Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods are performed to validate our results.

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通过有效的拟线性化贝塞尔方法逼近非线性 Troesch 问题的解

开发了两种利用新型贝塞尔多项式（具有正系数）的基于搭配的方法来解决非线性 Troesch 问题。在第一种方法中，通过将未知解及其二阶导数表达为贝塞尔矩阵形式以及一些搭配点，控制方程转换为非线性代数矩阵方程。在第二种方法中，首先采用拟线性化技术对模型问题进行线性化，然后将第一种搭配方法迭代应用于线性化方程的序列。在后一种方法中，我们需要在每次迭代中求解线性代数矩阵方程。此外，建立了贝塞尔级数解的误差分析。到底，提供了数值模拟和计算结果来说明所提出的搭配方法的实用性和适用性。与一些现有的可用方法进行数值比较以验证我们的结果。