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):
<Text-field style="Heading 2" layout="Heading 2">Euler's method</Text-field> 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 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 Lets try and solve the equation dsolve({diff(x(t),t)=x(t)^2-t^2,x(0)=1/2},x(t)); Ly1JInhHNiI2I0kidEdGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2KDJGJyIiISwkKihGJyIiIiwmKipJI1BpR0YqRjAsJiomLUkmR0FNTUFHNiRGKkkoX3N5c2xpYkdGJTYjIyIiJCIiJSIiI0Y+I0YwRj5GPkYzRjBGMEY2ISIjLUkoQmVzc2VsSUdGODYkIyEiJEY9LCQqJEYnRj5GP0YwI0YwRj0tSShCZXNzZWxLR0Y4NiRGO0ZGISIiRjAsJioqRjNGMEY0RjBGNkZALUZCNiRGSEZGRjBGSC1GSkZQRjBGTEZML0YnRi1GPzJGLUYnLCQqKEYnRjAsJioqRjNGMCwmRjVGQEYzRjBGMEY2RkBGQUYwI0ZMRj1GSUZMRjAsJioqRjNGMEZYRjBGNkZARk9GMEZZRlFGMEZMRkw= 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=-3..3,x=-4..4): display(DF); 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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.
<Text-field style="Heading 3" layout="Heading 3">Euler's Method</Text-field> 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; Zio2J0kiZkc2IkkjdDBHRiVJI3gwR0YlSSJoR0YlSSJOR0YlNiZJInRHRiVJInhHRiVJIm5HRiVJJHNvbEdGJUYlRiVDJj44JC1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjOSU+OCUtRjM2IzkmPjgnNyM3JEYxRjg/KDgmIiIiRkI5KEkldHJ1ZUdGNEMlPkY4LCZGOEZCKiY5J0ZCLTkkNiRGMUY4RkJGQj5GMSwmRjFGQkZJRkI+Rj03JC1JI29wR0Y0NiNGPUY/RiVGJUYl
t0:=0;tf:=3;x0:=1/2; IiIh IiIk IyIiIiIiIw== N:=3; IiIk h:=evalf((tf-t0)/N); JCIiIiIiIQ== f:=(t,x)->x^2-t^2; Zio2JEkidEc2IkkieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDklIiIjIiIiKiQ5JEYtISIiRiVGJUYl out:=FEuler(f,t0,x0,h,N); NyY3JCQiIiFGJSQiKysrKytdISM1NyQkIiIiRiUkIisrKysrdkYoNyQkIiIjRiUkIisrKytESkYoNyQkIiIkRiUkIStdUCUpKmUkISIq EulerApprox:=plot(out,color=green,thickness=2): display(EulerApprox,DF); 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