## Documentation Center |

Partial Differential Equation Toolbox™ software solves problems
in two space dimensions and time, whereas reality has three space
dimensions. The reduction to 2-D is possible when variations in the
third space dimension (taken to be *z*) can be accounted
for in the 2-D equation. In some cases, like the plane stress analysis,
the material parameters must be modified in the process of dimensionality
reduction.

When the problem is such that variation with *z* is
negligible, all *z*-derivatives drop out and the
2-D equation has exactly the same units and coefficients as in 3-D.

Slab geometries are treated by integration through the thickness.
The result is a 2-D equation for the *z*-averaged
solution with the thickness, say *D*(*x*,*y*),
multiplied onto all the PDE coefficients, *c*, *a*, *d*,
and *f*, etc. For instance, if you want to compute
the stresses in a sheet welded together from plates of different thickness,
multiply Young's modulus *E*, volume forces, and
specified surface tractions by *D*(*x*,*y*),
Similar definitions of the equation coefficients are called for in
other slab geometry examples and application modes.

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