General PDEs

Solve general linear and nonlinear PDEs for stationary, time-dependent, and eigenvalue problems

You can use Partial Differential Equation Toolbox™ to solve linear and nonlinear second-order PDEs for stationary, time-dependent, and eigenvalue problems that occur in common applications in engineering and science.

A typical workflow for solving a general PDE or a system of PDEs includes the following steps:

Convert PDEs to the form required by Partial Differential Equation Toolbox.
Create a PDE model container specifying the number of equations in your model.
Define 2-D or 3-D geometry and mesh it using triangular and tetrahedral elements with linear or quadratic basis functions.
Specify the coefficients, boundary and initial conditions. Use function handles to specify non-constant values.
Solve and plot the results at nodal locations or interpolate them to custom locations.

Contour and colored surface plot on a tilted plane

Functions

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PDE Model Setup

`createpde`	Create model
`applyBoundaryCondition`	Add boundary condition to `PDEModel` container
`specifyCoefficients`	Specify coefficients in PDE model
`setInitialConditions`	Give initial conditions or initial solution
`assembleFEMatrices`	Assemble finite element matrices
`solvepde`	Solve PDE specified in a PDEModel
`solvepdeeig`	Solve PDE eigenvalue problem specified in a PDEModel

Solutions at Nodal and Custom Locations

`evaluateGradient`	Evaluate gradients of PDE solutions at arbitrary points
`evaluateCGradient`	Evaluate flux of PDE solution
`interpolateSolution`	Interpolate PDE solution to arbitrary points

Visualization

`pdeplot`	Plot solution or mesh for 2-D problem
`pdeplot3D`	Plot solution or surface mesh for 3-D problem
`pdegplot`	Plot PDE geometry
`pdemesh`	Plot PDE mesh
`pdeviz`	Create and plot PDE visualization object (Since R2021a)

PDE Model Properties

`findBoundaryConditions`	Find boundary condition assignment for a geometric region
`findCoefficients`	Locate active PDE coefficients
`findInitialConditions`	Locate active initial conditions

Legacy Workflow

`createPDEResults`	Create solution object
`evaluate`	Interpolate data to selected locations
`pdeInterpolant`	Interpolant for nodal data to selected locations

Live Editor Tasks

Visualize PDE Results

Create and explore visualizations of PDE results in the Live Editor (Since R2022b)

Objects

`PDEModel`	PDE model object
`StationaryResults`	Time-independent PDE solution and derived quantities
`TimeDependentResults`	Time-dependent PDE solution and derived quantities
`EigenResults`	PDE eigenvalue solution and derived quantities

Properties

BoundaryCondition Properties	Boundary condition for PDE model
CoefficientAssignment Properties	Coefficient assignments
GeometricInitialConditions Properties	Initial conditions over a region or region boundary
NodalInitialConditions Properties	Initial conditions at mesh nodes
PDESolverOptions Properties	Algorithm options for solvers
PDEVisualization Properties	PDE visualization of mesh and nodal results (Since R2021a)

Topics

PDE Problem Setup

Solve Problems Using PDEModel Objects
Workflow describing how to set up and solve PDE problems using Partial Differential Equation Toolbox.
Specify Boundary Conditions

Set Dirichlet and Neumann conditions for scalar PDEs and systems of PDEs. Use functions when you cannot express your boundary conditions by constant input arguments.
f Coefficient for specifyCoefficients

Specify the coefficient f in the equation.
Set Initial Conditions

Set initial conditions for time-dependent problems or initial guess for nonlinear stationary problems.

Solutions and Their Gradients

Solution and Gradient Plots with pdeplot and pdeplot3D
Plot 2-D and 3-D PDE solutions and their gradients using pdeplot and pdeplot3D.
2-D Solution and Gradient Plots with MATLAB Functions
Plot 2-D PDE solutions and their gradients using surf, mesh, quiver, and other MATLAB^® functions.
3-D Solution and Gradient Plots with MATLAB Functions
Plot 3-D PDE solutions, their gradients, and streamlines using surf, contourslice, quiver, and other MATLAB functions.
Dimensions of Solutions, Gradients, and Fluxes
Dimensions of stationary, time-dependent, and eigenvalue results at mesh nodes and arbitrary locations.

Eigenvalue and Wave Problems

Eigenvalues and Eigenmodes of Square
Find the eigenvalues and eigenmodes of a square domain.
Eigenvalues and Eigenmodes of L-Shaped Membrane
Use command-line functions to find the eigenvalues and the corresponding eigenmodes of an L-shaped membrane.
Wave Equation on Square Domain
Solve a standard second-order wave equation.
Helmholtz Equation on Disk with Square Hole
Compute reflected waves from an object illuminated by incident waves.

Workflows Integrated with Other Toolboxes

Solve Poisson Equation on Unit Disk Using Physics-Informed Neural Networks
Solve a Poisson's equation with Dirichlet boundary conditions using PINN.
Medical Image-Based Finite Element Analysis of Spine (Medical Imaging Toolbox)
Estimate bone stress and strain in a vertebra bone under axial compression using finite element (FE) analysis.

Finite Element Method and Partial Differential Equations

Equations You Can Solve Using PDE Toolbox
Types of scalar PDEs and systems of PDEs that you can solve using Partial Differential Equation Toolbox.
Put Equations in Divergence Form
Transform PDEs to the form required by Partial Differential Equation Toolbox.
Finite Element Method Basics
Description of the use of the finite element method to approximate a PDE solution using a piecewise linear function.