# Documentation

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## Specify Scalar PDE Coefficients in Character Form

 Note:   THIS PAGE DESCRIBES THE LEGACY WORKFLOW. New features might not be compatible with the legacy workflow. For the corresponding step in the recommended workflow, see the recommended examples on the PDE Coefficients page.

Write a text expression using these conventions:

• `'x'`x-coordinate

• `'y'`y-coordinate

• `'z'`z-coordinate (3-D geometry)

• `'u'` — Solution of equation

• `'ux'` — Derivative of u in the x-direction

• `'uy'` — Derivative of u in the y-direction

• `'uz'` — Derivative of u in the z-direction (3-D geometry)

• `'t'` — Time (parabolic and hyperbolic equations)

• `'sd'` — Subdomain number (not used in 3-D geometry)

For example, you could use this vector of characters to represent a coefficient:

`'(x + y)./(x.^2 + y.^2 + 1) + 3 + sin(t)./(1 + u.^4)'`
 Note:   Use `.*`, `./`, and `.^` for multiplication, division, and exponentiation operations. The text expressions operate on row vectors, so the operations must make sense for row vectors. For 2-D geometry, the row vectors are the values at the triangle centroids in the mesh.

You can write MATLAB® functions for coefficients as well as plain text expressions. For example, suppose your coefficient `f` is given by the file `fcoeff.m`:

```function f = fcoeff(x,y,t,sd) f = (x.*y)./(1 + x.^2 + y.^2); % f on subdomain 1 f = f + log(1 + t); % include time r = (sd == 2); % subdomain 2 f2 = cos(x + y); % coefficient on subdomain 2 f(r) = f2(r); % f on subdomain 2```

Represent this function in the `parabolic` solver, for example:

`u1 = parabolic(u0,tlist,b,p,e,t,c,a,'fcoeff(x,y,t,sd)',d)`
 Caution   In function form, `t` represents triangles, and `time` represents time. In character form, `t` represents time, and triangles do not enter into the form.

There is a simple way to write a text expression for multiple subdomains without using `'sd'` or a function. Separate the formulas for the different subdomains with the `'!'` character. Generally use the same number of expressions as subdomains. However, if an expression does not depend on the subdomain number, you can give just one expression.

For example, an expression for an input (`a`, `c`, `f`, or `d`) with three subdomains:

```'2 + tanh(x.*y)!cosh(x)./(1 + x.^2 + y.^2)!x.^2 + y.^2'```

The coefficient c is a 2-by-2 matrix. You can give c in any of the following forms:

• Scalar or single vector of characters — The software interprets c as a diagonal matrix:

`$\left(\begin{array}{cc}c& 0\\ 0& c\end{array}\right)$`
• Two-element column vector or two-row text array — The software interprets c as a diagonal matrix:

`$\left(\begin{array}{cc}c\left(1\right)& 0\\ 0& c\left(2\right)\end{array}\right)$`
• Three-element column vector or three-row text array — The software interprets c as a symmetric matrix:

`$\left(\begin{array}{cc}c\left(1\right)& c\left(2\right)\\ c\left(2\right)& c\left(3\right)\end{array}\right)$`
• Four-element column vector or four-row text array — The software interprets c as a full matrix:

`$\left(\begin{array}{cc}c\left(1\right)& c\left(3\right)\\ c\left(2\right)& c\left(4\right)\end{array}\right)$`

For example, c as a symmetric matrix with `cos(xy)` on the off-diagonal terms:

```c = char('x.^2+y.^2',... 'cos(x.*y)',... 'u./(1+x.^2+y.^2)')```

To include subdomains separated by `'!'`, include the `'!'` in each row. For example,

```c = char('1 + x.^2 + y.^2!x.^2 + y.^2',... 'cos(x.*y)!sin(x.*y)',... 'u./(1 + x.^2 + y.^2)!u.*(x.^2 + y.^2)')```
 Caution   Do not use spaces when specifying coefficients in the PDE app. The parser can misinterpret a space as a vector separator, as when a MATLAB vector uses a space to separate elements of a vector.

For elliptic problems, when you include `'u'`, `'ux'`, `'uy'`, or `'uz'`, you must use the `pdenonlin` solver instead of `assempde`. In the PDE app, select Solve > Parameters > Use nonlinear solver.

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