Hybrid function method for solving Fredholm and Volterra integral equations of the second kind

Numerical solutions of Fredholm and Volterra integral equations of the second kind via hybrid functions, are proposed in this paper. Based upon some useful properties of hybrid functions, integration of the cross product, a special product matrix and a related coefficient matrix   with optimal order, are applied to solve these integral equations. The main characteristic of this technique is to convert an integral equation into an algebraic; hence, the solution procedures are either reduced or simplified accordingly. The advantages of hybrid functions are that the values of n$n$ and m$m$ are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function, for the solutions of integral equations. We propose that the available optimal values of n$n$ and m$m$ can minimize the relative errors of the numerical solutions. The high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples. The hybrid function method is superior to other piecewise constant orthogonal functions [W.F. Blyth, R.L. May, P. Widyaningsih, Volterra integral equations solved in Fredholm form using Walsh functions, Anziam J. 45 (E) (2004) C269–C282; M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200 (2007) 12–20] for these problems.