A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

Abstract

In this paper, a collocation method is given to solve singularly perturbated two-point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results.

A Muntz-Legendre Approach to Obtain Solutions of Singular Perturbed Problems

“…1-2; the exact solutions and the improved approximate solutions are drawn for () yx for different values of truncation limits. [11], the Chebyshev method (CM) [4], the Bessel collocation method [6]. As a result, it is seen from these examples that the present method gives effective results.…”

“…In Table 1 and Table 2; the actual absolute errors, the estimated absolute errors and the improved absolute errors are compared for (N,M)= (12,13) and 3 10    , 5 10    respectively. In Table 3, a comparison is given between the exact solution and the results of other methods such as seventh order numerical method (SNM) [10], the Chebyshev method (CM) [4], the Bessel collocation method [6] and the present method for () yx.On the other hand in Figs. 1-2; the exact solutions and the improved approximate solutions are drawn for () yx for different values of truncation limits.…”

“…Mathematics Section collocation method [2], the B-splines with artificial viscosity [3], the Chebysev method [4], the second order spline finite difference method [5], the Bessel collocation method [6], the exponential polyomial approach [7] and an algorithm of finite difference method [8]. On the otther hand, Müntz-Legendre polynomails are used for solutions of systems of differential equations with functional arguments [8].…”

A new approximation method to solve boundary value problems by using functional perturbation concepts

“…The other approach is designing numerical algorithms to solve equations. For example step difference schemes [20], collocation method [22,10,12] and tau method [17] are of this types. The third class of methods is 10 S. Pourghanbar and M. Ranjbar based on semi analytical approaches like Adomian decomposition method [8], variational iteration method [7], homotopy perturbation method [15], etc.…”

A numerical scheme for solutions of a class of nonlinear differential equations

“…For example, various nonlinear differential equations have been solved using the Taylor matrix method [9], the closed-form method [10], Chebyshev polynomial approximations [11], the variational iteration method [12], the subdomain finite element method [13], the differential quadrature method [14,15], the variational iteration method [16,17], He's variational iteration method [18], the cubic B-spline scaling functions and Chebyshev cardinal functions [19], the homotopy perturbation method [20,21], the variation of parameters method [22], the Adomian decomposition method [23,24], the quintic B-spline differential quadrature method [25], the variational iteration method, the power series method [26], the Adomian decomposition method [27], the Pade series method [28], the Legendre polynomial function approximation [29], the Taylor method [30], the Chebyshev series method [31] and the modified variational iteration method [32]. Recently, the Taylor, Chebyshev, Legendre, Bernstein and Bessel matrixcollocation methods have been used [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] to solve some types of differential, integral, integro-differential-difference equations. Additionally, Sheikholeslami et al…”