# Questions tagged [heat-equation]

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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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76 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: ...
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149 views

1 vote
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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 ...
126 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 ...
1 vote
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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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174 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 ...
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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 ...
• 115k
117 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 ...
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1 vote
122 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 ...
1 vote
76 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: ...
248 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 ...
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225 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 ...
237 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. ...
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1 vote
179 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 ...
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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 vote
161 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] == ...
71 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 ...
122 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: ...
1 vote
150 views

405 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 ...
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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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### Laplace's equation in spherical coordinates with Neumann b.c

I am trying to find the temperature field in a semi-infinite solid on whose surface there is an isotherm spherical cap sunken by a length p. For example: ...
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### 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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1 vote
98 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 ...
180 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: ...
124 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). ...
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1 vote
104 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 ...
850 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 ...
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1 vote
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### 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?
283 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 ...
281 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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### 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. ...
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### 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/...
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### 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 ...
149 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 ...
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1 vote
110 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 ...
208 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:...
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### 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 ($\nabla^2 T(x,y)=0$) ...
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### 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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1 vote
104 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 ...
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1 vote
161 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\\\...
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### 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/...
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### 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 ...
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### 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 ...
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