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

Abstract

In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically solve higher order linear Fredholm-Volterra integro differential equations (FVIDE). The approximate solutions are assumed in form of the truncated Pell-Lucas polynomial series. By using Pell-Lucas polynomials and relations of their derivatives, the solution form and its derivatives are brought to matrix forms. By applying the collocation method based on equally spaced collocation points, the method reduces the problem to a system of linear algebraic equations. Solution of this system determines the coefficients of assumed solution. Error estimation is made and also a method with the help of the obtained approximate solution is developed that finds approximate solution with better results. Then, the applications are made on five examples to show that the method is successful. In addition, the results are supported by tables and graphs and the comparisons are made with other methods available in the literature. All calculations in this study have been made using codes written in Matlab.

Lucas Polynomial Solution of the Single Degree of Freedom System

“…The method has also been used to solve nonlinear equations by . Yüzbaşı and Yıldırım (2020) proposed Pell-Lucas collocation method to solve high-order linear Fredholm-Volterra integrodifferential equations. In addition, Kübra et al (2021) , Yüzbaşı and Ismailov (2018), Kürkçü and Sezer (2022) proposed a different the matrix method.…”

Pell-Lucas Collocation Method to Solve Second-Order Nonlinear Lane-Emden Type Pantograph Differential Equations

“…On the other hand, there are studies by using the Pell-Lucas polynomials for many types of the differential equations [18], [49], [63], [64]. Since no study has yet been done by using the Pell-Lucas polynomials for the solutions of the nonlinear Lane-Emden type pantograph differential equation (LETPDE), the Pell-Lucas polynomials are used for the approximate solutions of the nonlinear second-order Lane-Emden type pantograph differential equation (LETPDE) in this study.…”

On Solutions of Linear Functional Integral and Integro-Differential Equations via Lagrange Polynomials

“…In recent years, various techniques have been used for solving FIDEs such as He's variational iteration technique [11], the homotopy perturbation method [12], He's homotopy perturbation method [13], the modified homotopy perturbation method [14], the rationalized Haar functions method [15], the Chebyshev cardinal functions method [16], the differential transformation method [17], the Tau method with error estimation [18], He's variational iteration method [19], the collocation method [20], the Adomian decomposition method [21], the Adomian-Pade technique [22], the discontinuous Galerkin method [23], the Legendre multiwavelets [24], the trigonometric wavelets [25], the spectral methods [26,27] the meshless method [28] and etc [35,40,41,[43][44][45][46][47][48][49][50]52]. In addition, the operation matrix method, the Galerkin-like method and the matrix-collocation methods based on Taylor, Chebyshev, Bessel, Bernoulli, Laguerre, Bernstein, Legendre, Chebyshev, Morgan-Voyce, Taylor-Lucas, Dickson and Lucas, polynomials, have been studied by some authors [29][30][31][32][33][34]42,51] to solve the mentioned type equations.…”