An efficient hybrid pseudo-spectral method for solving optimal control of Volterra integral systems

In this paper, a new pseudo-spectral (PS) method is developed for solving optimal controproblems governed by the non-linear Volterra integral equation(VIE). The novel method is based upon approximating the state and control variables by the hybrid of block pulse functions and Legendre polynomials. The properties of hybrid functions are presented. The numerical integration and collocation method is utilized to discretize the continuous optimal control problem and then the resulting large-scale finite-dimensional non-linear programming (NLP) is solved by the existing well-developed algorithm in Mathematica software. A set of sufficient conditions is presented under which optimal solutions of discrete optimal control problems converge to the optimal solution of the continuous problem. The error bound of approximation is also given. Numerical experiments confirm efficiency of the proposed method especially for problems with non-sufficiently smooth solutions belonging to class $C^1$ or $C^2$

Direct operational matrix approach for weakly singular Volterra integro-differential equations: application in theory of anomalous diffusion

In the current paper, we present an efficient direct scheme for weakly singular Volterra integro-differential equations arising in the theory of anomalous diffusion. The behavior of the system demonstrating the anomalous diffusion is significant for small times. The method is based on operational matrices of Chebyshev and Legendre polynomials with some techniques to reduce the total errors of the already existing schemes. The proposed scheme converts these equations into a linear system of algebraic equations. The main advantages of the method are high accuracy, simplicity of performing, and low storage requirement. The main focus of this study is to obtain an analytical explicit expression to estimate the error. Numerical results confirm the superiority and applicability of our scheme in comparison with other methods in the literature

Numerical solution of a non-linear Volterra integral equation

In this paper, a numerical method to solve non-linear integral equations based on a successive approximation technique is considered. A sequence of functions is produced which converges to the solution. The process includes a fixed point method, a quadrature rule, and an interpolation method. To find a total bound of the error, we investigate error bounds for each approximation and by combining them, we will derive an estimate for the total error. The accuracy and efficiency of the method is illustrated in some numerical examples