Equations You Can Solve Using PDE Toolbox

Partial Differential Equation Toolbox™ solves scalar equations of the form

$m \frac{\partial^{2} u}{\partial t^{2}} + d \frac{\partial u}{\partial t} - \nabla \cdot (c \nabla u) + a u = f$

and eigenvalue equations of the form

$\begin{array}{l} - \nabla \cdot (c \nabla u) + a u = λ d u \\ or \\ - \nabla \cdot (c \nabla u) + a u = λ^{2} m u \end{array}$

For scalar PDEs, there are two choices of boundary conditions for each edge or face:

Dirichlet — On the edge or face, the solutionusatisfies the equation

hu=r，

wherehandrcan be functions of space (x，y，and, in 3-D case,z), the solutionu，and time. Often, you takeh= 1, and setrto the appropriate value.
Generalized Neumann boundary conditions — On the edge or face the solutionusatisfies the equation

$\vec{n} \cdot (c \nabla u) + q u = g$

$\vec{n}$ is the outward unit normal.qandgare functions defined on ∂Ω, and can be functions ofx，y，and, in 3-D case,z，the solutionu，and, for time-dependent equations, time.

The toolbox also solves systems of equations of the form

$m \frac{\partial^{2} u}{\partial t^{2}} + d \frac{\partial u}{\partial t} - \nabla \cdot (c \otimes \nabla u) + a u = f$

and eigenvalue systems of the form

$\begin{array}{l} - \nabla \cdot (c \otimes \nabla u) + a u = λ d u \\ or \\ - \nabla \cdot (c \otimes \nabla u) + a u = λ^{2} m u \end{array}$

A system of PDEs withNcomponents isNcoupled PDEs with coupled boundary conditions. Scalar PDEs are those withN= 1，meaning just one PDE. Systems of PDEs generally meansN> 1. The documentation sometimes refers to systems as multidimensional PDEs or as PDEs with a vector solutionu. In all cases, PDE systems have a single geometry and mesh. It is onlyN，the number of equations, that can vary.

The coefficientsm，d，c，a，andfcan be functions of location (x，y，and, in 3-D,z), and, except for eigenvalue problems, they also can be functions of the solutionuor its gradient. For eigenvalue problems, the coefficients cannot depend on the solutionuor its gradient.

For scalar equations, all the coefficients exceptcare scalar. The coefficientcrepresents a 2-by-2 matrix in 2-D geometry, or a 3-by-3 matrix in 3-D geometry. For systems ofNequations, the coefficientsm，d，andaareN-by-Nmatrices,fis anN-by-1 vector, andcis a 2N-by-2Ntensor (2-D geometry) or a 3N-by-3Ntensor (3-D geometry). For the meaning of $c \otimes u$ ，seec Coefficient for specifyCoefficients.

When bothmanddare0，the PDE is stationary. When eithermordare nonzero, the problem is time-dependent. When any coefficient depends on the solutionuor its gradient, the problem is called nonlinear.

For systems of PDEs, there are generalized versions of the Dirichlet and Neumann boundary conditions:

hu=rrepresents a matrixhmultiplying the solution vectoru，and equaling the vectorr.
$n \cdot (c \otimes \nabla u) + q u = g$ . For 2-D systems, the notation $n \cdot (c \otimes \nabla u)$ means theN1矩阵(i，1)-component

$\sum_{j = 1}^{N} (\cos (α) c_{i ， j ， 1 ， 1} \frac{\partial}{\partial x} + \cos (α) c_{i ， j ， 1 ， 2} \frac{\partial}{\partial y} + \sin (α) c_{i ， j ， 2 ， 1} \frac{\partial}{\partial x} + \sin (α) c_{i ， j ， 2 ， 2} \frac{\partial}{\partial y}) u_{j}$

where the outward normal vector of the boundary $n = (\cos (α) ， \sin (α))$ .
For 3-D systems, the notation $n \cdot (c \otimes \nabla u)$ means theN-by-1 vector with (i，1)-component

$\begin{array}{l} \sum_{j = 1}^{N} (\sin (φ) \cos (θ) c_{i ， j ， 1 ， 1} \frac{\partial}{\partial x} + \sin (φ) \cos (θ) c_{i ， j ， 1 ， 2} \frac{\partial}{\partial y} + \sin (φ) \cos (θ) c_{i ， j ， 1 ， 3} \frac{\partial}{\partial z}) u_{j} \\ + \sum_{j = 1}^{N} (\sin (φ) \sin (θ) c_{i ， j ， 2 ， 1} \frac{\partial}{\partial x} + \sin (φ) \sin (θ) c_{i ， j ， 2 ， 2} \frac{\partial}{\partial y} + \sin (φ) \sin (θ) c_{i ， j ， 2 ， 3} \frac{\partial}{\partial z}) u_{j} \\ + \sum_{j = 1}^{N} (\cos (θ) c_{i ， j ， 3 ， 1} \frac{\partial}{\partial x} + \cos (θ) c_{i ， j ， 3 ， 2} \frac{\partial}{\partial y} + \cos (θ) c_{i ， j ， 3 ， 3} \frac{\partial}{\partial z}) u_{j} \end{array}$

where the outward normal vector of the boundary $n = (\sin (φ) \cos (θ) ， \sin (φ) \sin (θ) ， \cos (φ))$ .
For each edge or face segment, there are a total ofNboundary conditions.

Equations You Can Solve Using PDE Toolbox

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