Usbrth

Is it possible to ask for a hotel room without minibar/extra services?

Two different pronunciation of "понял"

How to colour the US map with Yellow, Green, Red and Blue to minimize the number of states with the colour of Green

How many spell slots should a Fighter 11/Ranger 9 have?

Working around an AWS network ACL rule limit

Stars Make Stars

Strange behaviour of Check

Classification of bundles, Postnikov towers, obstruction theory, local coefficients

Can smartphones with the same camera sensor have different image quality?

Direct Experience of Meditation

Active filter with series inductor and resistor - do these exist?

Estimate capacitor parameters

Using "nakedly" instead of "with nothing on"

What is the largest species of polychaete?

Why is "Captain Marvel" translated as male in Portugal?

Array/tabular for long multiplication

How to politely respond to generic emails requesting a PhD/job in my lab? Without wasting too much time

The following signatures were invalid: EXPKEYSIG 1397BC53640DB551

Windows 10: How to Lock (not sleep) laptop on lid close?

Unable to start mainnet node docker container

Why does tar appear to skip file contents when output file is /dev/null?

Antler Helmet: Can it work?

Keep going mode for require-package

Why don't the Weasley twins use magic outside of school if the Trace can only find the location of spells cast?

Adams-Bashforth algorithm

0

$begingroup$

So i got a project and I've been trying to solve it, but I just don't know how to start, and I haven't found an example similar to this one. So if you could guide me a little bit I would really appreciate it.

Giving the Cauchy
$$y'=-y^2$$ $$y(0)=1$$

which has the exact solution $y(x)=1/(1+x)$. Using the Adams-Bashforth procedure for $h=0.5$ and $0.25$ solve the problem for the following interval $[0,5]$.

I didn't learned this algorithm in school and online I can't find similar examples, so if you could guide me a bit it would really help me!

algorithms numerical-calculus

share|cite|improve this question

edited Apr 8 at 19:40

Jonas

415312

asked Apr 8 at 19:23

AlexandraAlexandra

33

New contributor

Alexandra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is your task here? Shall you implement the algorithm in a programming language? Or compute the first couple of steps by hand?
  $endgroup$
  – Jonas
  Apr 8 at 19:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have to implement it in MATLAB.
  $endgroup$
  – Alexandra
  Apr 8 at 19:39
  

  
  
  
  

add a comment | 

0

$begingroup$

So i got a project and I've been trying to solve it, but I just don't know how to start, and I haven't found an example similar to this one. So if you could guide me a little bit I would really appreciate it.

Giving the Cauchy
$$y'=-y^2$$ $$y(0)=1$$

which has the exact solution $y(x)=1/(1+x)$. Using the Adams-Bashforth procedure for $h=0.5$ and $0.25$ solve the problem for the following interval $[0,5]$.

I didn't learned this algorithm in school and online I can't find similar examples, so if you could guide me a bit it would really help me!

algorithms numerical-calculus

share|cite|improve this question

edited Apr 8 at 19:40

Jonas

415312

asked Apr 8 at 19:23

AlexandraAlexandra

33

New contributor

Alexandra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is your task here? Shall you implement the algorithm in a programming language? Or compute the first couple of steps by hand?
  $endgroup$
  – Jonas
  Apr 8 at 19:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have to implement it in MATLAB.
  $endgroup$
  – Alexandra
  Apr 8 at 19:39
  

  
  
  
  

add a comment | 

$begingroup$

So i got a project and I've been trying to solve it, but I just don't know how to start, and I haven't found an example similar to this one. So if you could guide me a little bit I would really appreciate it.

Giving the Cauchy
$$y'=-y^2$$ $$y(0)=1$$

which has the exact solution $y(x)=1/(1+x)$. Using the Adams-Bashforth procedure for $h=0.5$ and $0.25$ solve the problem for the following interval $[0,5]$.

I didn't learned this algorithm in school and online I can't find similar examples, so if you could guide me a bit it would really help me!

algorithms numerical-calculus

share|cite|improve this question

edited Apr 8 at 19:40

Jonas

415312

asked Apr 8 at 19:23

AlexandraAlexandra

33

New contributor

Alexandra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Apr 8 at 19:40

Jonas

415312

asked Apr 8 at 19:23

AlexandraAlexandra

33

New contributor

Alexandra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Apr 8 at 19:40

Jonas

415312

asked Apr 8 at 19:23

AlexandraAlexandra

33

New contributor

Alexandra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited Apr 8 at 19:40

Jonas

415312

edited Apr 8 at 19:40

Jonas

415312

asked Apr 8 at 19:23

AlexandraAlexandra

33

New contributor

Alexandra is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Apr 8 at 19:23

AlexandraAlexandra

33

1 Answer
1

active

oldest

votes

0

$begingroup$

You just need to write the recurrence. For example, if you look at AB method with $s=2$, the numerical scheme is the following:

$y_n+2=y_n+1-frack2f(t_n,y_n) +frac3k2f(t_n+1,y_n+1)$.

It's a second order method, and you can compute $y_1$ ($y_0$ is of course given, and it's $y_0=1$, since it's the initial data!) with explicit Euler method.

In your specific example, $f$ is given by $f(t,y)=f(y)=-y^2$, and $k$ is your step size $h$.

