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DiffusionPDETerm
✖
DiffusionPDETerm

✖

represents a diffusion term with model variables vars.

✖

represents a diffusion term with diffusion coefficient .

✖
DiffusionPDETerm[vars,c,pars]

uses model parameters pars.

Details

  • Diffusion is a central concept of physics that is used in a number of domains, such as thermodynamics, acoustics, structural mechanics and fluid dynamics.
  • Diffusion is also known as conduction.
  • Diffusion with a diffusion coefficient is the process of equilibration solely driven by a gradient of the dependent variable :
  • DiffusionPDETerm returns a differential operators term to be used as a part of partial differential equations:
  • DiffusionPDETerm can be used to model diffusion equations with dependent variable , independent variables and time variable .
  • Stationary model variables vars are vars={u[x1,…,xn],{x1,…,xn}}.
  • Time-dependent model variables vars are vars={u[t,x1,…,xn],{x1,…,xn}} or vars={u[t,x1,…,xn],t,{x1,…,xn}}.
  • The diffusion term in context with other PDE terms is given by:
  • During diffusion, the medium in which the diffusion happens remains stationary, in contrast to convection where the medium is the transport mechanism.
  • The diffusion coefficient can have the following forms:
  • cscalar , isotropic diffusion
    {c1,…,cn}
  • vector , orthotropic diffusion
    • {{c11,…,c1n},…,{cn1,…,cnn}}
  • matrix , anistropic diffusion
  • For a system of PDEs with dependent variables {u1,…,um}, the diffusion represents:
  • The diffusion term in context systems of PDE terms:
  • The diffusion coefficient is a tensor of rank 4 of the form where each submatrix is an matrix that can be specified in the same way as for a single dependent variable.
  • The diffusion coefficient can depend on time, space, parameters and the dependent variables.
  • The following parameters pars can be given:
  • parameterdefaultsymbol
    "RegionSymmetry"None
  • A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
  • "Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
  • dimensionreductionequation
    1D
    2D
  • The coefficient affects the meaning of NeumannValue.
  • All quantities that do not explicitly depend on the independent variables given are taken to have zero partial derivative.

Examples

open allclose all

Basic Examples  (6)Summary of the most common use cases

Define a stationary diffusion term:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-6a2ghy
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-oa75fr
Out[2]=2

Define a stationary diffusion term with a parametric diffusion coefficient replaced:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-y453ds
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-hnt4kr
Out[2]=2

Define a symbolic diffusion term:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-q26ow7
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-1485kk
Out[2]=2

Find the eigenvalues of a diffusion term:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-sbsmdw
Out[1]=1

Construct a Poisson equation from basic terms and solve it symbolically:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-sakglf
Out[1]=1

Solve a time-dependent diffusion equation with a Gaussian initial condition:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-2r8h1y
Out[1]=1

Visualize the smoothing of the initial condition through time:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-jmbppv
Out[2]=2

Check that the area under the curves remains constant:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-qhjc1q
Out[3]=3

Scope  (31)Survey of the scope of standard use cases

1D  (6)

Define a symbolic diffusion term:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-mq8yxf
Out[1]=1

Not specifying a diffusion coefficient will result in an identity matrix coefficient:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-oycuqa
Out[1]=1

Define a time-dependent diffusion term:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-l1q57y
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-nxbkl
Out[2]=2

Define a stationary diffusion term with a parametric diffusion coefficient replaced:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-kgqoxy
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-1b5da0
Out[2]=2

Use DiffusionPDETerm for modeling an eigenvalue problem:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-rodqku
Out[1]=1

Use DiffusionPDETerm to set up a 1D Poisson equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-baak2e
Out[1]=1

Visualize the result:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-u0jfow
Out[2]=2

1D Axisymmetric  (1)

Set up a 1D axisymmetric diffusion equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-i2eeu1
Out[1]=1

Apply Activate to the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-weq7mi
Out[2]=2

Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the diffusion equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-id1t1j
Out[1]=1

2D  (12)

Define a 2D stationary diffusion term:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-gmmpen
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-z0sc14
Out[2]=2

Set up a 2D stationary diffusion equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-9mpx31
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-tbuq4l
Out[2]=2

