طرائق شرائحية مجمعة من المرتبة الخامسة لحل معادلات تفاضلية خطية من المرتبة الثانية بشروط حدية
Abstract
This paper presents numerical methods for the solution of linear second-order boundary value problems. These methods are based on C2-quintic splines, that is, fifth Hermite interpolating polynomials with three collocation points. The error analysis and sufficient conditions of the convergence for the presented methods when applied to BVPs are considered. A study shows that the proposed methods consist of order five for (c1=1/2, c2=3/4). Moreover, if:
,
where ,
then the regions of absolute stability of the methods contain some neighborhood of infinity. They are also A-stable and possess unbounded regions of absolute stability. Four widely applied problems are solved to illustrate the order and stability of the proposed methods. The comparisons of the presented methods with other methods show that our results are more accurate.
يقدم هذا البحث طرائق عددية لحل مسائل القيم الحدية في المعادلات التفاضلية الخطية من المرتبة الثانية. إن الطرائق المقترحة تعتمد على كثيرات حدود هرمية الشرائحية من الدرجة الخامسة في الفضاء C2 و تحقق شروط المسألة في ثلاث نقاط مجمعة. حيث يتم تحليل الخطأ لهذه الطرائق بالإضافة إلى وضع الشروط الكافية لتقاربها لدى تطبيقها على مسائل القيم الحدية. تبين الدراسة أن الطرائق المذكورة تكون متجانسة من المرتبة الخامسة لأجل (c1=1/2, c2=3/4)، كما يشير تحليل الاستقرار إلى أنها تكون في حالة -A استقراراً وأن مناطق الاستقرار المطلق تشغل مساحات لانهائية في المستوي العقدي إذا تحققت المتراجعة:
,
علما بأن .
وقد تم اختبار الطرائق المقترحة باستخدامها لحل أربع مسائل مطبقة على نطاق واسع، وكانت النتائج التي تم التوصل إليها دقيقة بالمقارنة مع طرائق أخرى.
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