طريقة الدالة الشرائحية لحل مسائل القيم الحدية غير الخطية المعممة من المرتبة الثالثة
Abstract
In this paper, a numerical method is suggested for solving general a nonlinear third-order boundary value problem (BVP). In this method, the given nonlinear third-order BVP will be transformed into two third-order initial value problems (IVPs), then spline function approximations are applied to both two IVP for finding the Spline solution and its derivatives up to third order of the given BVP. The study shows that the spline solution of the BVP is existent and unique, and the convergence order of the spline method is fourth with a local truncation error . The presented algorithm is designed for solving a general BVP, where it is applied to some types of nonlinear third-order differential equations. Comparisons of the results obtained by spline method with other methods show the efficiency and highly accurate of the proposed method.
في هذا العمل تم اقتراح طريقة عددية لحل مسألة القيم الحدية غير الخطية المعممة من المرتبة الثالثة. هذه الطريقة ستحول مسألة القيمة الحدية المطروحة إلى مسألتين من مسائل القيم الابتدائية، وعندئذ تم تطبيق تقريبات دالة الشريحة على مسألتي القيم الابتدائية للحصول على الحل الشرائحي ومشتقاته حتى المرتبة الثالثة لمسألة القيمة الحدية. تبين الدراسة أنه يوجد حل شرائحي وحيد للمسألة المطروحة، بالإضافة إلى أن مرتبة التقارب للطريقة هي الرابعة وبخطأ مقتطع موضعي .
تم وضع خوارزمية للطريقة في الحالة العامة لحل مسألة القيم الحدية، حيث طُبِقتْ لحل بعض مسائل القيم الحدية في المعادلات التفاضلية غير الخطية، وتشير النتائج التي تم الحصول عليها والمقارنات مع بعض الطرائق الأخرى إلى الفعالية الحسابية والدقة العالية للطريقة المقترحة.
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