Optimal Control Approach for Solving Linear Volterra Integral Equations

Abstract

Abstract-In this paper we present a new approach for linear Volterra integral equations that is based on optimal control theory. S ome optimal control problems corresponding Volterra integral equation be introduced which we solve these problems by discretization methods and linear programming approaches. Finally, some examples are given to show the efficiency of approach.Index Terms -Volterra integral equations, Optimal control, Linear programming. I. INTRO DUCTIO NVo lterra integral equations arise in many physical applications, e.g., potential theory and Dirichlet problems and electrostatics. Also, Volterra integral equations are applied in the biology, chemistry, engineering, mathematical problems of radiat ion equilibriu m, the particle transport problems of astrophysics and reactor theory, and radiation heat transfer problems [1,2,3,4,5]. There exist the some valid appro ximate and numerical methods for solving Vo lterra integral equation such as Adomian decomposition method [6], Walsh functions method and mult igrid approach [7,8] [35,36,37,38,39]. In this paper, we present a d ifferent approach fro m above methods for solving linear volterra integral equations which is based on the optimal control theory [40,41,42]. Consider the following linear Vo lterra integral equations of second kind: ( ) ( ) ( , ) ( ) , [ , ],   

Investigation of singular ordinary differential equations by a neuroevolutionary approach

“…The unknown decision weights in error functions (17), (19) and 21are determined by using the novel ANN based FO-DPSO approach. We briefly present approximate solutions for cases 1, 2, and 3 in Eqs (22), (23) and (24). We show the best values of weights along with convergence plots and step sizes in Figs 4, 5, 6 for Problems 1, 2, and 3.…”

Solving Linear Volterra – Fredholm Integral Equation of the Second Type Using Linear Programming Method

“…Erfanian and Mostahsan in 2011 considered two stages of approximation to find an approximate solution for a class of nonlinear Volterra integral equations by: first convert the integral equation to a moment problem, and then modify the new problem to two classes of optimization problems (nonconstraint optimization and optimal control problems) (9). Effati and Skandari in 2012 presented a new approach for linear Volterra integral equations based on optimal control theory and some optimal control problems corresponding Volterra integral equation are introduced which solved by discretization methods and linear programming approaches (10).…”

Numerical Solution of System of Three Nonlinear Volterra Integral Equations Using Implicit Trapezoidal

“…In this section, we recall the main theorems (Effati, S. & NooriSkandari, M. H., 2012). Theorem 1.Consider the equation …”

“…In this section, we use the technique of the Volterra equation (Balakumar, V. & Murugesan, K., 2013;Effati, S. & NooriSkandari, M. H., 2012) to find an approximates the solution ( ) of (1) at the equally spaced points = 0 + ℎ for = 1, ⋯ , where 0 = 0 and is the total number of steps of size ℎ.…”

“…The iterates ( ) are computed until the infinity norm of the vector is less than a prescribed tolerance Tol. Then is assigned the value of the last iterate (Balakumar, V. & Murugesan, K., 2013;Effati, S. & NooriSkandari, M. H., 2012).…”