This chapter describes the PDE Modeler tool, which is used to solve spatially two-dimensional partial differential equations (PDE). It begins with a description of the standard forms of PDEs and its initial and boundary conditions that the tool uses. It is shown how various PDEs and boundary conditions can be represented in standard forms. Applications to the mechanics and tribology are presented in the final part of the chapter. They illustrate the use of PDE Modeler to solve the Reynolds equation describing the hydrodynamic lubrication, to implement the mechanical stress modeler application for a plate with an elliptical hole, to solve the transient heat equation with temperature-dependent material properties, and to study vibration of a rectangular membrane.
TopIntroduction
The pdepe command described in the preceding chapter solves one-dimensional PDEs only. To solve two-dimensional and some simplified object geometries of three-dimensional PDEs, the special PDE Modeler tool (formerly known as the PDE Tool) is designed. The modeler is part of the Partial Differential Equation ToolboxTM. Further, we study two-dimensional PDEs only. The chapter described the steps that should be used to solve 2D PDEs with various boundary and initial conditions and presents solutions for some examples and applications from the M&T fields. Among them are
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Reynolds equation describing the hydrodynamic lubrication of surfaces with hemispherical pores;
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The “Mechanical Stress” PDE Modeler option applied to a thin plate with an elliptical hole;
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The transient heat equation with temperature-dependent coefficients describing the material properties;
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The wave equation adopted to the problem of vibration of a rectangular membrane.
TopThe PDE toolbox provides the PDE Modeler which represents an interface for solving elliptic, parabolic, hyperbolic and eigenvalue types of scalar PDEs with Dirichlet or Neumann boundary conditions. The general standard forms of the PDE-types and boundary conditions (BC) that can be solved with this modeler, as well as examples of PDEs with its coefficients matching the standard form PDE or BC are given in Table 1.
Table 1. The standard forms of PDEs and BCs, designed to be solved using PDE Modeler
| Standard Form of the PDE or BC | PDE or BC Name | Example |
| PDE and His Name | Variables and Coefficients to Match Standard Form |
 | Elliptic equation, when m=0, d=0. Parabolic equation, when m=0.
Hyperbolic equation, when d=0. |  +s -
parabolic PDE | u=T,
m=0,
a=0,
f=s,
c=k,
d=1. |
or
 | Eigenvalue PDE. | -∇(∇u)=λu, | m=0,
c=1,
a=λ,
d=0. |
| hu=r | Dirichlet boundary:
the u value is given. | T=0
| T=u,
h=1,
r=0. |
 | Neumann boundary: given
the du/dx, du/dy values with/without the u values. | 
| T=u,
c=1,
q=0,
g=0 |