On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials

Abstract

In this study, we suggest a collocation method to solve a class of the nonlinear differential equations under the mixed conditions in terms of the Bessel polynomials. The method is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. To illustrate the usefulness of this technique we apply it to some test problems and the comparisons are made with existing results. The results show the efficiently and accuracy of the present work. All of the numerical computations have been performed on computer using a program written in Maple.

A new class of polynomial functions equipped with a parameter

“…2. Furthermore, the modified Boubaker polynomials [13] are defined asB n ðxÞ ¼ 6xU nÀ1 ðxÞ À 2U n ðxÞ, for n ! 1.…”

“…The mth-order non-linear differential equation where y is an unknown function, g and P k;r are continuous functions, and a j;k , b j;k , and c j are real or complex constants, is considered in [13]. By PFS collocation method, we can solve this problem.…”

“…m is selected in any positive integer number. The matrix form of the collocation method is discussed in [13]. In this method, a non-linear system with N þ 1 equations is constructed to find N þ 1 unknowns.…”

“…Afterwards, we can continue the matrix collocation method to find other unknowns like [13]. with the initial condition yð0Þ ¼ 1, [4,9,13], which is a special case of (1). The exact solution is yðxÞ ¼ 1=ð2 À e Àx Þ.…”

“…In numerous cases, we can find the solution of the linear or non-linear functional equations by Polynomial Expansion Scheme (PES) or matrix methods (in the sense of collocation scheme) based on famous polynomials such as Taylor, Chebyshev, Berstein, Legendre, Laguerre, Hermite, Bessel, Lucas, and Boubaker [1][2][3][10][11][12][13].…”