A numerical approach for a nonhomogeneous differential equation with variable delays

Abstract

In this study, we consider a linear nonhomogeneous differential equation with variable coefficients and variable delays and present a novel matrix-collocation method based on Morgan-Voyce polynomials to obtain the approximate solutions under the initial conditions. The method reduces the equation with variable delays to a matrix equation with unknown MorganVoyce coefficients. Thereby, the solution is obtained in terms of Morgan-Voyce polynomials. In addition, two test problems together with error analysis are performed to illustrate the accuracy and applicability of the method; the obtained results are scrutinized and interpreted by means of tables and figures.

Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method

“…In addition, we can obtain the general matrix forms of the nonlinear quadratic and cubic parts by similar operations as (2.1)-(2.4) [13,14], for p, q, r = 0, 1, 2, as follows:…”

“…In this section, we will give an error analysis based on the residual function [7,10,[12][13][14] for the present method.…”

“…Q pqr (t) y (p) (t) y (q) (t) y (r) is considered, where P k (t) , Q pq (t) , Q pqr (t) and g (t) are the given analytic functions defined on the interval a ≤ t ≤ b; λ j , a kj and b kj are the known rael coefficients. In order to solve the nonlinear problem (1.1)-(1.2), we utilize the matrix-collocation method, which have been developed by Sezer and Coworkers [7,10,[12][13][14], and research the numerical solution in the truncated Morgan-Voyce series form…”

“…. , N , N ≥ m, are the Morgan-Voyce polynomials [13][14][15]17] defined by, recursively [6, 13-15, 17, 18],…”

Lucas polynomial solution of nonlinear differential equations with variable delays

“…We propose a new matrix technique, developed by Sezer et.al. [9,15,20,32], to solve Eq. (1.1) with the initial conditions, in the finite Lucas series of the form…”