Equations You Can Solve Using Partial Differential Equation 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 solution u satisfies the equation
hu = r,
where h and r can be functions of space (x, y, and, in 3-D case, z), the solution u, and time. Often, you take h = 1, and set r to the appropriate value.
Generalized Neumann boundary conditions — On the edge or face the solution u satisfies the equation
$\vec{n} \cdot (c \nabla u) + q u = g$
$\vec{n}$ is the outward unit normal. q and g are functions defined on ∂Ω, and can be functions of x, y, and, in 3-D case, z, the solution u, 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 with N components is N coupled PDEs with coupled boundary conditions. Scalar PDEs are those with N = 1, meaning just one PDE. Systems of PDEs generally means N > 1. The documentation sometimes refers to systems as multidimensional PDEs or as PDEs with a vector solution u. In all cases, PDE systems have a single geometry and mesh. It is only N, the number of equations, that can vary.

Coefficients

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

For scalar equations, all the coefficients except c are scalar. The coefficient c represents a 2-by-2 matrix in 2-D geometry, or a 3-by-3 matrix in 3-D geometry. For systems of N equations, the coefficients m, d, and a are N-by-N matrices, f is an N-by-1 vector, and c is a 2N-by-2N tensor (2-D geometry) or a 3N-by-3N tensor (3-D geometry). For the meaning of $c \otimes u$ , see c Coefficient for specifyCoefficients.

When both m and d are 0, the PDE is stationary. When either m or d are nonzero, the problem is time-dependent. When any coefficient depends on the solution u or its gradient, the problem is called nonlinear.

Boundary Conditions

For a system of PDEs, the generalized version of the Dirichlet boundary condition is hu = r. The condition represents the matrix h multiplying the solution vector u, and equaling the vector r.

For a system of PDEs, the generalized version of the Neumann boundary condition is $n \cdot (c \otimes \nabla u) + q u = g$ . For example, in case of circumferential and spherical boundaries, the generalized versions of the Neumann boundary condition are as follows:

If the boundary is a circumference (2-D case), the outward normal vector of the boundary of the boundary is given by $n = (\cos (φ), \sin (φ))$ , the notation $n \cdot (c \otimes \nabla u)$ means the N-by-1 vector, for which the (i,1)-component is as follows:
$\sum_{j = 1}^{N} (\cos (φ) c_{i, j, 1, 1} \frac{\partial}{\partial x} + \cos (φ) c_{i, j, 1, 2} \frac{\partial}{\partial y} + \sin (φ) c_{i, j, 2, 1} \frac{\partial}{\partial x} + \sin (φ) c_{i, j, 2, 2} \frac{\partial}{\partial y}) u_{j}$
If the boundary is a spherical surface (3-D case), than the outward normal vector of the boundary is given by $n = (\sin (θ) \cos (φ), \sin (θ) \sin (φ), \cos (θ))$ , and the notation $n \cdot (c \otimes \nabla u)$ means the N-by-1 vector, for which the (i,1)-component is as follows:
$\begin{array}{l} \sum_{j = 1}^{N} (\sin (θ) \cos (φ) c_{i, j, 1, 1} \frac{\partial}{\partial x} + \sin (θ) \cos (φ) c_{i, j, 1, 2} \frac{\partial}{\partial y} + \sin (θ) \cos (φ) c_{i, j, 1, 3} \frac{\partial}{\partial z}) u_{j} \\ + \sum_{j = 1}^{N} (\sin (θ) \sin (φ) c_{i, j, 2, 1} \frac{\partial}{\partial x} + \sin (θ) \sin (φ) c_{i, j, 2, 2} \frac{\partial}{\partial y} + \sin (θ) \sin (φ) c_{i, j, 2, 3} \frac{\partial}{\partial z}) u_{j} \\ + \sum_{j = 1}^{N} (\cos (θ) c_{i, j, 3, 1} \frac{\partial}{\partial x} + \cos (θ) c_{i, j, 3, 2} \frac{\partial}{\partial y} + \cos (θ) c_{i, j, 3, 3} \frac{\partial}{\partial z}) u_{j} \end{array}$

For each edge or face segment, there are a total of N boundary conditions.