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Equations You Can Solve Using Recommended Functions

Partial Differential Equation Toolbox™ solves scalar equations of the form

m2ut2+dut·(cu)+au=f

and eigenvalue equations of the form

·(cu)+au=λduor·(cu)+au=λ2mu

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

    n·(cu)+qu=g

    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

m2ut2+dut·(cu)+au=f

and eigenvalue systems of the form

·(cu)+au=λduor·(cu)+au=λ2mu

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 cu, 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·(cu)+qu=g. For 2-D systems, the notation n·(cu) means the N-by-1 matrix with (i,1)-component

    j=1N(cos(α)ci,j,1,1x+cos(α)ci,j,1,2y+sin(α)ci,j,2,1x+sin(α)ci,j,2,2y)uj

    where the outward normal vector of the boundary n=(cos(α),sin(α)).

    For 3-D systems, the notation n·(cu) means the N-by-1 vector with (i,1)-component

    j=1N(sin(φ)cos(θ)ci,j,1,1x+sin(φ)cos(θ)ci,j,1,2y+sin(φ)cos(θ)ci,j,1,3z)uj+j=1N(sin(φ)sin(θ)ci,j,2,1x+sin(φ)sin(θ)ci,j,2,2y+sin(φ)sin(θ)ci,j,2,3z)uj+j=1N(cos(θ)ci,j,3,1x+cos(θ)ci,j,3,2y+cos(θ)ci,j,3,3z)uj

    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.

Related Examples

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