Math 2650 A. J. Meir Copyright (C) A. J. Meir. All rights reserved. This worksheet is for educational use only. No part of this publication may be reproduced or transmitted for profit in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system without prior written permission from the author. Not for profit distribution of the software is allowed without prior written permission, providing that the worksheet is not modified in any way and full credit to the author is acknowledged. restart:with(DEtools):with(plots): Warning, the name changecoords has been redefined Euler's method We now look at a very simple numerical method for approximating solutions of differential equations. It is essentially based on the direction field, and an approximation of the derivative as a difference quotient. Consider the equation NiMvKiYlI2R4RyIiIiUjZHRHISIiLCYqJCUieEciIiNGJiokJSJ0R0YsRig= NiMvLSUieEc2IyIiISomIiIiRikiIiMhIiI= Lets try and solve the equation dsolve({diff(x(t),t)=x(t)^2-t^2,x(0)=1/2},x(t)); NiMvLUkieEc2IjYjSSJ0R0YmLCQqKEYoIiIiLCYqKkkjUGlHSSpwcm90ZWN0ZWRHRi9GKywmKiYtSSZHQU1NQUc2JEYvSShfc3lzbGliR0YmNiMjIiIkIiIlIiIjRjojRitGOiEiI0YuRitGK0YyRjwtSShCZXNzZWxJR0Y0NiQjISIkRjksJCokRihGOkY7RisjISIiRjktSShCZXNzZWxLR0Y0NiRGN0ZCRkVGKywmKipGLkYrRjBGK0YyRjwtRj42JCNGK0Y5RkJGK0ZELUZHRkxGK0ZFRkU= This is a mess so lets look at the direction field to get some insight into the problem DF:=DEplot(diff(x(t),t)=x(t)^2-t^2,x(t),
t=-1..3,x=-4..4): display(DF); 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Fe\`l-%*THICKNESSG6#"""-%+AXESLABELSG6$Q"t6"Q%x(t)F_]`l-%%VIEWG6$;$!2-++++++?"!#;$"#KFd\`l;$!#WFd\`l$"#WFd\`l Lets try and use Euler's method to construct an approximate solution Below we write a little Maple procedure for the forward Euler's method. Euler's Method FEuler:=proc(f,t0,x0,h,N)
local t,x,n,sol:
t:=evalf(t0):
x:=evalf(x0):
sol:=[[t,x]]:
for n from 1 to N do
x:=x+h*f(t,x):
t:=t+h:
sol:=[op(sol),[t,x]]:
od;
end; NiM+SSdGRXVsZXJHNiJmKjYnSSJmR0YlSSN0MEdGJUkjeDBHRiVJImhHRiVJIk5HRiU2JkkidEdGJUkieEdGJUkibkdGJUkkc29sR0YlRiVGJUMmPjgkLUkmZXZhbGZHSSpwcm90ZWN0ZWRHRjc2IzklPjglLUY2NiM5Jj44JzcjNyRGNEY7Pyg4JiIiIkZFOShJJXRydWVHRjdDJT5GOywmRjtGRSomOSdGRS05JDYkRjRGO0ZFRkU+RjQsJkY0RkVGTEZFPkZANyQtSSNvcEdGNzYjRkBGQkYlRiVGJQ== t0:=0;tf:=3;x0:=1/2; NiM+SSN0MEc2IiIiIQ== NiM+SSN0Zkc2IiIiJA== NiM+SSN4MEc2IiMiIiIiIiM= N:=3; NiM+SSJORzYiIiIk h:=evalf((tf-t0)/N); NiM+SSJoRzYiJCIiIiIiIQ== f:=(t,x)->x^2-t^2; NiM+SSJmRzYiZio2JEkidEdGJUkieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDklIiIjIiIiKiQ5JEYwISIiRiVGJUYl out:=FEuler(f,t0,x0,h,N); NiM+SSRvdXRHNiI3JjckJCIiIUYpJCIrKysrK10hIzU3JCQiIiJGKSQiKysrKyt2Riw3JCQiIiNGKSQiKysrK0RKRiw3JCQiIiRGKSQhK11QJSkqZSQhIio= EulerApprox:=plot(out,color=green,thickness=2): display(EulerApprox,DF); 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We can also use Maple's built in numerical o.d.e. "solvers", but then we may not have precise control over the step size. ans:=dsolve({diff(x(t),t)=x(t)^2-t^2,x(0)=1/2},x(t),numeric,method=classical[foreuler],stepsize=h); 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 ans:=dsolve({diff(x(t),t)=x(t)^2-t^2,x(0)=1/2},x(t),numeric,method=classical[foreuler],stepsize=h); 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 approx:=odeplot(ans,[t,x(t)],0..3,color=orange,
thickness=2): display(DF,approx); 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