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.
Defining 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.
PDE Model Setup
|Add boundary condition to |
|Specify coefficients in a PDE model|
|Give initial conditions or initial solution|
|Assemble finite element matrices|
|Solve PDE specified in a PDEModel|
|Solve PDE eigenvalue problem specified in a PDEModel|
Solutions at Nodal and Custom Locations
|Evaluate gradients of PDE solutions at arbitrary points|
|Evaluate flux of PDE solution|
|Interpolate PDE solution to arbitrary points|
PDE Model Properties
|Find boundary condition assignment for a geometric region|
|Locate active PDE coefficients|
|Locate active initial conditions|
|Create solution object|
|Interpolate data to selected locations|
|Interpolant for nodal data to selected locations|
Live Editor Tasks
|Visualize PDE Results||Create and explore visualizations of PDE results in the Live Editor|
|PDE model object|
|Time-independent PDE solution and derived quantities|
|Time-dependent PDE solution and derived quantities|
|PDE eigenvalue solution and derived quantities|
|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|
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
- 2-D Solution and Gradient Plots with MATLAB Functions
Plot 2-D PDE solutions and their gradients using
quiver, and other MATLAB® functions.
- 3-D Solution and Gradient Plots with MATLAB Functions
Plot 3-D PDE solutions, their gradients, and streamlines using
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.