c Coefficient for specifyCoefficients

c Coefficient for `specifyCoefficients`

Overview of the c Coefficient

This topic describes how to write the coefficient c in equations such as

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

The topic applies to the recommended workflow for including coefficients in your model using specifyCoefficients.

For 2-D systems, c is a tensor with 4N² elements. For 3-D systems, c is a tensor with 9N² elements. For a definition of the tensor elements, see Definition of the c Tensor Elements. N is the number of equations, see Equations You Can Solve Using PDE Toolbox.

To write the coefficient c for inclusion in the PDE model via specifyCoefficients, give c as either of the following:

If c is constant, give a column vector representing the elements in the tensor.
If c is not constant, give a function handle. The function must be of the form
```
ccoeffunction(location,state)
```
solvepde or solvepdeeig pass the location and state structures to ccoeffunction. The function must return a matrix of size N1-by-Nr, where:
- N1 is the length of the vector representing the c coefficient. There are several possible values of N1, detailed in Some c Vectors Can Be Short. For 2-D geometry, 1 ≤ N1 ≤ 4N², and for 3-D geometry, 1 ≤ N1 ≤ 9N².
- Nr is the number of points in the location that the solver passes. Nr is equal to the length of the location.x or any other location field. The function should evaluate c at these points.

Definition of the c Tensor Elements

For 2-D systems, the notation $\nabla \cdot (c \otimes \nabla u)$ represents an N-by-1 matrix with an (i,1)-component

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

For 3-D systems, the notation $\nabla \cdot (c \otimes \nabla u)$ represents an N-by-1 matrix with an (i,1)-component

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

All representations of the c coefficient begin with a “flattening” of the tensor to a matrix. For 2-D systems, the N-by-N-by-2-by-2 tensor flattens to a 2N-by-2N matrix, where the matrix is logically an N-by-N matrix of 2-by-2 blocks.

$(\begin{matrix} c (1, 1, 1, 1) & c (1, 1, 1, 2) & c (1, 2, 1, 1) & c (1, 2, 1, 2) & \dots & c (1, N, 1, 1) & c (1, N, 1, 2) \\ c (1, 1, 2, 1) & c (1, 1, 2, 2) & c (1, 2, 2, 1) & c (1, 2, 2, 2) & \dots & c (1, N, 2, 1) & c (1, N, 2, 2) \\ c (2, 1, 1, 1) & c (2, 1, 1, 2) & c (2, 2, 1, 1) & c (2, 2, 1, 2) & \dots & c (2, N, 1, 1) & c (2, N, 1, 2) \\ c (2, 1, 2, 1) & c (2, 1, 2, 2) & c (2, 2, 2, 1) & c (2, 2, 2, 2) & \dots & c (2, N, 2, 1) & c (2, N, 2, 2) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ c (N, 1, 1, 1) & c (N, 1, 1, 2) & c (N, 2, 1, 1) & c (N, 2, 1, 2) & \dots & c (N, N, 1, 1) & c (N, N, 1, 2) \\ c (N, 1, 2, 1) & c (N, 1, 2, 2) & c (N, 2, 2, 1) & c (N, 2, 2, 2) & \dots & c (N, N, 2, 1) & c (N, N, 2, 2) \end{matrix})$

For 3-D systems, the N-by-N-by-3-by-3 tensor flattens to a 3N-by-3N matrix, where the matrix is logically an N-by-N matrix of 3-by-3 blocks.

