DerivativePDETerm[vars,γ]
represents a load derivative term 
with load derivative coefficient 
and model variables vars.
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. - A symbolic derivative coefficient can be specified through a VectorSymbol. »
- The load derivative coefficient
can depend on time, space, parameters and the dependent variables. - The following parameters pars can be given:
-
parameter default symbol "CoordinateChart" "Cartesian" 
"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
open all close allBasic Examples (3)
Define a time-independent derivative term:
DerivativePDETerm[{u[x], {x}}, {x}]Activate the derivative term:
Activate[%]Define a time-dependent derivative term:
DerivativePDETerm[{u[t, x], t, {x}}, {x}]Solve a diffusion equation with a nonlinear derivative term and a source term:
vars = {u[x, y], {x, y}};
NDSolveValue[{DiffusionPDETerm[vars, 1] + DerivativePDETerm[vars, {2u[x, y], u[x, y]}] == SourcePDETerm[vars, -1], DirichletCondition[u[x, y] == 0, x == 0 || y == 0]}, u[x, y], {x, y}∈Rectangle[]]ContourPlot[%, {x, y}∈Rectangle[]]Scope (16)
Define a time-independent 2D derivative term:
DerivativePDETerm[{u[x, y], {x, y}}, {2, 1}]Define a symbolic derivative term:
DerivativePDETerm[{u[t, x], t, {x}}, {a}]Define a stationary derivative term with a symbolic convection coefficient:
γ = VectorSymbol["γ", {1}];
DerivativePDETerm[{u[x], {x}}, γ]%//ActivateDefine a stationary derivative term with a symbolic convection coefficient replaced:
γ = VectorSymbol["γ", {1}];
DerivativePDETerm[{u[x], {x}}, γ, <|γ -> {1}|>]Define a time derivative term with a symbolic convection coefficient replaced:
γ = VectorSymbol["γ", {1}];
DerivativePDETerm[{u[t, x], t, {x}}, γ, <|γ -> {1}|>]Define a 1D axisymmetric time-independent derivative term:
DerivativePDETerm[{u[r], {r}}, {1}, <|"RegionSymmetry" -> "Axisymmetric"|>]Apply Activate to the term:
Activate[%]Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the derivative term:
Div[{1, 0, 0}, {r, theta, z}, "Cylindrical"]Define a 2D stationary derivative term:
DerivativePDETerm[{u[x, y], {x, y}}, {1, 0}]Define a derivative term with a coordinate chart:
DerivativePDETerm[{u[r, z], {r, z}}, {1, 0}, <|"CoordinateChart" -> "Polar"|>]Activate[%]Compare with a Div:
Div[{1, 0}, {r, z}, "Polar"]Define a 2D axisymmetric time-independent derivative term:
DerivativePDETerm[{u[r, z], {r, z}}, {Subscript[γ, 1][r, z], Subscript[γ, 3][r, z]}, <|"RegionSymmetry" -> "Axisymmetric"|>]Apply Activate to the term:
Activate[%]Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the derivative term:
Div[{Subscript[γ, 1][r, z], 0, Subscript[γ, 3][r, z]}, {r, theta, z}, "Cylindrical"]Define a nonlinear time-independent 2D derivative term:
DerivativePDETerm[{u[x, y], {x, y}}, {2, u[x, y] ^ 2}]Define a nonlinear time-dependent 2D derivative term:
DerivativePDETerm[{u[t, x, y], t, {x, y}}, {2, u[x, y] ^ 2}]Define a derivative with multiple dependent variables:
DerivativePDETerm[{{u[x, y], v[x, y]}, {x, y}}, {{{x, x * y}}, {{0, Sin[x]}}}]Define a nonlinear derivative with multiple dependent variables:
DerivativePDETerm[{{u[x, y], v[x, y]}, {x, y}}, {{{v[x, y], 0}}, {{0, 0}}}]Define a nonlinear 2D axisymmetric derivative with multiple dependent variables:
DerivativePDETerm[{{u[r, z], v[r, z]}, {r, z}}, {{{v[r, z], 0}}, {{0, 0}}}, <|"RegionSymmetry" -> "Axisymmetric"|>]The DerivativePDETerm can be used to compute the derivative of a differential equation component. Set up variables, a region and a boundary condition:
vars = {u[x], {x}};
Ω = Line[{{0}, {1}}];
bc = DirichletCondition[u[x] == 0, True];Set up a term for which the derivative is needed in the differential equation:
term = x ^ 2;if1 = NDSolveValue[{DiffusionPDETerm[vars, 1] + DerivativePDETerm[vars, {term}] == 1, bc}, u[x], {x}∈Ω]Solve the equation while explicitly computing the derivative of the term:
if2 = NDSolveValue[{DiffusionPDETerm[vars, 1] + D[term, x] == 1, bc}, u[x], {x}∈Ω]Show that the solutions are identical up to numerical precision:
Plot[if1 - if2, {x}∈Ω]Set up a system of derivative terms with DerivativePDETerm:
DerivativePDETerm[{{u[x], v[x]}, {x}}, {{{x}}, {{x ^ 2}}}]Text
Wolfram Research (2020), DerivativePDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/DerivativePDETerm.html (updated 2026).
CMS
Wolfram Language. 2020. "DerivativePDETerm." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2026. https://reference.wolfram.com/language/ref/DerivativePDETerm.html.
APA
Wolfram Language. (2020). DerivativePDETerm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DerivativePDETerm.html