This research aims to assemble two methodical spectral Legendre's derivative algorithms to numerically attack the Lane-Emden, Bratu's, and singularly perturbed type equations. We discretize the exact unknown solution as a truncated series of Legendre's derivative polynomials. Then, via tau and collocation methods for linear and nonlinear problems, respectively, we obtain linear/nonlinear systems of algebraic equations in the unknown expansion coefficients. Finally, with the aid of the Gaussian elimination technique in the linear case and Newton's iterative method for the non-linear case - with vanishing initial guess- we solve these systems to obtain the desired solutions. The stability and convergence analyses of the numerical schemes were studied in-depth. The schemes are convergent and accurate. Some numerical test problems are performed to verify the efficiency of the proposed algorithms.

本研究旨在组装两种有条理的谱勒让德导数算法，以数值攻击 Lane-Emden、Bratu 和奇异摄动型方程。我们将精确的未知解离散为勒让德导数多项式的截断级数。然后，分别通过线性和非线性问题的tau和搭配方法，我们得到未知展开系数下代数方程的线性/非线性系统。最后，借助线性情况下的高斯消元技术和非线性情况下的牛顿迭代方法（初始猜测消失），我们求解这些系统以获得所需的解。深入研究了数值方案的稳定性和收敛性分析。这些方案是收敛的和准确的。