avatar
1 vote
1 answer
107 views

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\...
Athanasios Paraskevopoulos's user avatar
2 votes
2 answers
141 views

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. ...
ConfuzzledStudent's user avatar
1 vote
1 answer
330 views

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: ...
Luigi's user avatar
  • 1,303
2 votes
1 answer
56 views

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 <...
Luigi's user avatar
  • 1,303
3 votes
2 answers
232 views

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 &...
Josè's user avatar
  • 31
2 votes
2 answers
298 views

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 ...
Athanasios Paraskevopoulos's user avatar
5 votes
2 answers
278 views

Inaccuracy for FEM for 3D Heat Equation

I'm simulating the heat transfer within a cylindrical rod, with external heating from its sides. ...
Lucas's user avatar
  • 103
1 vote
0 answers
85 views

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 ...
Lucas's user avatar
  • 103
1 vote
0 answers
83 views

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. ...
Lucas's user avatar
  • 103
5 votes
2 answers
306 views

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 ...
Avrana's user avatar
  • 297
4 votes
1 answer
452 views

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 ...
Avrana's user avatar
  • 297
7 votes
1 answer
322 views

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 ...
Avrana's user avatar
  • 297
2 votes
1 answer
163 views

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 ...
Avrana's user avatar
  • 297
3 votes
1 answer
153 views

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 ...
Avrana's user avatar
  • 297
8 votes
1 answer
391 views

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 : ...
ConfuzzledStudent's user avatar
2 votes
0 answers
113 views

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&...
Rz_PU's user avatar
  • 73
5 votes
1 answer
177 views

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 ...
Rz_PU's user avatar
  • 73
0 votes
1 answer
88 views

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}]}; ...
walid fssm's user avatar
3 votes
1 answer
152 views

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 ...
Giuseppe's user avatar
3 votes
1 answer
127 views

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 ...
deedeefive's user avatar
0 votes
1 answer
153 views

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: ...
麦克米兰's user avatar
3 votes
1 answer
240 views

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=...
我心永恒's user avatar
  • 1,572
0 votes
1 answer
262 views

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,...
Jeruwaho's user avatar
  • 125
0 votes
1 answer
181 views

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}{...
Aji Wibowo's user avatar
1 vote
0 answers
86 views

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 ...
Archie Watts-Farmer's user avatar
4 votes
1 answer
319 views

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 ...
Lion Sahara's user avatar
1 vote
1 answer
74 views

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 ...
jwmttsn's user avatar
  • 11
2 votes
1 answer
315 views

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 ...
Avrana's user avatar
  • 297
10 votes
2 answers
871 views

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 ...
Avrana's user avatar
  • 297
5 votes
2 answers
212 views

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 ...
Nasser's user avatar
  • 147k
3 votes
1 answer
217 views

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 ...
mgolunski's user avatar
1 vote
0 answers
142 views

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 ...
Robert Dickinson's user avatar
1 vote
1 answer
109 views

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: ...
walid fssm's user avatar
5 votes
2 answers
334 views

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 ...
ABCDEMMM's user avatar
  • 1,854
4 votes
1 answer
259 views

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 ...
Charmbracelet's user avatar
7 votes
2 answers
265 views

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. ...
Tobias's user avatar
  • 563
1 vote
1 answer
252 views

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 ...
Xyive's user avatar
  • 115
2 votes
1 answer
221 views

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 ...
Tobias's user avatar
  • 563
1 vote
1 answer
319 views

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] == ...
Henoc Zinga's user avatar
0 votes
0 answers
73 views

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 ...
Henoc Zinga's user avatar
2 votes
1 answer
284 views

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: ...
walid fssm's user avatar
1 vote
1 answer
155 views

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\}) \\ \...
walid fssm's user avatar
-1 votes
1 answer
166 views

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\}) \\ ...
walid fssm's user avatar
12 votes
2 answers
504 views

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 ...
umby's user avatar
  • 585