The problem descriptor file sets up the problem to be solved. Each file is composed of several sections that send information to the PDEase2D math engine. Sections are separated by predefined, reserved words in brackets, much like Windows INI files. Sections can include titles, coordinates, variables, definitions, initial values, equations, constraints, boundaries, and plots. Once a problem's description is set up, clicking on a single button in the MFE sends the appropriate commands to the PDEase2D math engine.

PDEase2D's features make solving PDEs deceptively simple. Notebooks make it easy to organize, share, and view both the input and output for any problem to be solved. Navigation within individual files and between different notebooks is facilitated by several MFE tools. PDEase2D automatically generates appropriate graphics when a series of PDEs is solved, but it is also simple to display, select, and plot a table of data using the MFE's DataView feature.

Figure 2

PDEase2D solves systems of linear and nonlinear partial differential equations by using the finite-element method. It is a companion product to Macsyma Inc.'s eponymous Macsyma software package. While Macsyma is useful for symbolic solutions to partial differential equations, PDEase2D enables users to solve a system of PDEs numerically.

The finite-element method is one strategy for numerically solving partial differential equations, and it is especially suited for regions that are irregular in shape. This method can be reduced to several essential steps :(1) find a functional that describes the partial differential equation, (2) subdivide elements into smaller regions, (3) interpolate to find values within each region, (4) assemble the element equations to get a system of equations, (5) adjust for boundary conditions, and (6) finally solve. PDEase2D automatically handles all of these steps to output approximate solutions to this series of PDEs.

One obstacle encountered when solving PDEs using the finite-element method is to determine how precisely to subdivide the irregular region into smaller regions. This grid must be sufficiently fine to come to an approximation within the limits of error set by the user, but computing time is greatly increased as smaller regions are used. PDEase2D automatically solves this problem with a fully automated adaptive grid refinement approach. PDEase2D starts with a coarse grid of triangular regions, and iteratively refines this grid until an acceptable solution is reached. Because it only uses fine gridding in the regions that require this (e.g., where the error limit exceeds the preset values) PDEase2D optimizes speed and accurate results when it converges on a solution.

Partial differential equations (PDEs) can be used to describe mathematically a variety of physical, mechanical, and chemical processes. For example, the process of diffusion is dependent on a number of factors. In addition to the properties of the diffusing molecule itself, other factors such as changing concentration with distance and other macromolecules that may bind (and therefore impede diffusion of the molecule being studied) must be considered. In a generic way, there are multiple processes that are linked or coupled, and the behavior of one may depend on the state of another. Partial differential equations really refer to equations that involve two or more independent variables that may affect the behavior of a dependent variable; PDEs are necessary to consider all of these processes simultaneously. Dr. Michael Stern describes the use of PDEase2D to study the diffusion of calcium in single muscle cells in this issue's Software Solutions column.

The MFE, or Macsyma front end, is your interface to both the Macsyma and PDEase2D math engines. It has the look and feel of a notebook, and is essentially an online document that contains all of the necessary information to execute a command in the math engine. More than one notebook file can be displayed in the MFE, and users can easily copy and paste between notebooks or Microsoft Word or PowerPoint. Individual notebooks' attributes can be controlled by opening the Notebook Options dialog box from the File menu. Other features such as font style, height of graphics, and DataViewer displays can also be changed quickly with a few mouse clicks and keystrokes.

The MFE is an application independent of Macsyma and PDEase2D. It can be launched without connecting to the math engines by selecting the MFE program item/shortcut. Specific templates for either Macsyma or PDEase2D can be opened from within the MFE. The displayed menus are dependent on the template type utilized as well as the current section selected within a Macsyma or PDEase2D notebook. For example, different menu items and toolbar buttons are displayed if one selects a graphic section or a data section in a particular notebook.

Figure 3

The online documentation is quite impressive. Users will learn not only how to use PDEase2D and Macsyma, but also learn general principles of mathematics. In addition to the standard help file, which provides comprehensive information, Macsyma Inc. has provided a number of ways to access this information. MathHelp is available for both Macsyma and PDEase2D via a browser window that makes it easy to navigate through a complex hierarchy of information. Demo files can be accessed through the help menus - important when learning to use PDEase2D. Users can search for PDEase2D sample problems by topic, keywords, or category of physical problem. These notebook files are then automatically opened in the MFE and can be modified to suit specific problems.