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Register a new userCollocation Spline method for solving Linear and Nonlinear Sixth-Order Boundary-Value Problems
طريقة الشريحة التجميعية لحل مسائل القيم الحدية الخطية و غير الخطية من المرتبة السادسة
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سليمان محمود( باحث )
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في هذا العمل تم تقديم طريقة الشريحة التجميعية للحل العددي لنوعين من المسائل. النوع الأول هو مسألة القيمة الحدية في المعادلات التفاضلية الخطية المعممة من المرتبة السادسة و النوع الثاني هو مسألة القيمة الابتدائية في المعادلات التفاضلية غير الخطية المعممة من المرتبة السادسة. تم إثبات أن الطريقة المذكورة عند تطبيقها لمثل هذه المسائل تكون موجودة بشكل وحيد بالإضافة إلى تقدير الأخطاء و تحليل التقارب. تبين الدراسة أن طريقة الشريحة بثلاث نقاط تجميعية تستطيع إيجاد الحلول العددية الشرائحية و مشتقاتها حتى المرتبة السادسة للمسائل الخطية و غير الخطية المطروحة و بالتالي فهي أداة فعالة للحل العددي لمثل هذه المسائل. تم إثبات فعالية وكفاءة الطريقة المقترحة بحل عدد من مسائل الاختبار و مقارنة النتائج التي تم التوصل إليها مع نتائج لطرائق أخرى.
In this paper, spline collocation method is considered for solving two forms of problems. The first form is general linear sixth-order boundary-value problem (BVP), and the second form is nonlinear sixth-order initial value problem (IVP). The existence, uniqueness, error estimation and convergence analysis of purpose methods are investigated. The study shows that proposed spline method with three collocation points can find the spline solutions and their derivatives up to sixth-order of the two BVP and IVP, thus is very effective tools in numerically solving such problems. Several examples are given to verify the reliability and efficiency of the proposed method. Comparisons are made to reconfirm the efficiency and accuracy of the suggested techniques.
References used
KASI VISWANADHAM K.N.S. and Y. SHOWRI RAJU, Quintic B-spline Collocation Method for Sixth Order Boundary Value Problems, Global Journal of Researches in Engineering, Vol. 12 , No. 1 , 2012
RASHIDINIA J., M. GHASEMI, B-Spline Collocation For Solution of Two-Point Boundary Value Problems, Journal of Computation and Applied Math., 235, pp. 2325–2342, 2011
LAMNII A., H. MRAOUI, D. SBIBIH, A. TIJINI and A. ZIDNA, Spline Collocation Method for Solving Linear Sixth-Order Boundary-Value Problems, International Journal of Computer Mathematics, Vol. 85, No. 11, (2008)1673-1684
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