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There are two main ways of solving differential equations on a computer
which are very different from both the technical and the philosophical
view points: symbolic solutions and numerical approximations.
A symbolic solution is possible whenever there exists an expression of
the solution by means of elementary functions and their composition. On
the other hand, we have numerical approximations of the solution, this
method is more general but does not provide a formula for the solution
it is just able to compute the solution at given points, outputing hence
a vector as a solution.
As an an example let us start with the following equation
y'(t) = t + y(t)
with initial condition
y(t0) = y0, for
some given t0.
Symbolic solution with dsolve
To compute the symbolic solution, with all the embelishments, we will use
and M-file. (Remember that you should write an M-file as soon as you find
it useful, sometimes it can be useful even if you don't find it so. Typing
in commands from the prompt to do projects in MATLAB is a very poor and
highly inefficient practice.)
The solution file is already written and can be found in symbolic_ode.m.
In this case dsolve is the main command.
Since we are dealing with symbolics, the output of dsolve is an expression
(the output is actually produced by Maple, not by MATLAB).
To transform the expression into something more manageable by MATLAB
we must first vectorize the expression
(to put "dots" where Maple doesn't do it) and then use the inline
command to produce a MATLAB function. Notice that the inline command constructs
a function of three arguments: one is the independent variable t
and the other two are the parameters given by the initial condition, namely
t0
and y0.
Once the solution family is given as a MATLAB function we can exploit
it for plots, tabulates, etc.
Numerical integration
MATLAB version 5.3 has a very efficient family of numerical solvers for
differential equations. The most simple example is given by int, which
is just the definite integral of a function. (Calculating the integral
of a function f can be viewed as the solution of the differential equation
y' = f(x).) For example, let us compute the (definite) integral of the
Gaussian distribution f(x) = e-x^2, between
the points -4 and 4. To do this we first must define the integrand function
with an M-file or with and inline command
f = inline('exp(-x.^2)','x')
and then approximate the definite integral using the quadrature rule quad:
quad('f',-4,4)
Numerical
solution of ODE (Ordinary Differential Equations)
Before we illustrate the MATLAB technique for approximating numerically
the solution of differential equations, let us make a small
Exercise Modify the M-file symbolic_ode.m
to solve the differential equation
y'(t) = t + |y(t)|2
with initial condition
y(0) = 1.
What is the outcome of such a modification? MATLAB is unable to compute
the solution in terms of elementary functions. This is a classical example
(of striking simpicity) of a differential equation whose solution is transcendental.
This example gives not only legitimacy, but shows that it is necessary,
to look at numerical algorithms in order to get understand the solution
of certain (most of them) equations.
Please download the M-file numeric_ode.m which
contains the script to solve the above equation.
Observations:
-
the maxt parameter in the M-file is important. Try to see what
happens if maxt is pushed all the way to a value of 1. (Modify the M-file
by doing an appropriate for loop.)
-
controlling the axes is important to visualize the "interesting" features
of the solution, a bad choice of axes scaling might hide interesting stuff.
-
the syntax of ode45 is delicate, you should always consult the helpdesk
while using it. Because:
... YOU NEVER END UP LEARNING MATLAB.
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