$(\begin{matrix} c (1, 1, 1, 1) & c (1, 1, 1, 2) & c (1, 1, 1, 3) & c (1, 2, 1, 1) & c (1, 2, 1, 2) & c (1, 2, 1, 3) & \dots & c (1, N, 1, 1) & c (1, N, 1, 2) & c (1, N, 1, 3) \\ c (1, 1, 2, 1) & c (1, 1, 2, 2) & c (1, 1, 2, 3) & c (1, 2, 2, 1) & c (1, 2, 2, 2) & c (1, 2, 2, 3) & \dots & c (1, N, 2, 1) & c (1, N, 2, 2) & c (1, N, 2, 3) \\ c (1, 1, 3, 1) & c (1, 1, 3, 2) & c (1, 1, 3, 3) & c (1, 2, 3, 1) & c (1, 2, 3, 2) & c (1, 2, 3, 3) & \dots & c (1, N, 3, 1) & c (1, N, 3, 2) & c (1, N, 3, 3) \\ c (2, 1, 1, 1) & c (2, 1, 1, 2) & c (2, 1, 1, 3) & c (2, 2, 1, 1) & c (2, 2, 1, 2) & c (2, 2, 1, 3) & \dots & c (2, N, 1, 1) & c (2, N, 1, 2) & c (2, N, 1, 3) \\ c (2, 1, 2, 1) & c (2, 1, 2, 2) & c (2, 1, 2, 3) & c (2, 2, 2, 1) & c (2, 2, 2, 2) & c (2, 2, 2, 3) & \dots & c (2, N, 2, 1) & c (2, N, 2, 2) & c (2, N, 2, 3) \\ c (2, 1, 3, 1) & c (2, 1, 3, 2) & c (2, 1, 3, 3) & c (2, 2, 3, 1) & c (2, 2, 3, 2) & c (2, 2, 3, 3) & \dots & c (2, N, 3, 1) & c (2, N, 3, 2) & c (2, N, 3, 3) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ c (N, 1, 1, 1) & c (N, 1, 1, 2) & c (N, 1, 1, 3) & c (N, 2, 1, 1) & c (N, 2, 1, 2) & c (N, 2, 1, 3) & \dots & c (N, N, 1, 1) & c (N, N, 1, 2) & c (N, N, 1, 3) \\ c (N, 1, 2, 1) & c (N, 1, 2, 2) & c (N, 1, 2, 3) & c (N, 2, 2, 1) & c (N, 2, 2, 2) & c (N, 2, 2, 3) & \dots & c (N, N, 2, 1) & c (N, N, 2, 2) & c (N, N, 2, 3) \\ c (N, 1, 3, 1) & c (N, 1, 3, 2) & c (N, 1, 3, 3) & c (N, 2, 3, 1) & c (N, 2, 3, 2) & c (N, 2, 3, 3) & \dots & c (N, N, 3, 1) & c (N, N, 3, 2) & c (N, N, 3, 3) \end{matrix})$

These matrices further get flattened into a column vector. First the N-by-N matrices of 2-by-2 and 3-by-3 blocks are transformed into "vectors" of 2-by-2 and 3-by-3 blocks. Then the blocks are turned into vectors in the usual column-wise way.

The coefficient vector c relates to the tensor c as follows. For 2-D systems,

$(\begin{matrix} c (1) & c (3) & c (4 N + 1) & c (4 N + 3) & \dots & c (4 N (N - 1) + 1) & c (4 N (N - 1) + 3) \\ c (2) & c (4) & c (4 N + 2) & c (4 N + 4) & \dots & c (4 N (N - 1) + 2) & c (4 N (N - 1) + 4) \\ c (5) & c (7) & c (4 N + 5) & c (4 N + 7) & \dots & c (4 N (N - 1) + 5) & c (4 N (N - 1) + 7) \\ c (6) & c (8) & c (4 N + 6) & c (4 N + 8) & \dots & c (4 N (N - 1) + 6) & c (4 N (N - 1) + 8) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ c (4 N - 3) & c (4 N - 1) & c (8 N - 3) & c (8 N - 1) & \dots & c (4 N^{2} - 3) & c (4 N^{2} - 1) \\ c (4 N - 2) & c (4 N) & c (8 N - 2) & c (8 N) & \dots & c (4 N^{2} - 2) & c (4 N^{2}) \end{matrix})$

Coefficient c(i,j,k,l) is in row (4N(j–1) + 4i + 2l + k – 6) of the vector c.

