The document discusses a presentation given by Dr. V.Gnanaraj on mathematical modeling using Maple. It provides an overview of Maple, a computer algebra system, and how it allows users to perform both numerical and analytical computations. It also describes the basics of working in Maple, including document organization in worksheets, syntax, solving equations, functions, and packages for tasks like differential equations, graph theory, and optimization.

1. MATHEMATICAL MODELING WITH MAPLE
8/4/2015 SFR COLLEGE PRESENTATION
By
Dr. V.GNANARAJ
ASSOCIATE PROFESSOR IN MATHEMATICS
THIAGARAJAR COLLEGE OF ENGINEERING
MADURAI
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Mathematical Modeling with Maple
• Computer algebra systems in scientific computing
• Maple – an interactive system, allowing one to carry out both
numerical and analytical computations. Maple contains 2D and 3D
graphical means, powerful programming language and an extensive
library of mathematical formulae
• A convenient working method, when all stages of mathematical,
physical or engineering research can be stored in a single document
- worksheet
• The worksheet automatically becomes a scientific article, report,
chapter in a book, etc.

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Document organization in Maple
• One should familiarize oneself with Maple GUI
• The best way is to learn use and syntax of Maple simultaneously
and by examples
• A worksheet (*.mws) consists of logical groups called „execution
groups“ Each group consists of several parts :
- Input Region
- Output Region
- Text Region (e.g. comments)
• The principal objects in Maple: numbers, constants, names
- numbers: integer, real, rational, radicals
- constants: Pi, I, E, Infinity, gamma, true, false
- names: sequences of symbols starting with literal (e.g. new, New,
new_old, etc.)

