معالجة عددية لمسألة فولكنر-سكان باستخدام تقريبات دالة شرائحية
Abstract
This paper presents an iterative method based on spline function approximations for the numerical solution of the Falkner–Skan equation (FSE) over a semi-infinite interval. This technique will transform the FSE into two initial value problems, so the solution of FSE will be reduced from the interval into [0,1]. Spline approximations are applied directly to the FSE without its reducing into a system of first-order differential equations, thus, the algorithm of spline method has a computational cost that is cost-effective. The spline solution of the FSE is existent and unique, and the convergence analysis for the spline method applied to the FSE is discussed. Numerical results are compared with those obtained by previous methods under various instances of the FSE. The comparisons show the accuracy and efficiency of the presented methodology.
تم تقديم طريقة تكرارية مبنية على تقريبات دالة شرائحية لحل معادلة فولكنر-سكان فوق مجال شبه لانهائي. هذه التقنية ستحول مسألة القيمة الحدية المطروحة التي حلها في المجال إلى مسألتين من مسائل القيم الابتدائية حليهما في المجال [0, 1]. نطبق التقريبات الشرائحية مباشرة على المسألة دون تخفيضها إلى جملة معادلات تفاضلية من المرتبة الأولى، وهذا بالتأكيد سيقلل من الكلفة الحسابية لخوارزمية الطريقة. الطريقة المقترحة تمكّننا دوماً من إيجاد الحل التقريبي ومشتقاته حتى المرتبة الثالثة للمسألة عند أي نقطة من مجال الحل. كما تم إثبات أن الحل الشرائحي للمسألة موجود بشكل وحيد، بالإضافة إلى مناقشة لتحليل تقارب الطريقة.
أخيراً نقدم الحلول العددية للمسألة وفق مختلف حالاتها الخاصة المعروفة وفوق مجالات متنوعة، حيث تشير النَتائِج العددية إلى دقة وفعالية الطرائق المقترحة مقارنة مع نتائج بعض الطرائق الأخرى.
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