Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThree Point Spline Collocation Method for Solving General Linear and Nonlinear Eighth-Order Boundary-Value Problems
طريقة تجميع شرائحية بثلاث نقاط لحل مسائل القيم الحدية الخطية وغير الخطية المعممة من المرتبة الثامنة
1783
0 113
0
(
0
)
Authors
سليمان محمود( باحث )
Created by
Shamra Editor
Ask ChatGPT about the research
يتم في هذا البحث تطوير طريقة شرائحية لإيجاد الحل العددي لمسائل القيم الحدية الخطية وغير الخطية في المعادلات التفاضلية المعممة من المرتبة الثامنة. الطريقة المقترحة تقدم الحل الشرائحي التقريبي باستخدام كثيرة حدود من الدرجة إحدى عشرة وتلك الحدودية تحقق المسائل الحدية والابتدائية المطروحة في ثلاث نقاط تجميع. تبين الدراسة أن الطريقة المقترحة عندما تطبق لحل هذه المسائل تكون موجودة ومعرفة بشكل وحيد. كما تظهر الدراسة التحليلية أن الطريقة تكون متجانسة ومتقاربة وأن الخطأ المقتطع الشامل من الرتبة إحدى عشرة. تم اختبار الطريقة الشرائحية بحل أربع مسائل مختلفة، حيث تشير المقارنات لنتائج طريقتنا مع نتائج الطرائق الأخرى إلى أفضلية الطريقة المقترحة من حيث الدقة والفعالية.
In this paper, a spline collocation method is developed for finding numerical solutions of general linear eighth-order boundary-value problems (BVPs) and nonlinear eighth-order initial value problems (IVPs). The presented collocation method affords the spline solution by the polynomial of degree eleventh which satisfies the BVPs and IVPs at three collocation points. The study shows that the spline collocation method when is applied such this problems is existent and unique. Moreover, the purposed method if applied to these systems will be consistent and the global truncation error equal eleventh. Numerical results are given for four examples to illustrate the implementation and efficiency of the method. Comparisons of the results obtained by the present method with results obtained by the other methods reveal that the present method is very effective and convenient.
References used
JALEB H., K. FARAJEYAN, Solution of eighth-order boundary-value problems using nonpolynomial spline, Mathematical Sciences, Vol. 2, No. 1 (2008) 33-45
RASHIDINIA J., R. JALILIAN; K. FARAJEYAN, Spline approximate solution of eighth-order boundary-value problems, International Journal of Computer Mathematics. Vol. 86, No. 8, (2009), 1319–1333
LAMNII A. and H. MRAOUI, Spline collocation method for solving boundary value problems, International Journal of Mathematical Modelling & Computations Vol. 03, No. 01, (2013), 11- 23
(KASI VISWANADHAM K.N.S. and Y. SHOWRI RAJU, Quintic B-spline Collocation Method for Eighth Order Boundary Value Problems, Advances in Computational Mathematics and its Applications, Vol. 1, No. 1, (2012
NOOR M. A. and S.T. MOHYUD-Din, Variational iteration decomposition method for solving eighth-order boundary value problems, Differential Equations and Nonlinear Mechanics, Vol. (2007), pp. 1-16
rate research
Read More
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 existen
ce, 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.
In this paper, we use polynomial splines of eleventh degree with three collocation
points to develop a method for computing approximations to the solution and its
derivatives up to ninth order for general linear and nonlinear ninth-order boundary-v
alue
problems (BVPs). The study shows that the spline method with three collocation points
when is applied to these problems is existent and unique. We prove that the proposed
method if applied to ninth-order BVPs is stable and consistent of order eleven, and it
possesses convergence rate greater than six.
Finally, some numerical experiments are presented for illustrating the theoretical
results and by comparing the results of our method with the other methods, we reveal that
the proposed method is better than others.
In this paper, the numerical solution of general linear fifth-order boundary-value problem (BVP) is considered. This problem is transformed into three initial value problems (IVPs) and then spline functions with four collocation points are applied to
the three IVPs. The presented spline method enables us to find the spline solution and derivatives up to fifth-order of BVP. By giving four examples and comparing with the other methods, the efficiency and highly accurate of the method will be shown.
In this paper, we introduce a numerical method for solving systems of high-index differential algebraic equations. This method is based on approximating the exact solution by spline polynomial of degree eight with five collocation points to find the
numerical solution in each step. The study shows that the method when applied to linear differential-algebraic systems with index equal one is stable and convergent of order 8, while it is stable and convergent of order 9-u for index equal u .
Numerical experiments for four test examples and comparisons with other available results are given to illustrate the applicability and efficiency of the presented method
In this paper, we develop spline collocation technique for the numerical solution of
general twelfth-order linear boundary value problems (BVPs). This technique based on
polynomial splines from order sixteenth as well as five collocation points at
every
subinterval of BVPs. The method developed not only approximates the solution of BVP,
but its higher order derivatives as well. We show that the spline collocation method is
existent and unique when it is applied into a test problem. Also, its global truncation error
is estimated. Moreover, the purposed spline method when applied to test problems will be
consistent and convergent from sixteenth order. Three numerical examples are given to
illustrate the applicability and efficiency of the new method. Comparisons of our results
with some other methods show that our method is very effective and successful.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