4. Maple Startup Menu
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5. Maple Worksheet with input and output
Maple Input
Maple Output
Maple Execution
Group
Maple Prompt
Time for
executing the
command
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6. Maple Help Menu
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7. Syntax and expressions
• After typing a command, you should inform the computer that the
command has ended and must be accepted. There are two types of
terminators: ; and : The colon is needed to suppress Return – the
output is not printed
• The symbol % is called “ditto”, it is used to refer to previous return.
Twofold ditto (%%) refers to the second last return, etc
• Expressions are the main object for the most of commands.
Expressions are composed of constants, variables, signs of
arithmetical and logical operations
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8. Typical syntax errors and remedies
• Forgetting a bracket or an operator – the error message comes after
Return (sometimes it appears meaningless)
• It is easier to type the brackets and some operators first, and then
insert functions, variables and parameters, e.g.
> ( )*(sqrt( ) + ( )^( ) )* exp( );
• It is better to start a new worksheet with the command restart –
this resets the system and variables used in a previous worksheet
do not cast their value into the new one
• Not all combinations of symbols are allowed in Maple (e.g. Pi and
I are already used by the system
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9. Solving equations with Maple
• Two different methods of finding roots:
1) analytical (symbolic); 2)numerical
• Maple provides two respective built-in functions:
1) solve; 2) fsolve
• Analytical approach: the equation is manipulated up to the form
when the unknown is expressed in terms of the knowns – solve
operates in this way. The variety of equations to be analytically
solved is determined by the set of techniques available in Maple
• Each new Maple version – new commands and packages.
Multipurpose commands of the xsolve type: dsolve for ODE,
pdsolve for PDE, rsolve for difference equations
• New libraries: DEtools, PDEtools, LREtools. Special packages
for implicit solutions: DESol, RESol (difference), RootOf (algebra)
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10. Analytical solutions
• The solve command – multipurpose, can be applied to systems of
algebraic equations, inequalities, functional equations, identities.
The form: solve (eqn, var), where eqn is an equation or a system
of equations, var is a variable or a group of variables. If the latter
parameter is absent, solve will look for solutions with respect to
all symbolic variables
• The system of equations and the group of variables are given as
sets (in curly brackets and separated by commas)
• Example: > s:=solve({6*x+y=8, 3*x-2*y=4},{x,y});
• If the equation is written without the equality sign it is viewed as
one part of the equality, the other being considered zero:
> sl:=solve(x^7+8*x,x);
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11. • Many equations encountered in science and engineering are hard
to treat analytically – they have to be solved numerically
• To solve differential equations numerically with the help of dsolve
command one must declare the parameter numeric. This can be
done by two manners:
a) dsolve(ODE, var, type=numeric, method)
b) dsolve(ODE, var, numeric, method)
• Equations to be solved and initial (or boundary) conditions are
defined as a set; var are lists of unknown functions; method
parameters define the numerical techniques invoked to solve ODE.
By default, the method is understood as Runge-Kutta-Fehlberg 4/5
(rfk45)
• Also by default, as a result of applying dsolve with the parameter
numeric a procedure is created to compute specific values
Numerical treatment of equations in Maple
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12.  s:=D(x)(t)=y(t),D(y)(t)=-a*y(t)-.8*sin(x(t));
 ic:=x(0)=0,y(0)=2;
 F:=dsolve({s,ic},{x(t),y(t)},numeric);
 a:=.2;evalf(F(1.5),5);
>plots[odeplot](F,[[t,x(t)],[t,y(t)]],0..30,labels=[t,"x,y"],axes=boxed);
• To visualize numerical solutions of this Cauchy problem, one may
use the graphics commands from the DEtools package
• The obtained solution may serve as a source of all the necessary
information about the system
An example of numerical solution of ODE
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13. Functions in Maple
 f := x^2;
 g := x -> 2 * x^3 * f;
 g(x); g(2);
 f := x -> x^2;
 g := 2 * (x -> x^3) * f;
 g(x); g(2);
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14. Suppose we want to evaluate our mathematical
function at a point, say at 1.
 f := x^2 - 3*x-10;
 g := x -> x^2-3*x-10;
 subs( x=1, f );
 eval( f, x=1 );
 g(1);
 f(1);
 subs( x=1, g );
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15. • eval( f, x=1 ); subs( x=1, f );
• Think of reading eval( f, x=1) as " evaluate f
at x=1" and think of reading subs (x=1,f) as "
substitute x=1 into f".
• factor( f );
• factor( g(x) );
• factor( f (x) );
• factor( g );
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16.  diff( f, x );
 D( g );
• diff command needed a reference to x in it but the D
command did not.
 D( f );
 diff( g, x );
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17. Let us do an example of combining two mathematical functions f and g by
composing them to make a new function .
For the expression:
 f := x^2 + 3*x;
 g := x + 1;
 h := subs( x=g, f );
 subs(x=1, h);
))(()( xgfxh 
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18. Example: representing a mathematical
function of two variables.
• f := (x^2+y^2)/(x+x*y);
• g := (x,y) -> (x^2+y^2)/(x+x*y);
• subs( x=1, y=2, f );
• eval( f, {x=1, y=2} );
• g(1,2);
• simplify( f );
• simplify( g(x,y) );
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19. Here is how we compute partial derivatives of
the expression.
• diff( f, x );
• simplify( % );
• diff( f, y );
• simplify( % );
• D[1](g);
• simplify( %(x,y) );
• D[2](g);
• simplify( %(x,y) );
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22. Sl.No Package Name Commands/ Tools
1 CodeGeneration tools for translating Maple code to other languages
2 CurveFitting commands that support curve-fitting
3 DEtools tools for manipulating, solving, and plotting systems
of differential
equations
4 GraphTheory collction of routines for creating, drawing,
manipulating, and testing graphs
5 finance commands for financial computations
6 IntegrationTools tools used for manipulation of integrals
7 LinearAlgebra commands for manipulating Matrices and Vectors
8 IntegrationTools tools used for manipulation of integrals
9 numtheory commands for classic number theory
10 Optimization commands for numerically solving optimization
theory problems
11 PDEtools tools for solving partial differential equations
12 Statistics tools for mathematical statistics and data analysis
Some of the Maple Packages
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Differential Package
Graph Theory Package
Optimization Package
EXPLORING THE PACKAGES

24. Thank you
for your
attention