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: 0.0000 362248 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0425 1211680 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0425 1211680 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1866 4039424 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1881 4067216 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1889 4084928 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1889 4101936 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1898 4109672 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1898 4109672 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2345 4353648 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2360 4371048 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2366 4391976 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2463 4464752 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2464 4465008 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Finding Perturbations Expansions to Algebraic Equations

If a system is perturbed by an amountwe can assume a solution of the formand solve for the coefficientsto obtain the solution.

Example: Find a perturbation expansion for the solution to

Substituteto obtain

Grouping together coefficients offor eachgives

Then

Then

(1) can be solved exactly to giveand

The Taylor expansion of the first coincides with the perturbation expansion ifbut the second root isand is not present in the unperturbed equationso cannot be given by the perturbed equation. We can find a perturbation expansion for the other root by puttingto obtain the equationand assuming an expansion

Example: Find a perturbation expansion for the root nearof the equationup to the term in

Assume

Grouping powers ofgives

Then