It's just a for loop where you update at each time-step $y_n+1,y_n$.

share|cite|improve this answer

answered Apr 8 at 19:46

VoBVoB

801513

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  function [y t] = abm4(f,a,b,ya,n) h = (b - a) / n; h24 = h / 24; y(1,:) = ya; t(1) = a; m = min(3,n); for i = 1 : m t(i+1) = t(i) + h; s(i,:) = f(t(i), y(i,:)); s2 = f(t(i) + h / 2, y(i,:) + s(i,:) * h /2); s3 = f(t(i) + h / 2, y(i,:) + s2 * h /2); s4 = f(t(i+1), y(i,:) + s3 * h); y(i+1,:) = y(i,:) + (s(i,:) + s2+s2 + s3+s3 + s4) * h / 6; end; for i = m + 1 : n s(i,:) = f(t(i), y(i,:)); y(i+1,:) = y(i,:) + (55 * s(i,:) - 59 * s(i-1,:) + 37 * s(i-2,:) - 9 * s(i-3,:)) * h24; t(i+1) = t(i) + h; y(i+1,:) = y(i,:) + (9 * f(t(i+1), y(i+1,:)) + 19 * s(i,:) - 5 * s(i-1,:) + s(i-2,:)) * h24; end;
  $endgroup$
  – Alexandra
  Apr 9 at 6:35
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

Alexandra is a new contributor. Be nice, and check out our Code of Conduct.

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3180086%2fadams-bashforth-algorithm%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

0

$begingroup$

You just need to write the recurrence. For example, if you look at AB method with $s=2$, the numerical scheme is the following:

$y_n+2=y_n+1-frack2f(t_n,y_n) +frac3k2f(t_n+1,y_n+1)$.

It's a second order method, and you can compute $y_1$ ($y_0$ is of course given, and it's $y_0=1$, since it's the initial data!) with explicit Euler method.

In your specific example, $f$ is given by $f(t,y)=f(y)=-y^2$, and $k$ is your step size $h$.

It's just a for loop where you update at each time-step $y_n+1,y_n$.

share|cite|improve this answer

answered Apr 8 at 19:46

VoBVoB

801513

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  function [y t] = abm4(f,a,b,ya,n) h = (b - a) / n; h24 = h / 24; y(1,:) = ya; t(1) = a; m = min(3,n); for i = 1 : m t(i+1) = t(i) + h; s(i,:) = f(t(i), y(i,:)); s2 = f(t(i) + h / 2, y(i,:) + s(i,:) * h /2); s3 = f(t(i) + h / 2, y(i,:) + s2 * h /2); s4 = f(t(i+1), y(i,:) + s3 * h); y(i+1,:) = y(i,:) + (s(i,:) + s2+s2 + s3+s3 + s4) * h / 6; end; for i = m + 1 : n s(i,:) = f(t(i), y(i,:)); y(i+1,:) = y(i,:) + (55 * s(i,:) - 59 * s(i-1,:) + 37 * s(i-2,:) - 9 * s(i-3,:)) * h24; t(i+1) = t(i) + h; y(i+1,:) = y(i,:) + (9 * f(t(i+1), y(i+1,:)) + 19 * s(i,:) - 5 * s(i-1,:) + s(i-2,:)) * h24; end;
  $endgroup$
  – Alexandra
  Apr 9 at 6:35
  

  
  
  
  

add a comment | 

0

$begingroup$

You just need to write the recurrence. For example, if you look at AB method with $s=2$, the numerical scheme is the following:

$y_n+2=y_n+1-frack2f(t_n,y_n) +frac3k2f(t_n+1,y_n+1)$.

It's a second order method, and you can compute $y_1$ ($y_0$ is of course given, and it's $y_0=1$, since it's the initial data!) with explicit Euler method.

In your specific example, $f$ is given by $f(t,y)=f(y)=-y^2$, and $k$ is your step size $h$.

It's just a for loop where you update at each time-step $y_n+1,y_n$.

share|cite|improve this answer

answered Apr 8 at 19:46

VoBVoB

801513

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  function [y t] = abm4(f,a,b,ya,n) h = (b - a) / n; h24 = h / 24; y(1,:) = ya; t(1) = a; m = min(3,n); for i = 1 : m t(i+1) = t(i) + h; s(i,:) = f(t(i), y(i,:)); s2 = f(t(i) + h / 2, y(i,:) + s(i,:) * h /2); s3 = f(t(i) + h / 2, y(i,:) + s2 * h /2); s4 = f(t(i+1), y(i,:) + s3 * h); y(i+1,:) = y(i,:) + (s(i,:) + s2+s2 + s3+s3 + s4) * h / 6; end; for i = m + 1 : n s(i,:) = f(t(i), y(i,:)); y(i+1,:) = y(i,:) + (55 * s(i,:) - 59 * s(i-1,:) + 37 * s(i-2,:) - 9 * s(i-3,:)) * h24; t(i+1) = t(i) + h; y(i+1,:) = y(i,:) + (9 * f(t(i+1), y(i+1,:)) + 19 * s(i,:) - 5 * s(i-1,:) + s(i-2,:)) * h24; end;
  $endgroup$
  – Alexandra
  Apr 9 at 6:35
  

  
  
  
  

add a comment | 

$begingroup$

You just need to write the recurrence. For example, if you look at AB method with $s=2$, the numerical scheme is the following:

$y_n+2=y_n+1-frack2f(t_n,y_n) +frac3k2f(t_n+1,y_n+1)$.

It's a second order method, and you can compute $y_1$ ($y_0$ is of course given, and it's $y_0=1$, since it's the initial data!) with explicit Euler method.

In your specific example, $f$ is given by $f(t,y)=f(y)=-y^2$, and $k$ is your step size $h$.

It's just a for loop where you update at each time-step $y_n+1,y_n$.

share|cite|improve this answer

answered Apr 8 at 19:46

VoBVoB

801513

$endgroup$

share|cite|improve this answer

answered Apr 8 at 19:46

VoBVoB

801513

answered Apr 8 at 19:46

VoBVoB

801513

answered Apr 8 at 19:46

VoBVoB

801513