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

Abstract

In this paper, a collocation method based on Bessel functions of the first kind is presented to compute the approximate solutions of a class of high-order nonlinear differential equations under the initial and boundary conditions. First, the matrix forms of the Bessel functions of the first kind and their derivatives are constructed. Second, by using these matrix forms, collocation points and the matrix operations, a nonlinear differential equation problem is converted to a system of nonlinear algebraic equations. The solutions of this system give the coefficients of the assumed approximate solution. To demonstrate the validity and applicability of the technique, numerical examples are included and comparisons are made with existing results. The results show the efficiency and accuracy of the present work.

Accurate numerical method for Liénard nonlinear differential equations

“…To describe the methods, let's denote u (1) = u and u (2) = v, then Equation (2) can be written as: (1) , u (2) ), for j = 1, 2…”

“…subject to u (1) 0 = α, u (2) 0 = β. Now, divide the interval [x 0 , L] into N equal subintervals of mesh length h and integrating Equation (3) on the interval [x i , x i+1 ], we obtain: (1) , u (2) )dx,…”

“…To formulate the methods, we approximate f (j) (x, u (1) , u (2) ), for j = 1, 2 by a suitable interpolation polynomials. Case 1:…”

“…Most of the nonlinear differential equations are complicated to be solved using analytical techniques, but these problems are tackled using approximate analytical or numerical methods. For example, various nonlinear differential equations have been solved using the Taylor matrix method, the closed-form method, Chebyshev polynomial approximations, the variational iteration method, the subdomain finite element method, the differential quadrature method, the homotopy perturbation method, the quitic B-spline differential quadrature method, the power series method, the Pade series method, the Legendre polynomial function approximation [2], predictor-corrector method [3], etc.…”

Legendre wavelet solution of high order nonlinear ordinary delay differential equations

“…These methods can be listed as: Aboodh transformation method [2], Adomian decomposition method [8,14,30], power series method [7], decomposition method [43], differential transform method [26], Hermite wavelet-based method [34], variational iteration method [22], power and Padé series-based method [21], spectral method [3], variable multistep methods [25], quasilinearization technique [31], the Runge-Kutta-Fehlberg methods [29], polynomial least squares method [12], homotopy perturbation method [33], variational approach [1], Bessel functions of first kind [36,37], shifted Legendre polynomials [38,42], and first Boubaker polynomial approach [13]. There are also studies on the solutions of nonlinear ordinary differential equations solved by collocation methods which are based on Bessel polynomials [39][40][41].…”