For 3-D systems,

$(\begin{matrix} c (1) & c (4) & c (7) & c (9 N + 1) & c (9 N + 4) & c (9 N + 7) & \dots & c (9 N (N - 1) + 1) & c (9 N (N - 1) + 4) & c (9 N (N - 1) + 7) \\ c (2) & c (5) & c (8) & c (9 N + 2) & c (9 N + 5) & c (9 N + 8) & \dots & c (9 N (N - 1) + 2) & c (9 N (N - 1) + 5) & c (9 N (N - 1) + 8) \\ c (3) & c (6) & c (9) & c (9 N + 3) & c (9 N + 6) & c (9 N + 9) & \dots & c (9 N (N - 1) + 3) & c (9 N (N - 1) + 6) & c (9 N (N - 1) + 9) \\ c (10) & c (13) & c (16) & c (9 N + 10) & c (9 N + 13) & c (9 N + 16) & \dots & c (9 N (N - 1) + 10) & c (9 N (N - 1) + 13) & c (9 N (N - 1) + 16) \\ c (11) & c (14) & c (17) & c (9 N + 11) & c (9 N + 14) & c (9 N + 17) & \dots & c (9 N (N - 1) + 11) & c (9 N (N - 1) + 14) & c (9 N (N - 1) + 17) \\ c (12) & c (15) & c (18) & c (9 N + 12) & c (9 N + 15) & c (9 N + 18) & \dots & c (9 N (N - 1) + 12) & c (9 N (N - 1) + 15) & c (9 N (N - 1) + 18) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ c (9 N - 8) & c (9 N - 5) & c (9 N - 2) & c (18 N - 8) & c (18 N - 5) & c (18 N - 2) & \dots & c (9 N^{2} - 8) & c (9 N^{2} - 5) & c (9 N^{2} - 2) \\ c (9 N - 7) & c (9 N - 4) & c (9 N - 1) & c (18 N - 7) & c (18 N - 4) & c (18 N - 1) & \dots & c (9 N^{2} - 7) & c (9 N^{2} - 4) & c (9 N^{2} - 1) \\ c (9 N - 6) & c (9 N - 3) & c (9 N) & c (18 N - 6) & c (18 N - 3) & c (18 N) & \dots & c (9 N^{2} - 6) & c (9 N^{2} - 3) & c (9 N^{2}) \end{matrix})$

Coefficient c(i,j,k,l) is in row (9N(j–1) + 9i + 3l + k – 12) of the vector c.

Some c Vectors Can Be Short

Often, your tensor c has structure, such as symmetric or block diagonal. In many cases, you can represent c using a smaller vector than one with 4N² components for 2-D or 9N² components for 3-D. The following sections give the possibilities.

2-D Systems
3-D Systems

2-D Systems

Scalar c, 2-D Systems
Two-Element Column Vector c, 2-D Systems
Three-Element Column Vector c, 2-D Systems
Four-Element Column Vector c, 2-D Systems
N-Element Column Vector c, 2-D Systems
2N-Element Column Vector c, 2-D Systems
3N-Element Column Vector c, 2-D Systems
4N-Element Column Vector c, 2-D Systems
2N(2N+1)/2-Element Column Vector c, 2-D Systems
4N²-Element Column Vector c, 2-D Systems

Scalar c, 2-D Systems. The software interprets a scalar c as a diagonal matrix, with c(i,i,1,1) and c(i,i,2,2) equal to the scalar, and all other entries 0.

$(\begin{matrix} c & 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & c & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & c & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & c \end{matrix})$

Two-Element Column Vector c, 2-D Systems. The software interprets a two-element column vector c as a diagonal matrix, with c(i,i,1,1) and c(i,i,2,2) as the two entries, and all other entries 0.

$(\begin{matrix} c (1) & 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & c (2) & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c (1) & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & c (2) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c (1) & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & c (2) \end{matrix})$

Three-Element Column Vector c, 2-D Systems. The software interprets a three-element column vector c as a symmetric block diagonal matrix, with c(i,i,1,1) = c(1), c(i,i,2,2) = c(3), and c(i,i,1,2) = c(i,i,2,1) = c(2).

$(\begin{matrix} c (1) & c (2) & 0 & 0 & \dots & 0 & 0 \\ c (2) & c (3) & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c (1) & c (2) & \dots & 0 & 0 \\ 0 & 0 & c (2) & c (3) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c (1) & c (2) \\ 0 & 0 & 0 & 0 & \dots & c (2) & c (3) \end{matrix})$

Four-Element Column Vector c, 2-D Systems. The software interprets a four-element column vector c as a block diagonal matrix.

$(\begin{matrix} c (1) & c (3) & 0 & 0 & \dots & 0 & 0 \\ c (2) & c (4) & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c (1) & c (3) & \dots & 0 & 0 \\ 0 & 0 & c (2) & c (4) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c (1) & c (3) \\ 0 & 0 & 0 & 0 & \dots & c (2) & c (4) \end{matrix})$

N-Element Column Vector c, 2-D Systems. The software interprets an N-element column vector c as a diagonal matrix.

