A reliable algorithm based on the shifted orthonormal Bernstein polynomials for solving Volterra–Fredholm integral equations

Abstract

This paper deals with the numerical solution of Volterra-Fredholm integral equations. In this work, we approximate the unknown functions based on the shifted orthonormal Bernstein polynomials, in conjunction with the least-squares approximation method. The method is using a simple computational manner to obtain a quite acceptable approximate solution. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of the method for solving these equations are high. The convergence analysis of proposed method have been discussed through some theorems. Moreover, we will obtain an estimation of error bound for this algorithm. Finally, some examples are given to show the capability of presented method in comparison with four well-known algorithms in the literature namely the Legendre collocation method, Taylor collocation method, Taylor polynomial method and Lagrange collocation method. ARTICLE HISTORY

Pell-Lucas collocation method to solve high-order linear Fredholm-Volterraintegro-differential equations and residual correction

“…According to the method in Section (3), collocation points are computed as Table 9. Comparison of the absolute errors for (N, M ) = (5, 6), (5,9), (8,10), (8,13) of the prob. (5.15)- (5.16) Absolute errors for the Estimated errors for the Absolute errors for the corrected approximate solution for approximate solution approximate solution…”

“…Maleknejad, Mahmoudi, Yalçınbaş and Sezer have studied on the Taylor polynomial approach for linear and nonlinear FVIDEs [21,33]. Also, various methods [7,8,20,29,37,38,40] such as compact the finite difference method [43], the rationalized Haar functions method [22,27], the CAS wavelet method [5], the differential transform method [6], the improved homotopy perturbation method [35], the sine-cosine wavelet method [18,32], the homotopy perturbation method [11,18], the hybrid function method [17], the sinc method [42], the Legendre method [23,25,26,28], the Bernstein method [9,10,36] and the combined Laplace transform-Adomian decomposition method [41] have been studied for solving linear and nonlinear FVIDEs.…”

Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto–Sivashinsky equation

“…The accuracy of the RKHSM for the problem is controllable. Many scientific properties of the RKHSM can be seen in [30][31][32][33][34][35][36][37][38][39][40].…”