Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

Abstract

This paper describes an algorithm for obtaining approximate solutions to a variety of well-known Lane-Emden type equations. The algorithm expands the desired solution y(x) ≃ yN(x), in terms of shifted Chebyshev polynomials of first kind such that yN(i)(0) = y(i)(0) (i = 0, 1, ..., N). The derivative values y(j)(0) for j = 2, 3, ..., are computed by using the given differential equation and its initial conditions. This makes approximate solutions more consistent with the exact solutions of given differential equations. The explicit expressions of the expansion coefficients of yN(x) are obtained. The suggested method is much simpler compared to any other method for solving this initial value problem. An excellent agreement between the exact and the approximate solutions is found in the given examples. In addition, the error analysis is presented.

Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions

“…Therefore, instead of solving (8) governed by ( 9) and (10), we can solve the modified equation (13) governed by the nonlocal condition (14) and homogeneous boundary condition (15).…”

“…The Chebyshev Petrov-Galerkin method [12] is a numerical technique employed to solve differential equations, particularly those encountered in various scientific and engineering applications. This method is characterized by its reliance on Chebyshev polynomials [13][14][15][16][17], which serve as basis functions, and the Petrov-Galerkin formulation to approximate the solutions of these equations. In essence, the Chebyshev Petrov-Galerkin method begins by expressing the solution of a differential equation as a linear combination of Chebyshev polynomials, which are chosen due to their advantageous properties, such as rapid convergence and inherent orthogonality.…”

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

“…The same polynomials were utilized to solve some BVPs in [40]. In [41], CPs of the first kind were utilized to solve the singular Lane-Emden equation. The authors of [42,43] utilized second-kind CPs to treat some types of DEs.…”

Enhanced shifted Jacobi operational matrices of integrals: spectral algorithm for solving some types of ordinary and fractional differential equations

“…There has been a lot of research on numerical methods for solving IVPs and BVPs in ordinary differential equations and partial differential equations (e.g., [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). To numerically solve various types of DEs, OMs constructed from orthogonal and nonorthogonal polynomials have been extensively used [25][26][27][28][29][30][31][32].…”