The Wolfram Language has powerful functionality based on the finite element method and the numerical method of lines for solving a wide variety of partial differential equations. The symbolic capabilities of the Wolfram Language make it possible to efficiently compute solutions from PDE models expressed as equations.

The PDE Models Overview provides a collection of numerical application examples from a wide range of fields solved with the finite element method.

D  ▪  Grad  ▪  Div  ▪  Curl  ▪  Laplacian  ▪  ...

Inactive — represent an operator in an inactive form

NDSolve — numerical solution to partial differential equations over a region

NDEigensystem — numerical eigenvalues and eigenfunctions to PDE over a region

NDSolveValue  ▪  ParametricNDSolveValue  ▪  NDEigenvalues  ▪  ...

DSolve — symbolic solution to partial differential equations over a region

DEigensystem — symbolic eigenvalues and eigenfunctions to PDE over a region

DSolveValue  ▪  DEigenvalues  ▪  ...

Boundary Conditions

DirichletCondition — specify Dirichlet conditions for partial differential equations

NeumannValue — specify Neumann and Robin conditions

NeumannBoundaryUnitNormal — specify a unit normal in Neumann conditions

PeriodicBoundaryCondition — specify periodic boundary conditions

Geometric Regions »

{x,y,…}∈Ω — specify the region for the independent variables

Disk  ▪  Ball  ▪  ImplicitRegion  ▪  MeshRegion  ▪  BoundaryMeshRegion  ▪  ...