# 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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### Multiphysics - Electrostatics and Heat Transfer - Joule Heating boundary conditions

I am working on a multiphysics problem involving electrostatics (direct current) and heat transfer. I was messing around with the Joule Heating tutorial case and need to answer some questions to apply ...
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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 ...
65 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 ...
101 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] == ...
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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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### 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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### 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 ...
132 views

### 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 ...
431 views

### 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: ...
121 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 ...
60 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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### 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: ...
93 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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### 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 ...
564 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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### 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?
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### 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 ...
202 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 ...
88 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. ...
98 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/...
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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 ...
115 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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### 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 ...
109 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:...
225 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 ($\nabla^2 T(x,y)=0$) ...
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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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### 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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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 $... 4answers 794 views ### How to model diffusion through a membrane? This is a follow-up on How to handle discontinuity in diffusion coefficient? Consider diffusion of$u(t,x)$on the domain$x \in [0,2]$with some simple boundary conditions such as$u(0) = 2, u(2) = ...
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: $$\partial_t u = \partial_x (D \partial_x u),$$ ...