Collocation Orthonormal Bernstein Polynomials Method for Solving Integral Equations

Abstract

In this paper, we use a combination of Orthonormal Bernstein functions on the interval 0,1 for degree 5, and 6 to produce a new approach implementing Bernstein operational matrix of derivative as a method for the numerical solution of linear Fredholm integral equations and Volterra integral equations of the second kind. The method converges rapidly to the exact solution and gives very accurate results even by low value of m. Illustrative examples are included to demonstrate the validity and efficiency of the technique and convergence of the method to the exact solution. Keywords: Bernstein polynomials, Operational Matrix of Derivative, Linear Fredholm Integral Equations of the Second Kind and Volterra Integral Equations. ‫برنشتن‬ ‫حدود‬ ‫لمتعددة‬ ‫العد‬ ‫طريقة‬ ‫المتعامدة‬ ‫التكاملية‬ ‫المعادالت‬ ‫لحل‬ :‫الخالصة‬ ‫الفترة‬ ‫ضمن‬ ‫المتعامدة‬ ‫لبرنشتن‬ ‫متعددات‬ ‫مجموعة‬ ‫استخدام‬ ‫تم‬ ‫البحث‬ ‫ھذا‬ ‫في‬ ] 0,1 [ ‫للدرجة‬ 5,6 ‫وايجاد‬ ‫النوع‬ ‫من‬ ‫وفولتيرا‬ ‫فردھولم‬ ‫بنوعيھا‬ ‫الخطية‬ ‫التكاملية‬ ‫المعادالت‬ ‫لحل‬ ‫واستخدامھا‬ ‫للمشتقات‬ ‫جديدة‬ ‫مصفوفة‬ ‫قيمة‬ ‫انخفاض‬ ‫عند‬ ‫حتى‬ ‫للغاية‬ ‫دقيقة‬ ‫نتائج‬ ‫لتعطي‬ ‫الثاني‬ m ‫ھذه‬ ‫وكفاءة‬ ‫صحة‬ ‫على‬ ‫دليل‬ ‫التوضيحية‬ ‫االمثلة‬ , .‫المضبوط‬ ‫للحل‬ ‫والتقارب‬ ‫التقنية‬