A composite collocation method for the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

“…ADM possesses a great potential in solving different kinds of functional equations. Application of ADM to different types of integral equations was discussed by many authors for example [6][7][8][9][10]. In this paper we consider the two dimensional nonlinear Fredholm-Volterra integral equation u(x, t) = f (x, t) + λ 1 t 0 k 1 (t, τ )f 1 (u(x, τ )) dτ + λ 2 b a k 2 (x, ζ )f 2 (u(t, ζ )) dζ.…”

An efficient algorithm for solving nonlinear Volterra–Fredholm integral equations

“…And many methods have been proposed for solving them such as modified decomposition method [2,3], reproducing kernel Hilbert space method [4], Legendre wavelets method [5], homotopy perturbation method [6], a composite collocation method [7], rationalized Haar functions method [8], variational iteration method [9], collocation method based on radial basis functions [10], method based on Bernstein operational matrices [11], hybrid of block-pulse functions and Taylor series method [12], sinc method [13] etc. The comprehensive view of nonlinear Volterra-Fredholm integral equations can be found in Ref.…”

Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator

“…In this study we used Chebyshev cardinal functions which are special cases of Lagrange-interpolating polynomials based on zeros of the Chebyshev polynomials of the first kind to overcome this problem. In this paper, we present the properties of hybrid functions which consist of block-pulse functions plus Chebyshev cardinal functions similar to [27,29,30]. The hybrid functions are first introduced, and the operational matrices of integration and product are then derived.…”

“…Marzban et al in [27,29,30] used the hybrid of block-pulse functions and Lagrange-interpolating polynomials based on zeros of the Legendre polynomials. But no explicit formulas are known for the zeros of the Legendre polynomials.…”

“…The hybrid of block-pulse functions and Taylor series is employed in [29] to solve the linear quadratic optimal control with delay systems. Application of hybrid of blockpulse functions and Lagrange polynomials for solving the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations is investigated in [30].…”