# DerivativePDETerm

DerivativePDETerm[vars,γ]

represents a load derivative term with load derivative coefficient and model variables vars.

DerivativePDETerm[vars,γ,pars]

uses model parameters pars.

# Details   • A load derivative is typically used to model derivatives of a source or sink.
• Computing a derivative with a source derivative coefficient is the process of adding the derivative of a source in a model by:
• • DerivativePDETerm returns a differential operators term to be used as a part of partial differential equations:
• • DerivativePDETerm can be used to model derivatives in 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 source derivative term in context with other PDE terms is given by:
• • The coefficient affects the meaning of NeumannValue.
• The source derivative coefficient has the following form:
•  {γ1,…,γn} vector • For a system of PDEs with dependent variables {u1,,um}, the load derivative represents:
• • The derivative term in context systems of PDE terms:
• • The load derivative coefficient is a tensor of rank 3 of the form where each subvector is a vector of length that is specified in the same way as for a single dependent variable.
• The load derivative coefficient can depend on time, space, parameters and the dependent variables.
• The following parameters pars can be given:
•  parameter default symbol "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:
•  dimension reduction equation 1D  2D  • All quantities that do not explicitly depend on the independent variables given are taken to have zero partial derivative.

# Examples

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## Basic Examples(3)

Define a time-independent derivative term:

Activate the derivative term:

Define a time-dependent derivative term:

Solve a diffusion equation with a nonlinear derivative term and a source term:

Visualize the result:

## Scope(9)

Define a symbolic derivative term:

Define a 1D axisymmetric time-independent derivative term:

Apply Activate to the term:

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

Define a 2D stationary derivative term:

Define a 2D axisymmetric time-independent derivative term:

Apply Activate to the term:

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

Define a derivative with multiple dependent variables:

Define a nonlinear derivative with multiple dependent variables:

Define a nonlinear 2D axisymmetric derivative with multiple dependent variables:

The DerivativePDETerm can be used to compute the derivative of a differential equation component. Set up variables, a region and a boundary condition:

Set up a term for which the derivative is needed in the differential equation:

Solve the equation:

Solve the equation while explicitly computing the derivative of the term:

Show that the solutions are identical up to numerical precision:

Set up a system of derivative terms with DerivativePDETerm: