Euler's Method Tutorials

Agnishom · 2014-05-04 02:41:04

In this thread, we will be solving Differential Equations through the Euler's Method

bobbym · 2014-05-04 02:52:53

Hi;

You meant numerically solving them of course. We would start on an easy one like

anonimnystefy · 2014-05-04 02:53:28

http://www.math.upenn.edu/~wilf/DeturckWilf.pdf

Page 23.

bobbym · 2014-05-04 02:55:29

I have that book but I am not going to be using it here.

For the general first order DE:

y′ = f(x, y)

y(x0) = y0 and we want to find y(xn)

We need to define h, this will be easier with an example.

The page I am looking at was nice enough to provide code for the problem!

euler[f_,{x_,x0_,xn_},{y_,y0_},steps_]:=

Block[{
    xold=x0,
    yold=y0,
    sollist={{x0,y0}},
    x,y,h
  },

  h=N[(xn-x0)/steps];

  Do[ xnew=xold+h;
    ynew=yold+h*(f/.{x->xold,y->yold});
    sollist=Append[sollist,{xnew,ynew}];
    xold=xnew;
    yold=ynew,
    {steps}
  ];

  Return[sollist]
]

We will use it! I will give credit to the page at the end of the posting.

The example we will use is

with y(0) = 1.

Are you okay up to here?

Agnishom · 2014-05-04 04:06:03

That is very procedural.

Is y(0) = 1 a part of the problem?

bobbym · 2014-05-04 04:09:16

Yes, it is the initial condition. It says that when x = 0 that y is 1.

Yes it is procedural. Most people learn that paradigm first so they try to treat M that way.

Ready to use the proggie?

Agnishom · 2014-05-06 02:03:23

Please tell me what solving a DE means.

bobbym · 2014-05-06 02:08:33

I am sure the Topo boys will just shrug their shoulders and smirk to each other over this def but it is how I understand it.

When you solve an equation you find the values of the variable that make the equation true. When you solve a DE you find the functions that make the DE true.

Agnishom · 2014-05-07 01:00:08

I am running the code.

What should I key in as f_,{x_,x0_,xn_},{y_,y0_},steps_ ?

bobbym · 2014-05-07 01:04:30

euler[y, {x, 0, 1}, {y, 1}, 2]

Agnishom · 2014-05-07 01:09:26

The first argument is the derivative of the function? What are steps?

Agnishom · 2014-05-07 01:11:29

In[6]:= euler[y, {x, 0, 1}, {y, 1}, 2]

Out[6]= {{0, 1}, {0.5, 1.5}, {1., 2.25}}

What does this mean?

bobbym · 2014-05-07 01:17:50

You know that at x=0 (x0, the initial condition) that y = 1. That is the first list {0, 1}. That was easy, it is given in the parameters.

We want the y value at x = 1

The number of steps is 2, notice the program computed two y values.

{0.5, 1.5} the program thinks that at x = .5, y = 1.5

and

{1., 2.25} the program thinks that at x = 1, y = 2.25

Agnishom · 2014-05-07 01:26:35

Agnishom alone can think. Mathematica cannot.

In[5]:= euler[f_, {x_, x0_, xn_}, {y_, y0_}, steps_] := 
 Block[{xold = x0, yold = y0, sollist = {{x0, y0}}, x, y, h}, 
  h = N[(xn - x0)/steps];
  Do[xnew = xold + h;
   ynew = yold + h*(f /. {x -> xold, y -> yold});
   sollist = Append[sollist, {xnew, ynew}];
   xold = xnew;
   yold = ynew, {steps}];
  Return[sollist]]

In[6]:= euler[y, {x, 0, 1}, {y, 1}, 2]

Out[6]= {{0, 1}, {0.5, 1.5}, {1., 2.25}}

In[7]:= euler[y, {x, 0, 1}, {y, 1}, 3]

Out[7]= {{0, 1}, {0.333333, 1.33333}, {0.666667, 1.77778}, {1., 
  2.37037}}

In[10]:= euler[y, {x, 0, 1}, {y, 1}, 50]

Out[10]= {{0, 1}, {0.02, 1.02}, {0.04, 1.0404}, {0.06, 
  1.06121}, {0.08, 1.08243}, {0.1, 1.10408}, {0.12, 1.12616}, {0.14, 
  1.14869}, {0.16, 1.17166}, {0.18, 1.19509}, {0.2, 1.21899}, {0.22, 
  1.24337}, {0.24, 1.26824}, {0.26, 1.29361}, {0.28, 1.31948}, {0.3, 
  1.34587}, {0.32, 1.37279}, {0.34, 1.40024}, {0.36, 1.42825}, {0.38, 
  1.45681}, {0.4, 1.48595}, {0.42, 1.51567}, {0.44, 1.54598}, {0.46, 
  1.5769}, {0.48, 1.60844}, {0.5, 1.64061}, {0.52, 1.67342}, {0.54, 
  1.70689}, {0.56, 1.74102}, {0.58, 1.77584}, {0.6, 1.81136}, {0.62, 
  1.84759}, {0.64, 1.88454}, {0.66, 1.92223}, {0.68, 1.96068}, {0.7, 
  1.99989}, {0.72, 2.03989}, {0.74, 2.08069}, {0.76, 2.1223}, {0.78, 
  2.16474}, {0.8, 2.20804}, {0.82, 2.2522}, {0.84, 2.29724}, {0.86, 
  2.34319}, {0.88, 2.39005}, {0.9, 2.43785}, {0.92, 2.48661}, {0.94, 
  2.53634}, {0.96, 2.58707}, {0.98, 2.63881}, {1., 2.69159}}

In[9]:= euler[y, {x, 0, 1}, {y, 1}, 100]

Out[9]= {{0, 1}, {0.01, 1.01}, {0.02, 1.0201}, {0.03, 1.0303}, {0.04, 
  1.0406}, {0.05, 1.05101}, {0.06, 1.06152}, {0.07, 1.07214}, {0.08, 
  1.08286}, {0.09, 1.09369}, {0.1, 1.10462}, {0.11, 1.11567}, {0.12, 
  1.12683}, {0.13, 1.13809}, {0.14, 1.14947}, {0.15, 1.16097}, {0.16, 
  1.17258}, {0.17, 1.1843}, {0.18, 1.19615}, {0.19, 1.20811}, {0.2, 
  1.22019}, {0.21, 1.23239}, {0.22, 1.24472}, {0.23, 1.25716}, {0.24, 
  1.26973}, {0.25, 1.28243}, {0.26, 1.29526}, {0.27, 1.30821}, {0.28, 
  1.32129}, {0.29, 1.3345}, {0.3, 1.34785}, {0.31, 1.36133}, {0.32, 
  1.37494}, {0.33, 1.38869}, {0.34, 1.40258}, {0.35, 1.4166}, {0.36, 
  1.43077}, {0.37, 1.44508}, {0.38, 1.45953}, {0.39, 1.47412}, {0.4, 
  1.48886}, {0.41, 1.50375}, {0.42, 1.51879}, {0.43, 1.53398}, {0.44, 
  1.54932}, {0.45, 1.56481}, {0.46, 1.58046}, {0.47, 1.59626}, {0.48, 
  1.61223}, {0.49, 1.62835}, {0.5, 1.64463}, {0.51, 1.66108}, {0.52, 
  1.67769}, {0.53, 1.69447}, {0.54, 1.71141}, {0.55, 1.72852}, {0.56, 
  1.74581}, {0.57, 1.76327}, {0.58, 1.7809}, {0.59, 1.79871}, {0.6, 
  1.8167}, {0.61, 1.83486}, {0.62, 1.85321}, {0.63, 1.87174}, {0.64, 
  1.89046}, {0.65, 1.90937}, {0.66, 1.92846}, {0.67, 1.94774}, {0.68, 
  1.96722}, {0.69, 1.98689}, {0.7, 2.00676}, {0.71, 2.02683}, {0.72, 
  2.0471}, {0.73, 2.06757}, {0.74, 2.08825}, {0.75, 2.10913}, {0.76, 
  2.13022}, {0.77, 2.15152}, {0.78, 2.17304}, {0.79, 2.19477}, {0.8, 
  2.21672}, {0.81, 2.23888}, {0.82, 2.26127}, {0.83, 2.28388}, {0.84, 
  2.30672}, {0.85, 2.32979}, {0.86, 2.35309}, {0.87, 2.37662}, {0.88, 
  2.40038}, {0.89, 2.42439}, {0.9, 2.44863}, {0.91, 2.47312}, {0.92, 
  2.49785}, {0.93, 2.52283}, {0.94, 2.54806}, {0.95, 2.57354}, {0.96, 
  2.59927}, {0.97, 2.62527}, {0.98, 2.65152}, {0.99, 2.67803}, {1., 
  2.70481}}

In[11]:= euler[y, {x, 0, 1}, {y, 1}, 200]

