HamCall (October 1991)/HamCall (Whitehall Publishing)(1991).bin/math/calculas/readme.ode

Text File | 1989-02-17 | 6KB | 138 lines

                        ORDINARY DIFFERENTIAL EQUATIONS

This slide show consists of various examples from ODEs.

When viewing the slides, the following keys are operational:

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These keys do NOT operate like that while you are reading this document.

A.  One parameter family of curves

                   2      2
     The function X  + c/X   is sketched for  c = 1, .5, .25, 0, -.25, -.5, -1.

B.  The US Population and logistic growth
     The population of the US from 1790 is frequently modeled using logistic
growth.  Here these data are shown (in millions) for the period 1790 - 1950 in
ten year intervals.  Using 1790, 1850, and 1910 as exact values the appropriate
logistic curve is drawn.  Notice how well it fits the data points between 1790
and 1910, and how well it predicted the population until 1950 (with a minor
perturbation in 1940 - was anything happening then to cause the population to
drop?)  Then the actual population from 1950 to 1980 is added to the first
slide, and the same three data points are used, and the logistic curve is again
overlaid.  Surprise!
     For completeness here is the population of the US in millions from 1790 to
1980, in ten year intervals, taken from the World Almanac.

          1790    3.929
          1800    5.308
          1810    7.240
          1820    9.638
          1830   12.861
          1840   17.063
          1850   23.192
          1860   31.443
          1870   38.558
          1880   50.189
          1890   62.980
          1900   76.212
          1910   92.228
          1920  106.022
          1930  123.203
          1940  132.165
          1950  151.326
          1960  179.324
          1970  203.302
          1980  226.549

C.  The cooling of coffee
     The temperature of a cup of coffee was recorded every minute for 14
minutes, when the ambient temperature was 22°C.  The first slide shows this.
Assuming Newton's law of cooling governs this process, the experimental results
at 2 minutes, 14 minutes, and the ambient temperature were used to generate the
theoretical curve in slide 2.  Then a least squares fit was done on the
experimental results at 2, 4, 6, 8, 10, 12, and 14 minutes, which were then used
to produce the theoretical curve in slide 3.
     For completeness here is the coffee temp in degrees C, taken at one minute
intervals (taken from Computer Simulation Methods, by Gould and Tobochnik,
Addison-Wesley 1987).

          Time    Temp

            1     77.7
            2     75.1
            3     73.0
            4     71.1
            5     69.4
            6     67.8
            7     66.4
            8     64.7
            9     63.4
           10     62.1
           11     61.0
           12     59.9
           13     58.7
           14     57.8

D.  Numerical Methods - Euler
     This deals with solving the logistic equation  y' = 10y(1-y) subject to the
initial condition y(0) = .1 in the region  0 < x < 10.  First the exact solution
is displayed - notice that as x gets large the solution goes to 1.  Then the
numerical solution that would be obtained by using Euler's method is
superimposed for different values of "h" (.18, .23, .25, .3).  A bit worrying,
isn't it!  The idea for this demonstration comes from Fundamentals Of
Differential Equations, by Nagle and Saff, Benjamin/Cummings 1989.
     Finally a slide is shown that explains what is behind this.  It shows a
plot of "h" versus the numerical solutions from h = .15 to h = .3. The idea for
this demonstration comes from Chet Weiss, an undergraduate at the University of
Arizona.

E.  Numerical Methods - Runge Kutta 4
     This deals with solving the logistic equation  y' = 10y(1-y) subject to the
initial condition y(0) = .1 in the region  0 < x < 10.  First the exact solution
is displayed - notice that as x gets large the solution goes to 1.  Then the
numerical solution that would be obtained by using the Runga Kutta 4 method is
superimposed for different values of "h" (.25, .3, .325. .35. .3675, .38).  A
bit worrying, isn't it!  The idea for this demonstration comes from Fundamentals
Of Differential Equations, by Nagle and Saff, Benjamin/Cummings 1989.
     Finally a slide is shown that explains what is behind this.  It shows a
plot of "h" versus the numerical solutions from h = .2 to h = .4.  The idea for
this demonstration comes from Chet Weiss, an undergraduate at the University of
Arizona.

F.  Damped free vibrations
     The solutions of the differential equation
                              X" + 2aX' + 64X = 0
subject to the initial conditions
                             X(0) = 1,  X'(0) = 0,
are displayed for various values of the constant a.  Critical damping occurs at
a = 8.

G.  Series solution
     This is associated with the series solution of the differential equation
xy" + y' + xy = 0 subject to the initial condition y(0) = 1.  The idea is to
build the function from the series.  First we show the 1st term of the series
expansion, then the 1st and 2nd terms, then the 2nd and 3rd terms, and so on.
Can you see the function being created? In fact, you are looking at Jo(x), the
Bessel function of the first kind of type 0, which is overlaid on the final
slide.

H.  Bessel Function
     This graphs Jo(x), the Bessel function of the first kind of type 0, in the
interval 0 < x < 8, and then overlays it with the polynomials
     1
     1 - x^2/4
     1 - x^2/4 + (x^2/4)^2/(2!)^2
     1 - x^2/4 + (x^2/4)^2/(2!)^2 - (x^2/4)^3/(3!)^2
     1 - x^2/4 + (x^2/4)^2/(2!)^2 - (x^2/4)^3/(3!)^2 + (x^2/4)^4/(4!)^2
and so on, down to
                  ... - (x^2/4)^7/(7!)^2
The radius of convergence of the Taylor series is infinity.

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