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Equations You Can Solve Using PDE Toolbox

Partial Differential Equation Toolbox™ solves scalar equations of the form

`$m\frac{{\partial }^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla ·\left(c\nabla u\right)+au=f$`

and eigenvalue equations of the form

`$\begin{array}{l}-\nabla ·\left(c\nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla ·\left(c\nabla u\right)+au={\lambda }^{2}mu\end{array}$`

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

`$\stackrel{\to }{n}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\nabla u\right)+qu=g$`

$\stackrel{\to }{n}$ 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

`$m\frac{{\partial }^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla ·\left(c\otimes \nabla u\right)+au=f$`

and eigenvalue systems of the form

`$\begin{array}{l}-\nabla ·\left(c\otimes \nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla ·\left(c\otimes \nabla u\right)+au={\lambda }^{2}mu\end{array}$`

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 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.

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.

• $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)+qu=g$. For 2-D systems, the notation $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$ means the N-by-1 matrix with (i,1)-component

`$\sum _{j=1}^{N}\left(\mathrm{cos}\left(\alpha \right){c}_{i,j,1,1}\frac{\partial }{\partial x}+\mathrm{cos}\left(\alpha \right){c}_{i,j,1,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\alpha \right){c}_{i,j,2,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\alpha \right){c}_{i,j,2,2}\frac{\partial }{\partial y}\right)\text{\hspace{0.17em}}{u}_{j}$`

where the outward normal vector of the boundary $n=\left(\mathrm{cos}\left(\alpha \right),\mathrm{sin}\left(\alpha \right)\right)$.

For 3-D systems, the notation $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$ means the N-by-1 vector with (i,1)-component

`$\begin{array}{l}\sum _{j=1}^{N}\left(\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,3}\frac{\partial }{\partial z}\right){u}_{j}\\ +\sum _{j=1}^{N}\left(\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,3}\frac{\partial }{\partial z}\right){u}_{j}\\ +\sum _{j=1}^{N}\left(\mathrm{cos}\left(\theta \right){c}_{i,j,3,1}\frac{\partial }{\partial x}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,2}\frac{\partial }{\partial y}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,3}\frac{\partial }{\partial z}\right){u}_{j}\end{array}$`

where the outward normal vector of the boundary $n=\left(\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right),\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right),\mathrm{cos}\left(\phi \right)\right)$.

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

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