# 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

## 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.