$(\begin{matrix} c (1) & 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & c (1) & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c (2) & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & c (2) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c (N) & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & c (N) \end{matrix})$

Caution

If N = 2, 3, or 4, the 2-, 3-, or 4-element column vector form takes precedence over the N-element form. For example, if N = 3, and you have a c matrix of the form

$(\begin{matrix} c 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & c 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & c 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & c 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & c 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & c 3 \end{matrix})$

you cannot use the N-element form of c. Instead, you must use the 2N-element form. If you give c as the vector [c1;c2;c3], the software interprets c as a 3-element form:

$(\begin{matrix} c 1 & c 2 & 0 & 0 & 0 & 0 \\ c 2 & c 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & c 1 & c 2 & 0 & 0 \\ 0 & 0 & c 2 & c 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & c 1 & c 2 \\ 0 & 0 & 0 & 0 & c 2 & c 3 \end{matrix})$

Instead, use the 2N-element form [c1;c1;c2;c2;c3;c3].

2N-Element Column Vector c, 2-D Systems. The software interprets a 2N-element column vector c as a diagonal matrix.

$(\begin{matrix} c (1) & 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & c (2) & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c (3) & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & c (4) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c (2 N - 1) & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & c (2 N) \end{matrix})$

Caution

If N = 2, the 4-element form takes precedence over the 2N-element form. For example, if your c matrix is

$(\begin{matrix} c 1 & 0 & 0 & 0 \\ 0 & c 2 & 0 & 0 \\ 0 & 0 & c 3 & 0 \\ 0 & 0 & 0 & c 4 \end{matrix})$

you cannot give c as [c1;c2;c3;c4], because the software interprets this vector as the 4-element form

$(\begin{matrix} c 1 & c 3 & 0 & 0 \\ c 2 & c 4 & 0 & 0 \\ 0 & 0 & c 1 & c 3 \\ 0 & 0 & c 2 & c 4 \end{matrix})$

Instead, use the 3N-element form [c1;0;c2;c3;0;c4] or the 4N-element form [c1;0;0;c2;c3;0;0;c4].

3N-Element Column Vector c, 2-D Systems. The software interprets a 3N-element column vector c as a symmetric block diagonal matrix.

$(\begin{matrix} c (1) & c (2) & 0 & 0 & \dots & 0 & 0 \\ c (2) & c (3) & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c (4) & c (5) & \dots & 0 & 0 \\ 0 & 0 & c (5) & c (6) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c (3 N - 2) & c (3 N - 1) \\ 0 & 0 & 0 & 0 & \dots & c (3 N - 1) & c (3 N) \end{matrix})$

Coefficient c(i,j,k,l) is in row (3i + k + l – 4) of the vector c.

4N-Element Column Vector c, 2-D Systems. The software interprets a 4N-element column vector c as a block diagonal matrix.

$(\begin{matrix} c (1) & c (3) & 0 & 0 & \dots & 0 & 0 \\ c (2) & c (4) & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & c (5) & c (7) & \dots & 0 & 0 \\ 0 & 0 & c (6) & c (8) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & c (4 N - 3) & c (4 N - 1) \\ 0 & 0 & 0 & 0 & \dots & c (4 N - 2) & c (4 N) \end{matrix})$

Coefficient c(i,j,k,l) is in row (4i + 2l + k – 6) of the vector c.

2N(2N+1)/2-Element Column Vector c, 2-D Systems. The software interprets a 2N(2N+1)/2-element column vector c as a symmetric matrix. In the following diagram, • means the entry is symmetric.

$(\begin{matrix} c (1) & c (2) & c (4) & c (6) & \dots & c ((N - 1) (2 N - 1) + 1) & c ((N - 1) (2 N - 1) + 3) \\ • & c (3) & c (5) & c (7) & \dots & c ((N - 1) (2 N - 1) + 2) & c ((N - 1) (2 N - 1) + 4) \\ • & • & c (8) & c (9) & \dots & c ((N - 1) (2 N - 1) + 5) & c ((N - 1) (2 N - 1) + 7) \\ • & • & • & c (10) & \dots & c ((N - 1) (2 N - 1) + 6) & c ((N - 1) (2 N - 1) + 8) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ • & • & • & • & \dots & c (N (2 N + 1) - 2) & c (N (2 N + 1) - 1) \\ • & • & • & • & \dots & • & c (N (2 N + 1)) \end{matrix})$

Coefficient c(i,j,k,l), for i < j, is in row (2j² – 3j + 4i + 2l + k – 5) of the vector c. For i = j, coefficient c(i,j,k,l) is in row (2i² + i + l + k – 4) of the vector c.

