Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362552 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.1108 1213432 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.1108 1213432 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1830 4200168 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1844 4227864 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1852 4245576 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1854 4276096 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1864 4298816 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1864 4298816 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2357 4984608 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2395 5176760 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2401 5197688 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2515 5414088 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2516 5414344 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Adams - Bashforth Methods

The Euler (one step method) method only uses values atto computeWe can get better estimates forif we make use of some of the previous values.

Ifthen

Using the Euler method givesfor step sizeIf we weight the values ofand then

We findandsuch that the above is exact for linear functions.

Putrespectively.

We solve the simultaneous equations

(1)

(2)

Sub (1) into (2) andcancels to givethen from (1)

Euler's equation becomesThis is a two step method.

We can also derive a three step method.

Putrespectively.

We solve the simultaneous equations

(3)

(4)

(5)

Rearrange (4) to giveand substitute (3) to give(6)

Rearrange (5) to giveand substitute (5) and (6) to give(7)

(7)-(6) givesthen from (6)and finally from (3)

Then we have

An alternative method of finding the coefficients exists using Lagrange polynomials.is the Lagrange polynomial such thatthen