Numerical Solution of Volterra Integral Equations by Using Some Positive Operators

The achievement of this research is bifurcated. Firstly, for the numerical solution of the first kind linear Fredholm and Volterra integral equations with smooth kernels a numerical method by using Modified Bernstein-Kantorovich operators is given. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein-Kantorovich operators. Hence, by applying discretization the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which implies the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein-Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the applicability of the proposed method on second kind Volterra integral equations with smooth kernels is also demonstrated with examples. Secondly we give hybrid positive linear operators which are defined by using the Bernstein-Kantorovich and Modified Bernstein-Kantorovich operators on certain subintervals of [0,1]. Additionally, we consider second kind linear Volterra integral equations with weak singular kernels of the form (x−t) −vKe(x,t), 0 < v < 1, where Ke is a smooth function. It is well known that the solution usually possess singularities at the initial point. Subsequently, we develop a combined method which uses the proposed hybrid operators and approximates the solution on the constructed subintervals. Two algorithms are developed through the given combined method and applied on some examples from the literature. Furthermore, numerical validation of the combined method is also given on first kind integral equations, by first utilizing regularization. Eventually, it is shown that the proposed combined method hence, the given computational algorithms are numerically stable and give acceptable accurate approximations to solutions with singularities.