4N²-Element Column Vector c, 2-D Systems. The software interprets a 4N²-element column vector c as a matrix.

Coefficient c(i,j,k,l) is in row (4N(j–1) + 4i + 2l + k – 6) of the vector c.

3-D Systems

Scalar c, 3-D Systems
Three-Element Column Vector c, 3-D Systems
Six-Element Column Vector c, 3-D Systems
Nine-Element Column Vector c, 3-D Systems
N-Element Column Vector c, 3-D Systems
3N-Element Column Vector c, 3-D Systems
6N-Element Column Vector c, 3-D Systems
9N-Element Column Vector c, 3-D Systems
3N(3N+1)/2-Element Column Vector c, 3-D Systems
9N²-Element Column Vector c, 3-D Systems

Scalar c, 3-D Systems. The software interprets a scalar c as a diagonal matrix, with c(i,i,1,1), c(i,i,2,2), and c(i,i,3,3) equal to the scalar, and all other entries 0.

$(\begin{matrix} c & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & c & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & c & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & c & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & c \end{matrix})$

Three-Element Column Vector c, 3-D Systems. The software interprets a three-element column vector c as a diagonal matrix, with c(i,i,1,1), c(i,i,2,2), and c(i,i,3,3) as the three entries, and all other entries 0.

$(\begin{matrix} c (1) & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & c (2) & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & c (3) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (1) & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c (2) & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c (3) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (1) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & c (2) & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & c (3) \end{matrix})$

Six-Element Column Vector c, 3-D Systems. The software interprets a six-element column vector c as a symmetric block diagonal matrix, with

c(i,i,1,1) = c(1)
c(i,i,2,2) = c(3)
c(i,i,1,2) = c(i,i,2,1) = c(2)
c(i,i,1,3) = c(i,i,3,1) = c(4)
c(i,i,2,3) = c(i,i,3,2) = c(5)
c(i,i,3,3) = c(6).

In the following diagram, • means the entry is symmetric.

$(\begin{matrix} c (1) & c (2) & c (4) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ • & c (3) & c (5) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ • & • & c (6) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (1) & c (2) & c (4) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & • & c (3) & c (5) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & • & • & c (6) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (1) & c (2) & c (4) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & • & c (3) & c (5) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & • & • & c (6) \end{matrix})$

Nine-Element Column Vector c, 3-D Systems. The software interprets a nine-element column vector c as a block diagonal matrix.

$(\begin{matrix} c (1) & c (4) & c (7) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ c (2) & c (5) & c (8) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ c (3) & c (6) & c (9) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (1) & c (4) & c (7) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (2) & c (5) & c (8) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (3) & c (6) & c (9) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (1) & c (4) & c (7) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (2) & c (5) & c (8) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (3) & c (6) & c (3) \end{matrix})$

N-Element Column Vector c, 3-D Systems. The software interprets an N-element column vector c as a diagonal matrix.

$(\begin{matrix} c (1) & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & c (1) & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & c (1) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (2) & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c (2) & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c (2) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (N) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & c (N) & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & c (N) \end{matrix})$

Caution

If N = 3, 6, or 9, the 3-, 6-, or 9-element column vector form takes precedence over the N-element form. For example, if N = 3, and you have a c matrix of the form

$(\begin{matrix} c (1) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & c (1) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c (1) & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c (2) & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c (2) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c (2) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c (3) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c (3) & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c (3) \end{matrix})$

you cannot use the N-element form of c. If you give c as the vector [c1;c2;c3], the software interprets c as a 3-element form:

$(\begin{matrix} c (1) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & c (2) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c (3) & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c (1) & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c (2) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c (3) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c (1) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c (2) & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c (3) \end{matrix})$

Instead, use one of these forms:

6N-element form — [c1;0;c1;0;0;c1;c2;0;c2;0;0;c2;c3;0;c3;0;0;c3]
9N-element form — [c1;0;0;0;c1;0;0;0;c1;c2;0;0;0;c2;0;0;0;c2;c3;0;0;0;c3;0;0;0;c3]

3N-Element Column Vector c, 3-D Systems. The software interprets a 3N-element column vector c as a diagonal matrix.