Set up a 2D time-dependent diffusion equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-wlbr0z
Out[1]=1

Activate the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-1db6sl
Out[2]=2

Not specifying a diffusion coefficient will result in an identity matrix coefficient:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-7k5vqe
Out[1]=1

Define a 2D orthotropic stationary diffusion term with a vector diffusion coefficient:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-c0ibpf
Out[1]=1

Define a 2D stationary diffusion term with an anisotropic diffusion matrix:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-h6n90o
Out[1]=1

Define a 2D diffusion term with an anisotropic diffusion matrix:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-q20qls
Out[1]=1

Use DiffusionPDETerm to set up a 2D Poisson equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-futv7o
Out[1]=1

Visualize the result:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-edayxf
Out[2]=2

Construct a Poisson equation from basic PDE terms and solve it numerically:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-rix32r
Out[1]=1

Visualize the result:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-fo25oj
Out[2]=2

Use a vector-valued diffusion coefficient that has a larger diffusion constant in the direction than the direction:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-st3h3f
Out[1]=1

Visualize the result:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ifwdbk
Out[2]=2

Use a vector-valued diffusion coefficient that has a larger diffusion constant in the direction than the direction:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-wrln97
Out[1]=1

Visualize the result:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-xwvygy
Out[2]=2

Use an anisotropic diffusion coefficient:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-twdq7z
Out[1]=1

Visualize the result:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ra7zn7
Out[2]=2

2D Axisymmetric  (3)

Set up a 2D axisymmetric diffusion equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-e4v8gb
Out[1]=1

Apply Activate to the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-g75pae
Out[2]=2

Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the diffusion equation:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-m3nsq
Out[2]=2

Set up a 2D axisymmetric diffusion equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-58qw8l
Out[1]=1

Apply Activate to the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-6rpvj9
Out[2]=2

Set up a 2D axisymmetric time-dependent diffusion equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ge4kw4
Out[1]=1

Apply Activate to the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-dtt6us
Out[2]=2

3D  (1)

Use DiffusionPDETerm to set up a 3D Poisson equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-c60l9u
Out[1]=1

Visualize the result:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-cg4ia
Out[3]=3

Coupled  (5)

Define a diffusion term with multiple dependent variables:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-4r1hdd
Out[1]=1

Define a diffusion term with multiple dependent variables and multiple diffusion coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-6ayqdv
Out[1]=1

Define a diffusion term with multiple dependent variables and anisotropic diffusion coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-zyryfd
Out[1]=1

Define a diffusion term with multiple dependent variables and all coefficients specified:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-rrzw71
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-3wy5a3
Out[2]=2

Off-diagonal diffusion coefficients are possible in coupled PDEs:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-q4os6r

Solve the equations:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-bxqjzn

Visualize the results:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-dlnd81
Out[3]=3

Coupled Axisymmetric  (3)

Use DiffusionPDETerm to set up a 1D axisymmetric equation with multiple dependent variables:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-nc7lfk
Out[2]=2

Specify boundary conditions:

In[12]:=12
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ks02iq

Solve the equations numerically:

In[14]:=14
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-q4ht4
Out[14]=14

Solve the same equation symbolically:

In[11]:=11
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-boufji
Out[11]=11

Visualize the difference between the results:

In[15]:=15
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ld6cu7
Out[15]=15

Define an axisymmetric diffusion term with multiple dependent variables:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-unc25
Out[1]=1

Apply Activate to the term:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-bxj18b
Out[2]=2

Define a 2D diffusion axisymmetric term with multiple dependent variables and multiple diffusion coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ea96io
In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-f8xpbs

Solve the coupled PDEs:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-cwwqcl

Visualize the results:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-j7gnv2
Out[4]=4

Applications  (9)Sample problems that can be solved with this function

Use DiffusionPDETerm with a variable diffusion coefficient:

In[8]:=8
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-3dnjo6
Out[8]=8

Solve the equation:

In[9]:=9
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-v3xa4k
Out[9]=9

Compute the flux:

In[10]:=10
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-mtucqa
Out[10]=10

Visualize the result:

In[11]:=11
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-7o73xy
Out[11]=11

Use DiffusionPDETerm to model conductive heat transfer using an axisymmetric geometry.

