Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods

Preliminaries

Author(s): Costica MOROSANU

Pp: 3-42 (40)

DOI: 10.2174/978160805350611201010003

* (Excluding Mailing and Handling)

Abstract

The aim of this Chapter is to provide some basic results that we will use frequently in the chapters that follow. Most results presented here are given without proof. We refer the reader to the appropriate monographs or textbooks where details can be found.


Keywords: real functions, functions of bounded variation, integral formulas, abstract approximation theory, nonlinear functional analysis, distributions, operator theory, semigroups of nonlinear operators, optimal control, numerical analysis, finite difference methods, finite element method, nonlinear systems, fractional steps method, phase changes, mathematical modeling, applications in industry.

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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1518
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 1022 and `b`.`banner_from_date` <= 1739834617 and `b`.`banner_to_date` >= 1739834617 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.09msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 1022
      • 3: 1739834617
      • 4: 1739834617
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='cd40582b2150d48a8a658460c1d80c2a'
      230μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    17Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
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        uc_cart_id
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        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/3332" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
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        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
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        request_query
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        request_request
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        ClearShow all
        Date ↕MethodURLData
        #12025-02-17 23:23:38GET/chapter/33321121819