MME 523

Dec 2,2003

 

Numerical Solutions of Differential Equations

Euler’s Method

 

Scope:

            We are interested in solving first order differential equations with intial conditions. This means a problem of the form

 

 

                        dy/dt =  f(y,t)             y(0) = y0  (given)

 

 

            This in turn means that we know the slope of the tangent line at any location, and that we know where we began.  We would like to be able to approximately predict where we will be at any time, t.

 

Comment:

            This is not the same as  numerical integration (Trapezoidal Rule etc) because the variable y may also appear on the right side.

 

Interpretation:

            we are given the initial position as well as the velocity at all times and wish to determine the location at any time.  Useful if you happen to be launching rockets at terrorists, or water balloons from rooftops.

 

Euler’s Method

 

            pick an increment in time,  Dt

            assume, as Newton did, that for a brief time the curve is straight and its slope is therefore given by

 

                                    f(y0,0)

 

            which means that Dt units of time later the y value will be approximately

 

                                    y1 = y0 + f(y0,0) Dt

 

            At this location   (y1, Dt)  play the same game again assuming a constant slope and getting a new y value at         t = 2 Dt  of

                                    y2 = y1 + f(y1, Dt) Dt

 

            This process may be continued indefinitely iteratively generating a sequence of points of the form

 

                                    yi+1 = yi +  f(yi, Dt) Dt                for i=0,1,2, . . . .

 

Your job

 

            work this into an Excel spreadsheet and apply it  to the following problems:

 

 

            Problem 1:

 

                        Estimate the solution to

 

                                    dy/dt = 1 – t + 4y               y(0) =1

 

                        using  Dt = .05   providing y values up to  t = 2.0  seconds.

 

                       

                        Optional: can you get Excel to plot the estimated solution??

 

            Problem 2:

 

                        Revise your work from Problem 1 by reducing   Dt  to  .01

 

            Problem 3:

 

                        Revise your work from Problem 1 by reducing   Dt  to  .001

 

           

            At this point let me know and I will provide you with the exact solution so you may compare results. Specifically work up a spreadsheet with the estimates from Problems 1,2 and 3 in columns for t=.1, .2, .3,…

2.0  seconds  with an additional column showing the exact values  Hence there should be 5 columns of data.

 

Please attempt the following:  Can you make any statement as to how the error after 2 seconds depends on Dt  ??

In other words, as Dt  decreases, how does the error decrease?

 

You may need to add additional, error, columns to your spreadsheet to do this.

 

 

 

            Problem 4:

                        Use Euler’s Method to estimate the solution  for t between 0 and 3 seconds for the problem

 

                        dy/dt = (t + y)1/2     y(0) = 3

 

                        taking  Dt = .025