The region of analysis is a 2D region. Instead of defining the full 2D region in Cartesian coordinates , you can define a region with truncated cylindrical coordinates in 1D . The cylindrical coordinate variables and vanish because the system is rotationally symmetric around the axis.

Set up the equation:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ddauin
Out[1]=1

Specify boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-5ise49

Solve the equation:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-c47tj1
Out[3]=3

Or solve it symbolically with DSolveValue:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-dom6fn
Out[4]=4

Visualize the difference between the results:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-kw0f8r
Out[5]=5

Use DiffusionPDETerm to model species diffusion under a dam. Set up the region:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ntfh44

Set up the model:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-1wzugl
Out[2]=2

Solve the equation:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-h8juq4
Out[3]=3

Compute the flux:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-2tis1m
Out[4]=4

Visualize the result:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-0zwblu
Out[5]=5

Find the concentration of species under the dam. Construct the model:

In[6]:=6
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-b9km2u

Solve the equation:

In[7]:=7
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-swulzj
Out[7]=7

Visualize the species concentration:

In[8]:=8
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-dw51b5
Out[8]=8

Define a Helmholtz model:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-1w36

Solve for the eigenvalues of the Helmholtz equation:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-cg765y
Out[2]=2

Solve the Helmholtz equation with a source term:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-bfm8gc
Out[3]=3

Visualize the solution:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-dxhrsa
Out[4]=4

Use DiffusionPDETerm to model conductive heat transfer using an axisymmetric geometry. The region of analysis is a 3D hollow cylinder. Instead of defining the full 3D region in Cartesian coordinates , you can define a region with truncated cylindrical coordinates in 2D . The cylindrical coordinate variable vanishes because the system is rotationally symmetric around the axis.

Set up the region:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-h0s39

Set up the PDE model:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ic22hc
Out[2]=2

Solve the axisymmetric equation:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-iqsrmt
Out[3]=3

Visualize the result:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-htitkz
Out[4]=4

Visualize the result in 3D space for part of the cylinder:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-bt46u6
Out[5]=5

Use DiffusionPDETerm to model a nonlinear conductive heat transfer using an axisymmetric geometry.

Set up the region:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-gdty9

Find the thermal conductivity for air:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-pg3osk

Set up the PDE model:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-g5w6rt

Solve the equation and measure the time and memory needed to do so:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-kz2q8

Visualize the axisymmetric result:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-oevjd
Out[5]=5

Print the total time to do the computation and number of megabytes used during the evaluation:

In[6]:=6
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-k7k1yy

Use DiffusionPDETerm to set up a plane stress operator. Set up the coupled coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-mm0icq

Set up the model:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-guenwu
Out[2]=2

Set up a region:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-g77gn8
Out[3]=3

Solve the equations:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-oef3fb
Out[4]=4

Visualize the results:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-nfaofz
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-xeh5mr
Out[6]=6

Compute the flux:

In[7]:=7
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-kmit6b
Out[7]=7

Define a Stokes-flow model:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-kqfq17

Set up the equation:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-piftwv
Out[2]=2

Define a narrowing region:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-lkdoer

Set up a boundary condition:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-63imdl

Solve the equation:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-l2dvxi

Visualize the solution:

In[6]:=6
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-s73evl
Out[6]=6

Extend a Stokes-flow model to a Navier–Stokes-flow model. Define a Stokes-flow model:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-r6w4c3

Define a Navier–Stokes-flow model:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-7anqg3

Set up the equation:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-5co8b4
Out[3]=3

Define a region:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-07p7u2

Set up a boundary condition:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-72s594

Solve the equation:

In[6]:=6
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ftjmbu

Visualize the solution:

In[7]:=7
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ng3ezz
Out[7]=7

Properties & Relations  (3)Properties of the function, and connections to other functions

Verify that anisotropic diffusion is the same as :

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-dxq7mi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-k0c95l
Out[2]=2

Visualize that there is no difference in the two solutions:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-su02vt
Out[3]=3

Solve a time-dependent diffusion equation with a Gaussian initial condition:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-3kw5jc
Out[1]=1

