MTN Volume 2 Number 1, Spring 1995

Maple Functionality

Tips for Maple Users and Programmers
M.B. Monagan

This issue's column addresses the following topics:

• How to use collect to perform controlled transformations of collected expressions.
• How to plot and integrate a function which is defined as a root of an equation.
• How assume really works.
• Automatic simplifications in Maple.

Solving Linear Differential Equations in Maple
G. Labahn

This article describes some of the major improvements to Maple's abilities in finding closed form solutions of linear differential equations that were included in Maple V Release 3. These improvements include the addition of two decision procedures for determining the existence of certain types of solutions for both the linear and Ricatti type of equations. In addition, Maple now has the ability to return partial solutions in the case where not all solutions can be found in a closed form. Other improvements such as the use of RootOf in dsolve along with an improved version of reduction of order are also included in the article.

Algorithms for Indefinite Summation of Rational Functions in Maple
R. Pirastu

This article gives an informal introduction to the problem of indefinite summation of rational functions and describes by examples several algorithms for solving it in a computer algebra system like Maple. The presentation wants to be more "pictorial" than formal, since the problem hides an intuitive combinatorial structure. In particular the algorithm of Moenck, on which the Maple implementation is based and which showed some gaps, is repaired. On the other hand, the author suggests to use a modification of Abramov's algorithm which admits a more efficient implementation in Maple.

Maple Applications

Source Code Generation for Linear and Non-linear Numerical Models using Maple
K.W. Ayotte

The article demonstrates a method of automated source code generation for the solution of coupled partial differential equations. In this context the method is used to generate the source code for a model of turbulent atmospheric flow over gentle terrain.

Population Genetics: Estimation of Distributions Through Systems of Non-linear Differential Equations
N.E. Abrouk, R.J. Lopez

This article uses Maple to solve a coupled set of nonlinear ordinary differential equations that arise in population genetics. The equation governing the density function for the population gene frequency is converted to an infinite set of differential equations by a moment technique. The set is rendered finite by A Hankel truncation, and solved with the ODE package of the Share Library. From these approximate moments the underlying probability measure of the gene frequency is then computed, and the approach to steady state is studied numerically and graphically.

Calculation of the Potential Distribution for a Three-Layer Spherical Volume Conductor
Z. Shung-Ren, J. Grotendorst, H. Halling

This article considers a three-layer spherical volume conductor model and calculate the dipole-induced potential by analytical methods. This is an interdisciplinary work in the field of Electroencephalography (EEG).

Maple in Education

Numerical, Graphical and Symbolic Analysis of Bernoulli Equations
D.B. Meade

This article is based on a homework problem for a sophomore-level course in differential equations. The original problem is very routine; in particular, it does not call for much thought on the part of the student. The article contains an illustration of how Maple can be used to transform this exercise into an experience which demonstrates the complementary roles of numerical, graphical, and theoretical techniques in the analysis of differential equations. This is one example of the curricular modifications resulting from the author's project with the Lilly Endowment Teaching Fellows Program.

EM Field Modeling using Maple
P. Mohseni

The article describes a simplified electrostatic problem and how Maple is used to model it, solve it and analyze the solution. Both numerical and graphical results are presented. Similar methods can be used to model more complicated problems in electromagnetics. A "self-discovery" path is taken to show the value of Maple as a useful tool in education.

Identification of Diffusion Coefficients in the Heat Equation using Maple
P.W. Hammer

This article outlines a specific use for Maple in an undergraduate mathematics course in partial differential equations. A Newton-type algorithm is developed for the identification of an unknown thermal diffusion coefficient in the heat equation based on given data. With Maple in use, the development and application of the algorithm provide challenging yet manageable assignments for the students.