استيفاء كثيرة الحدود للحل العددي لمعادلات فولتيرا، فريدهولم وفولتيرا-فريدهولم التكاملية المجزأة من النوع الثاني باستخدام تقريب برنشتاين

Abstract

This paper aims to study the numerical solution of Volterra, Fredholm, and Volterra-Fredholm integral equations of the second kind and its polynomial interpolation. To do this, a numerical method based on Bernstein polynomials was proposed and applied to the three types of integral equations to transform the integral equation into a linear system of equations that can be solved algebraically Leading us to the approximate solution. The proposed algorithm was applied to the Mathematica program to create a unified formula that enables us to find an approximate solution to the Volterra integral equation of the second kind, the Fredholm integral equation of the second kind, and also the disjoint form of the Volterra-Fredholm integral equation of the second kind, and polynomial interpolation of the solution at the same time. Several numerical examples were applied to the three types of equations and compared with other numerical methods that contributed to solving the same kind of equations. The numerical results showed the effectiveness of the proposed method in finding an approximate solution and interpolating the polynomial of the solution, with minimal error and high speed in performance.