General PDEs
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.
Functions
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 |
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.
Heat Transfer and Structural Problems
- Nonlinear Heat Transfer in Thin Plate
Perform a heat transfer analysis of a thin plate. - Deflection of Piezoelectric Actuator
Solve a coupled elasticity-electrostatics problem. - Clamped Square Isotropic Plate with Uniform Pressure Load
Calculate the deflection of a structural plate acted on by a pressure loading. - Dynamic Analysis of Clamped Beam
Analyze the dynamic behavior of a beam clamped at both ends and loaded with a uniform pressure load. - Vibration of Circular Membrane
Find vibration modes of a circular membrane.
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 a physics-informed neural network (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 Partial Differential Equation 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.
Teaching Resources
Applied Partial Differential Equations
MathWorks Courseware module that teaches topics suitable for a first class in partial differential equations using interactive live scripts