$(\begin{matrix} c (1) & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & c (2) & 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & c (3) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (4) & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c (5) & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c (6) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (3 N - 2) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & c (3 N - 1) & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0 & c (3 N) \end{matrix})$

Caution

If N = 3, the 9-element form takes precedence over the 3N-element form. For example, if your c matrix is

$(\begin{matrix} c (1) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & c (2) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c (3) & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c (4) & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c (5) & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c (6) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c (7) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c (8) & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c (9) \end{matrix})$

you cannot give c as [c1;c2;c3;c4;c5;c6;c7;c8;c9], because the software interprets this vector as the 9-element form

$(\begin{matrix} c (1) & c (4) & c (7) & 0 & 0 & 0 & 0 & 0 & 0 \\ c (2) & c (5) & c (8) & 0 & 0 & 0 & 0 & 0 & 0 \\ c (3) & c (6) & c (9) & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c (1) & c (4) & c (7) & 0 & 0 & 0 \\ 0 & 0 & 0 & c (2) & c (5) & c (8) & 0 & 0 & 0 \\ 0 & 0 & 0 & c (3) & c (6) & c (9) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c (1) & c (4) & c (7) \\ 0 & 0 & 0 & 0 & 0 & 0 & c (2) & c (5) & c (8) \\ 0 & 0 & 0 & 0 & 0 & 0 & c (3) & c (6) & c (3) \end{matrix})$

Instead, use one of these forms:

6N-element form — [c1;0;c2;0;0;c3;c4;0;c5;0;0;c6;c7;0;c8;0;0;c9]
9N-element form — [c1;0;0;0;c2;0;0;0;c3;c4;0;0;0;c5;0;0;0;c6;c7;0;0;0;c8;0;0;0;c9]

6N-Element Column Vector c, 3-D Systems. The software interprets a 6N-element column vector c as a symmetric block diagonal matrix. In the following diagram, • means the entry is symmetric.

$(\begin{matrix} c (1) & c (2) & c (4) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ • & c (3) & c (5) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ • & • & c (6) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (7) & c (8) & c (10) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & • & c (9) & c (11) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & • & • & c (12) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (6 N - 5) & c (6 N - 4) & c (6 N - 2) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & • & c (6 N - 3) & c (6 N - 1) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & • & • & c (6 N) \end{matrix})$

Coefficient c(i,j,k,l) is in row (6i + k + 1/2l(l–1) – 6) of the vector c.

9N-Element Column Vector c, 3-D Systems. The software interprets a 9N-element column vector c as a block diagonal matrix.

$(\begin{matrix} c (1) & c (4) & c (7) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ c (2) & c (5) & c (8) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ c (3) & c (6) & c (9) & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (10) & c (13) & c (16) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (11) & c (14) & c (17) & \dots & 0 & 0 & 0 \\ 0 & 0 & 0 & c (12) & c (15) & c (18) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (9 N - 8) & c (9 N - 5) & c (9 N - 2) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (9 N - 7) & c (9 N - 4) & c (9 N - 1) \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & c (9 N - 6) & c (9 N - 3) & c (9 N) \end{matrix})$

Coefficient c(i,j,k,l) is in row (9i + 3l + k – 12) of the vector c.

3N(3N+1)/2-Element Column Vector c, 3-D Systems. The software interprets a 3N(3N+1)/2-element column vector c as a symmetric matrix. In the following diagram, • means the entry is symmetric.

$(\begin{matrix} c (1) & c (2) & c (4) & c (7) & c (10) & c (13) & \dots & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 1 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 4 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 7 \\ • & c (3) & c (5) & c (8) & c (11) & c (14) & \dots & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 2 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 5 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 8 \\ • & • & c (6) & c (9) & c (12) & c (15) & \dots & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 3 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 6 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 9 \\ • & • & • & c (16) & c (17) & c (19) & \dots & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 10 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 13 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 16 \\ • & • & • & • & c (18) & c (20) & \dots & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 11 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 14 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 17 \\ • & • & • & • & • & c (21) & \dots & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 12 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 15 & c (3 (N - 1) (3 (N - 1) + 1) / 2 + 18 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ • & • & • & • & • & • & \dots & c (3 N (3 N + 1) / 2 - 5) & c (3 N (3 N + 1) / 2 - 4) & c (3 N (3 N + 1) / 2 - 2) \\ • & • & • & • & • & • & \dots & • & c (3 N (3 N + 1) / 2 - 3) & c (3 N (3 N + 1) / 2 - 1) \\ • & • & • & • & • & • & \dots & • & • & c (3 N (3 N + 1) / 2) \end{matrix})$