The analytical solution is an infinite series:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-f7zr24
Out[2]=2

Extract a few terms from the Inactive sum:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-24gjj2

Visualize the difference between the numerical and the analytical solutions:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-gsyn4l
Out[4]=4

The equilibration property of the diffusion term manifests itself in the smoothing of a discontinuous initial condition:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-mcyo3a
Out[1]=1

Visualize the discontinuous initial condition and the smoothed evolution of the initial condition:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-e8e3mm
Out[2]=2

Possible Issues  (5)Common pitfalls and unexpected behavior

The negative sign in the operator does not need to be given:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-36w55v
Out[1]=1

A symbolic diffusion coefficient is interpreted as a matrix diffusion coefficient:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-0692w2
Out[1]=1

A subsequent substitution must account for that:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-uy2whj
Out[2]=2

An alternative is to specify the symbolic diffusion coefficient as a matrix:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-5t1q7k
Out[3]=3

A numeric diffusion coefficient will automatically be multiplied with an IdentityMatrix of proper dimensions:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ycmeqd
Out[4]=4

While solving the differential equation with a scalar diffusion coefficient has been made to work for convenience, subsequent operations may rely on a proper setup:

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-b33xxy
Out[1]=1

Solve the equation:

In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-kjzd3p

Note that the result tries to take the dot product of a scalar with the gradient of the solution:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-hgm8k8
Out[3]=3

Visualizing the result does not work:

In[4]:=4
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-nxuhhv
Out[4]=4

One alternative is to directly specify the diffusion coefficient. In the case of a number, the diffusion coefficient is multiplied with the IdentityMatrix:

In[5]:=5
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-wjuswg
Out[5]=5

Solve the equation:

In[6]:=6
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-na1e86

Compute the flux:

In[7]:=7
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-ouy4h
Out[7]=7

Visualize the result:

In[8]:=8
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-9r5vzy
Out[8]=8

Yet another alternative is to specify the substituted value as a matrix of dimensions of the number of independent variables:

In[9]:=9
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-dgdm9o
Out[9]=9

DiffusionPDETerm models , not :

In[1]:=1
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-pjcisp
In[2]:=2
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-sv8g46

Visualize the difference in the solutions:

In[3]:=3
✖
https://wolfram.com/xid/0bzqzcro9ea1fovbu-cc4zlm
Out[3]=3
Wolfram Research (2020), DiffusionPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/DiffusionPDETerm.html (updated 2022).
✖
Wolfram Research (2020), DiffusionPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/DiffusionPDETerm.html (updated 2022).

Text

Wolfram Research (2020), DiffusionPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/DiffusionPDETerm.html (updated 2022).

✖
Wolfram Research (2020), DiffusionPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/DiffusionPDETerm.html (updated 2022).

CMS

Wolfram Language. 2020. "DiffusionPDETerm." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/DiffusionPDETerm.html.

✖
Wolfram Language. 2020. "DiffusionPDETerm." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/DiffusionPDETerm.html.

APA

Wolfram Language. (2020). DiffusionPDETerm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiffusionPDETerm.html

✖
Wolfram Language. (2020). DiffusionPDETerm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiffusionPDETerm.html

BibTeX

@misc{reference.wolfram_2025_diffusionpdeterm, author="Wolfram Research", title="{DiffusionPDETerm}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/DiffusionPDETerm.html}", note=[Accessed: 29-April-2025 ]}

✖
@misc{reference.wolfram_2025_diffusionpdeterm, author="Wolfram Research", title="{DiffusionPDETerm}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/DiffusionPDETerm.html}", note=[Accessed: 29-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_diffusionpdeterm, organization={Wolfram Research}, title={DiffusionPDETerm}, year={2022}, url={https://reference.wolfram.com/language/ref/DiffusionPDETerm.html}, note=[Accessed: 29-April-2025 ]}

✖
@online{reference.wolfram_2025_diffusionpdeterm, organization={Wolfram Research}, title={DiffusionPDETerm}, year={2022}, url={https://reference.wolfram.com/language/ref/DiffusionPDETerm.html}, note=[Accessed: 29-April-2025 ]}