An exponential method to solve linear Fredholm–Volterra integro-differential equations and residual improvement

Abstract

In this paper, a collocation approach based on exponential polynomials is introduced to solve linear Fredholm-Volterra integro-differential equations under the initial boundary conditions. First, by constructing the matrix forms of the exponential polynomials and their derivatives, the desired exponential solution and its derivatives are written in matrix forms. Second, the differential and integral parts of the problem are converted into matrix forms based on exponential polynomials. Later, the main problem is reduced to a system of linear algebraic equations by aid of the collocation points, the matrix operations, and the matrix forms of the conditions. The solutions of this system give the coefficients of the desired exponential solution. An error estimation method is also presented by using the residual function and the exponential solutions are improved by the estimated error function. Numerical examples are solved to show the applicability and the effectiveness of the method. In addition, the results are compared with the results of other methods.

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

“…According to these comparisons, the presented method gives better results than CASW [5], EM [37] and DT [6] methods, while it gives close results with IHP [35] and BC [39] methods. Matrix operations in the presented method are less and for higher order problems, it can be calculated in a short time with the help of Matlab.…”

“…In Table (7) and Figure (8), the actual errors, estimated errors, and improved errors of the problem (5.11)-(5.12) are compared for various N and M values. In Table ( 6), the actual absolute errors of the problem (5.11)-(5.12) are compared with CASW [5], IHP [35], BC [39], EM [37] and DT [6] methods.…”

“…According to these results, it can be said that the method in Section (3) and Section (4) is quite effective, and the calculations in Matlab show that it is reliable. CASWM [5] DTM [6] IHPM [35] BCM [39] EM [37] PM…”

“…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.…”

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

“…Moreover, for second-order impulsive integro-differential equations, a class of three-point boundary value problems in Banach space have been developed in [19]. Yüzbaşi et al in [50][51][52][53][54][55][56] used the non-polynomial functions to solve differential equations that have been based on non-polynomial functions set {1, e -t , e -2t , . .…”