Coefficient c(i,j,k,l), for i < j, is in row (9(j–1)(j–2)/2 + 6(j–1) + 9i + 3l + k – 12) of the vector c. For i = j, coefficient c(i,j,k,l) is in row (9(i–1)(i–2)/2 + 15(i–1) + 1/2l(l–1) + k) of the vector c.

9N²-Element Column Vector c, 3-D Systems. The software interprets a 9N²-element column vector c as a matrix.

Coefficient c(i,j,k,l) is in row (9N(j–1) + 9i + 3l + k – 12) of the vector c.

Functional Form

If your c coefficient is not constant, represent it as a function of the form

ccoeffunction(location,state)

solvepde or solvepdeeig pass the location and state structures to ccoeffunction. The function must return a matrix of size N1-by-Nr, where:

N1 is the number of coefficients you pass to the solver. There are several possible values of N1, detailed in Some c Vectors Can Be Short. For 2-D geometry, 1 ≤ N1 ≤ 4N², and for 3-D geometry, 1 ≤ N1 ≤ 9N².
Nr is the number of points in the location that the solver passes. Nr is equal to the length of the location.x or any other location field. The function should evaluate c at these points.

Pass the coefficient to specifyCoefficients as a function handle, such as

specifyCoefficients(model,'c',@ccoeffunction,...)

location is a structure with these fields:
- location.x
- location.y
- location.z
- location.subdomain
The fields x, y, and z represent the x-, y-, and z- coordinates of points for which your function calculates coefficient values. The subdomain field represents the subdomain numbers, which currently apply only to 2-D models. The location fields are row vectors.
state is a structure with these fields:
- state.u
- state.ux
- state.uy
- state.uz
- state.time
The state.u field represents the current value of the solution u. The state.ux, state.uy, and state.uz fields are estimates of the solution’s partial derivatives (∂u/∂x, ∂u/∂y, and ∂u/∂z) at the corresponding points of the location structure. The solution and gradient estimates are N-by-Nr matrices. The state.time field is a scalar representing time for time-dependent models.

For example, suppose N = 3, and you have 2-D geometry. Suppose your c matrix is of the form

$c = [\begin{matrix} 1 & 2 \\ 2 & 8 \\ 1 + x^{2} + y^{2} & \frac{u (2)}{1 + u {(1)}^{2} + u {(3)}^{2}} \\ \frac{u (2)}{1 + u {(1)}^{2} + u {(3)}^{2}} & 1 + x^{2} + y^{2} \\ s_{1} (x, y) & - 1 \\ - 1 & s_{1} (x, y) \end{matrix}]$

where unlisted elements are zero. Here s₁(x,y) is 5 in subdomain 1, and is 10 in subdomain 2.

This c is a symmetric, block-diagonal matrix with different coefficients in each block. So it is natural to represent c as a 3N-Element Column Vector c, 2-D Systems:

For that form, the following function is appropriate.

function cmatrix = ccoeffunction(location,state)

n1 = 9;
nr = numel(location.x);
cmatrix = zeros(n1,nr);
cmatrix(1,:) = ones(1,nr);
cmatrix(2,:) = 2*ones(1,nr);
cmatrix(3,:) = 8*ones(1,nr);
cmatrix(4,:) = 1+location.x.^2 + location.y.^2;
cmatrix(5,:) = state.u(2,:)./(1 + state.u(1,:).^2 + state.u(3,:).^2);
cmatrix(6,:) = cmatrix(4,:);
cmatrix(7,:) = 5*location.subdomain;
cmatrix(8,:) = -ones(1,nr);
cmatrix(9,:) = cmatrix(7,:);

To include this function as your c coefficient, pass the function handle @ccoeffunction:

specifyCoefficients(model,'c',@ccoeffunction,...

Documentation