Documentation

# 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.

• 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.

## Functions

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 `createpde` Create model `applyBoundaryCondition` Add boundary condition to PDEModel container `specifyCoefficients` Specify coefficients in a 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
 `evaluateGradient` Evaluate gradients of PDE solutions at arbitrary points `evaluateCGradient` Evaluate flux of PDE solution `interpolateSolution` Interpolate PDE solution to arbitrary points
 `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
 `findBoundaryConditions` Find boundary condition assignment for a geometric region `findCoefficients` Locate active PDE coefficients `findInitialConditions` Locate active initial conditions
 `createPDEResults` Create solution object `evaluate` Interpolate data to selected locations `pdecont` Shorthand command for contour plot `pdesurf` Shorthand command for surface plot `pdeInterpolant` Interpolant for nodal data to selected locations

## Objects

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

### Solutions and Their Gradients

Plot 2-D Solutions and Their Gradients

Visualize a 2-D PDE solution and its gradient at nodal and arbitrary locations.

Plot 3-D Solutions and Their Gradients

Visualize a 3-D PDE solution and its gradient at nodal and arbitrary locations.

Dimensions of Solutions, Gradients, and Fluxes

Dimensions of stationary, time-dependent, and eigenvalue results at mesh nodes and arbitrary locations.

### Eigenvalue 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.

### 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.

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