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

Approximation of The Phase Field Transition System

Author(s): Costica MOROSANU

Pp: 43-106 (64)

DOI: 10.2174/978160805350611201010043

* (Excluding Mailing and Handling)

Abstract

In this chapter we will deal, on one hand, with the study of the existence, uniqueness, regularity and estimates of the solution of phase-field transition system, and on the other hand, we will analyze the convergence of some approximating schemes of fractional steps type, associated to this nonlinear system.

system. In the first Section the existence, uniqueness and regularity of the solution of the phase-field transition system subject to the non-homogeneous Cauchy-Neumann boundary conditions (Theorem 2.1) are studied. Such kind of conditions allow the phase-field system (Caginalp’s model) to be considered as a model of heat transfer from the surface of product to the environment, when we assume that heat is extracted by convection and conduction in the continuous casting process.....


Keywords: nonlinear parabolic systems, abstract approximation theory, nonlinear functional analysis, distributions, operator theory, dissipative operators, degree theory, semigroups of nonlinear operators, numerical analysis, fractional steps method (Lie-Trotter product formula), stability and convergence of numerical methods.

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      • 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` <= 1739834712 and `b`.`banner_to_date` >= 1739834712 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.08msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 1022
      • 3: 1739834712
      • 4: 1739834712
      • 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='0a4f6da223b447dedf7613234a58dc88'
      220μsalphaeurekaselec_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
    11Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        iJLcvEum1h9MAcWhqi2xHLsTgq7WoNTB8dHcMXjV
        uc_cart_id
        0a4f6da223b447dedf7613234a58dc88
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/3333" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/3333
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "fpestid=lbwvGiXYriJVwA0g4i4Z5mNBVQ18k4lyIQO0xc9NQFqIW1_8-KR3YjdX9XGmJUZ0OicdEg; wayfindertarget=eyJpdiI6InZQS05STGsrMWthZVpDTUlmSDJIblE9PSIsInZhbHVlIjoidkFWdWVjZ0svYmFqWWpVSmg4UGpNMlhBK3ZUbVdkdERRVDhZblQyTzRTeFo5QXVUT3RVakZiTEg3WUMzRmoxcyIsIm1hYyI6ImIzNTE5MDZkMjZkODljYzFhYzg5ZDgyN2FmZTU4ZGQ2YjNhNThjZDhkNDZiZjRlOTU4NDFhN2NjZDMxN2QwMjciLCJ0YWciOiIifQ%3D%3D; XSRF-TOKEN=eyJpdiI6ImxIT05wTmZGcGhpYXFtTVl4MFVRZFE9PSIsInZhbHVlIjoiWEFnclZUZlNYQU9WU08rQ3k4K0JoZjNDQUxGZGl6Q0RkbTAwVHVxb25vKzYrVWxleE1hUy9BOVFXVFRySkZMNkdmRjZtNEk5VlMrN05YVWQ4UlBzVXNsWjNWNGpmZm00RE1DVW5PTTJ3anlPdVJDOFBnTy83TWFCRSt0VS9lY0MiLCJtYWMiOiIxZDdjOTAzYjAxOWM2YjkxYjdlZWEzNGY1MDQ1ODNhNWY1OTU4ZmJiMWU1MmRkYmZjZjQyODgxYTVjMjY0MzM3IiwidGFnIjoiIn0%3D; alphaeurekaselectnet_session=eyJpdiI6Inl3WlJ6bTh3cUl5c2ZhL3E3N3BYWHc9PSIsInZhbHVlIjoiK0c2NzI1UU1LT29HYWROUU55NVZEN3BDbURtRU9ZTDEzYjluSVRPc2MvVmJWWCtYQ2NiZFFURFIzTFNxdk1Gbm5WYTlwYmVTUlZrbWx2TTM1aXBCdmVwRzVUY0owRDlWWnN5WUtOK1NJbjNpVi9Vc0FHVDI5dDRIVWdLZkJHNzUiLCJtYWMiOiI3MDcxYmEzOGU0OGQ0ODJiMGE5NTBmMjg5NzFkOTgzYzc4OWU5NGUwYTNkYWQ5MmJhMzMzZjYxMjMxOWM5YWIwIiwidGFnIjoiIn0%3Dfpestid=lbwvGiXYriJVwA0g4i4Z5mNBVQ18k4lyIQO0xc9NQFqIW1_8-KR3YjdX9XGmJUZ0OicdEg; wayfindertarget=eyJpdiI6InZQS05STGsrMWthZVpDTUlmSDJIblE9PSIsInZhbHVlIjoidkFWdWVj" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
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        response_headers
        0 of 0
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        session_attributes
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        array:5 [ "_token" => "iJLcvEum1h9MAcWhqi2xHLsTgq7WoNTB8dHcMXjV" "uc_cart_id" => "0a4f6da223b447dedf7613234a58dc88" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/3333" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-17 23:25:12GET/chapter/33331121813