Out[11]= {{0, 1}, {0.005, 1.005}, {0.01, 1.01003}, {0.015, 
  1.01508}, {0.02, 1.02015}, {0.025, 1.02525}, {0.03, 
  1.03038}, {0.035, 1.03553}, {0.04, 1.04071}, {0.045, 
  1.04591}, {0.05, 1.05114}, {0.055, 1.0564}, {0.06, 1.06168}, {0.065,
   1.06699}, {0.07, 1.07232}, {0.075, 1.07768}, {0.08, 
  1.08307}, {0.085, 1.08849}, {0.09, 1.09393}, {0.095, 1.0994}, {0.1, 
  1.1049}, {0.105, 1.11042}, {0.11, 1.11597}, {0.115, 1.12155}, {0.12,
   1.12716}, {0.125, 1.1328}, {0.13, 1.13846}, {0.135, 
  1.14415}, {0.14, 1.14987}, {0.145, 1.15562}, {0.15, 1.1614}, {0.155,
   1.16721}, {0.16, 1.17304}, {0.165, 1.17891}, {0.17, 
  1.1848}, {0.175, 1.19073}, {0.18, 1.19668}, {0.185, 1.20266}, {0.19,
   1.20868}, {0.195, 1.21472}, {0.2, 1.22079}, {0.205, 1.2269}, {0.21,
   1.23303}, {0.215, 1.2392}, {0.22, 1.24539}, {0.225, 
  1.25162}, {0.23, 1.25788}, {0.235, 1.26417}, {0.24, 
  1.27049}, {0.245, 1.27684}, {0.25, 1.28323}, {0.255, 
  1.28964}, {0.26, 1.29609}, {0.265, 1.30257}, {0.27, 
  1.30908}, {0.275, 1.31563}, {0.28, 1.32221}, {0.285, 
  1.32882}, {0.29, 1.33546}, {0.295, 1.34214}, {0.3, 1.34885}, {0.305,
   1.35559}, {0.31, 1.36237}, {0.315, 1.36918}, {0.32, 
  1.37603}, {0.325, 1.38291}, {0.33, 1.38982}, {0.335, 
  1.39677}, {0.34, 1.40376}, {0.345, 1.41078}, {0.35, 
  1.41783}, {0.355, 1.42492}, {0.36, 1.43204}, {0.365, 1.4392}, {0.37,
   1.4464}, {0.375, 1.45363}, {0.38, 1.4609}, {0.385, 1.46821}, {0.39,
   1.47555}, {0.395, 1.48292}, {0.4, 1.49034}, {0.405, 
  1.49779}, {0.41, 1.50528}, {0.415, 1.51281}, {0.42, 
  1.52037}, {0.425, 1.52797}, {0.43, 1.53561}, {0.435, 
  1.54329}, {0.44, 1.55101}, {0.445, 1.55876}, {0.45, 
  1.56655}, {0.455, 1.57439}, {0.46, 1.58226}, {0.465, 
  1.59017}, {0.47, 1.59812}, {0.475, 1.60611}, {0.48, 
  1.61414}, {0.485, 1.62221}, {0.49, 1.63032}, {0.495, 1.63848}, {0.5,
   1.64667}, {0.505, 1.6549}, {0.51, 1.66318}, {0.515, 
  1.67149}, {0.52, 1.67985}, {0.525, 1.68825}, {0.53, 
  1.69669}, {0.535, 1.70517}, {0.54, 1.7137}, {0.545, 1.72227}, {0.55,
   1.73088}, {0.555, 1.73953}, {0.56, 1.74823}, {0.565, 
  1.75697}, {0.57, 1.76576}, {0.575, 1.77459}, {0.58, 
  1.78346}, {0.585, 1.79238}, {0.59, 1.80134}, {0.595, 1.81035}, {0.6,
   1.8194}, {0.605, 1.82849}, {0.61, 1.83764}, {0.615, 
  1.84682}, {0.62, 1.85606}, {0.625, 1.86534}, {0.63, 
  1.87467}, {0.635, 1.88404}, {0.64, 1.89346}, {0.645, 
  1.90293}, {0.65, 1.91244}, {0.655, 1.922}, {0.66, 1.93161}, {0.665, 
  1.94127}, {0.67, 1.95098}, {0.675, 1.96073}, {0.68, 
  1.97054}, {0.685, 1.98039}, {0.69, 1.99029}, {0.695, 2.00024}, {0.7,
   2.01024}, {0.705, 2.02029}, {0.71, 2.0304}, {0.715, 
  2.04055}, {0.72, 2.05075}, {0.725, 2.061}, {0.73, 2.07131}, {0.735, 
  2.08167}, {0.74, 2.09207}, {0.745, 2.10253}, {0.75, 
  2.11305}, {0.755, 2.12361}, {0.76, 2.13423}, {0.765, 2.1449}, {0.77,
   2.15563}, {0.775, 2.1664}, {0.78, 2.17724}, {0.785, 
  2.18812}, {0.79, 2.19906}, {0.795, 2.21006}, {0.8, 2.22111}, {0.805,
   2.23221}, {0.81, 2.24338}, {0.815, 2.25459}, {0.82, 
  2.26587}, {0.825, 2.27719}, {0.83, 2.28858}, {0.835, 
  2.30002}, {0.84, 2.31152}, {0.845, 2.32308}, {0.85, 2.3347}, {0.855,
   2.34637}, {0.86, 2.3581}, {0.865, 2.36989}, {0.87, 
  2.38174}, {0.875, 2.39365}, {0.88, 2.40562}, {0.885, 
  2.41765}, {0.89, 2.42974}, {0.895, 2.44188}, {0.9, 2.45409}, {0.905,
   2.46636}, {0.91, 2.4787}, {0.915, 2.49109}, {0.92, 
  2.50354}, {0.925, 2.51606}, {0.93, 2.52864}, {0.935, 
  2.54129}, {0.94, 2.55399}, {0.945, 2.56676}, {0.95, 2.5796}, {0.955,
   2.59249}, {0.96, 2.60546}, {0.965, 2.61848}, {0.97, 
  2.63158}, {0.975, 2.64473}, {0.98, 2.65796}, {0.985, 
  2.67125}, {0.99, 2.6846}, {0.995, 2.69803}, {1., 2.71152}}

