# High-Order Shifted Gegenbauer Integral Pseudospectral Method for Solving Differential Equations of Lane-Emden Type

## TL;DR

This paper introduces a high-order, exponentially convergent shifted Gegenbauer integral pseudospectral method for efficiently solving Lane-Emden differential equations with mixed boundary conditions, demonstrating superior accuracy and stability.

## Contribution

The paper develops a novel integral pseudospectral method using shifted Gegenbauer polynomials, including new algorithms and rigorous error analysis for solving Lane-Emden equations.

## Key findings

- Exhibits exponential convergence and high accuracy.
- Demonstrates effectiveness through five numerical examples.
- Outperforms existing methods in stability and efficiency.

## Abstract

We present a novel, high-order, efficient, and exponentially convergent shifted Gegenbauer integral pseudospectral method (SGIPSM) to solve numerically Lane-Emden equations provided with some mixed Neumann and Robin boundary conditions. The framework of the proposed method includes: (i) recasting the problem into its integral formulation, (ii) collocating the latter at the shifted flipped-Gegenbauer-Gauss-Radau (SFGGR) points, and (iii) replacing the integrals with accurate and well-conditioned numerical quadratures constructed via SFGGR-based shifted Gegenbauer integration matrices. The integral formulation is eventually discretized into linear/nonlinear system of equations that can be solved easily using standard direct system solvers. The implementation of the proposed method is further illustrated through four efficient computational algorithms. Moreover, we furnish rigorous error and convergence analyses of the SGIPSM. Five numerical test examples are presented to verify the effectiveness, accuracy, exponential convergence, and numerical stability of the proposed method. The numerical simulations are associated with extensive numerical comparisons with other rival methods in the literature to demonstrate further the power of the proposed method. The SGIPSM is broadly applicable and represents a strong addition to common numerical methods for solving linear/nonlinear differential equations when high-order approximations are required using a relatively small number of collocation points.