equations you can solve using pde toolbox -凯发k8网页登录

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 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.

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 .

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.

for systems of pdes, there are generalized versions of the dirichlet and neumann boundary conditions:

hu = r represents a matrix h multiplying the solution vector u, and equaling the vector r.
$n \cdot (c \otimes \nabla u) q u = g$ . for 2-d systems, the notation $n \cdot (c \otimes \nabla u)$ means the n-by-1 matrix with (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 the n-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 of n boundary conditions.

equations you can solve using pde toolbox -凯发k8网页登录

equations you can solve using pde toolbox

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