طرائق شرائحية تجميعية في C2 لحل مسائل القيمة الابتدائية في المعادلات التفاضلية الجبرية ذات الدليل العالي
Abstract
In this paper, a class of quintic C2- spline collocation methods when applied to differential-algebraic systems with index greater than or equal one is presented.
These methods do not in general attain the same order of accuracy for higher index differential-algebraic systems as they do for index-1 systems. We prove that the proposed methods if applied to index-1 systems are stable and consistent of order five, while they are stable and consistent of order four for index greater than one. Necessary and sufficient conditions on parameters of the methods are derived to ensure that the methods applied to index-v systems are strictly stable. By giving four numerical examples and comparing with other methods, the applicability and efficiency of the methods are shown.
تم في هذا البحث تقديم طرائق شرائحية تجميعية من الدرجة الخامسة في C2 عند تطبيقها للحل العددي للمعادلات التفاضلية الجبرية ذات دليل أكبر أو يساوي الواحد. تبين الدراسة أن الطرائق لا تملك في الحالة العامة نفس الرتبة من الدقة عند تطبيقها لمعادلات تفاضلية جبرية دليلها أكبر من الواحد؛ فالطرائق تكون مستقرة ومتناسقة ومتقاربة من الرتبة الخامسة عند تطبيقها لأنظمة دليلها يساوي الواحد، بينما هذا الاستقرار والتناسق والتقارب يكون من الرتبة الرابعة إذا كان دليل هذه الأنظمة أكبر من الواحد. نحدد بعض الشروط الضرورية والكافية على وسيطيْ الطرائق في المجال لضمان الاستقرار الأكيد للطرائق المقدمة.
وقد تم اختبار فعالية الطرائق المقدمة بحل أربع مسائل ذات أدلة مختلفة حيث تشير النَتائِج العددية إلى فعالية وكفاءة هذه الطرائق مقارنة مع بعض الطرائق الأخرى.
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