Are you sure it will converge?

bobbym · 2014-05-07 01:31:33

Sometimes we use the phrase thinks in that manner. It means, that is what it got as an answer. It is customary to proceed with a calculation like this with powers of 2.

euler[y, {x, 0, 1}, {y, 1}, 2]

euler[y, {x, 0, 1}, {y, 1}, 4]

euler[y, {x, 0, 1}, {y, 1}, 8]
.
.
.
euler[y, {x, 0, 1}, {y, 1}, 2^14]//Last

Are you sure it will converge?

Surely, you are joking.

Now here is where Brad comes in!

Agnishom · 2014-05-07 01:47:18

Who's Brad?

Its slow convergence.

bobbym · 2014-05-07 01:49:00

Recognize that number do you, hmmm?

Agnishom · 2014-05-07 01:52:18

In[14]:= N[E, 10]

Out[14]= 2.718281828

bobbym · 2014-05-07 01:55:04

So, what do you think it is?

Agnishom · 2014-05-07 01:59:23

Euler's number

bobbym · 2014-05-07 02:02:44

Agnishom wrote:

Who's Brad?

You are Brad. The new and improved Brad. You have jumped out of the box.

http://www.mathisfunforum.com/viewtopic … 08#p302808

We can solve that DE analytically quite easily but this is what M gets:

DSolve[{y'[x] == y[x], y[0] == 1}, y[x], x]

Agnishom · 2014-05-07 02:23:45

I will run the code a little later, but it seems that f(x) = e^x

I stil do not get it

bobbym · 2014-05-07 02:26:42

Get what? The numerical solution? The analytical one?

Agnishom · 2014-05-07 02:31:33

I do not get the whole thing about Brad.

bobbym · 2014-05-07 02:34:16

That is another thread and another whole concept.

Do you understand what we have done so far concerning the numerical solution of a DE by Euler's method?

Math Is Fun Forum

#1 2014-05-04 02:41:04

Euler's Method Tutorials

#2 2014-05-04 02:52:53

Re: Euler's Method Tutorials

#3 2014-05-04 02:53:28

Re: Euler's Method Tutorials

#4 2014-05-04 02:55:29

Re: Euler's Method Tutorials

#5 2014-05-04 04:06:03

Re: Euler's Method Tutorials

#6 2014-05-04 04:09:16

Re: Euler's Method Tutorials

#7 2014-05-06 02:03:23

Re: Euler's Method Tutorials

#8 2014-05-06 02:08:33

Re: Euler's Method Tutorials

#9 2014-05-07 01:00:08

Re: Euler's Method Tutorials

#10 2014-05-07 01:04:30

Re: Euler's Method Tutorials

#11 2014-05-07 01:09:26

Re: Euler's Method Tutorials

#12 2014-05-07 01:11:29

Re: Euler's Method Tutorials

#13 2014-05-07 01:17:50

Re: Euler's Method Tutorials

#14 2014-05-07 01:26:35

Re: Euler's Method Tutorials

#15 2014-05-07 01:31:33

Re: Euler's Method Tutorials

#16 2014-05-07 01:47:18

Re: Euler's Method Tutorials

#17 2014-05-07 01:49:00

Re: Euler's Method Tutorials

#18 2014-05-07 01:52:18

Re: Euler's Method Tutorials

#19 2014-05-07 01:55:04

Re: Euler's Method Tutorials

#20 2014-05-07 01:59:23

Re: Euler's Method Tutorials

#21 2014-05-07 02:02:44

Re: Euler's Method Tutorials

#22 2014-05-07 02:23:45

Re: Euler's Method Tutorials

#23 2014-05-07 02:26:42

Re: Euler's Method Tutorials

#24 2014-05-07 02:31:33

Re: Euler's Method Tutorials

#25 2014-05-07 02:34:16

Re: Euler's Method Tutorials

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