AI 中文总结
本文提出移位Horadam配点法，用于求解满足两点边界条件的非线性四阶常微分方程，通过算例验证其性能优于现有方法。
AI 中文摘要
本文采用基于移位Horadam多项式的配点法，求解一类非线性常微分方程的数值解。所研究的问题为四阶问题，满足一类两点边界条件。我们首先讨论Horadam多项式的定义和基本性质，然后给出其新的有用微分公式，并从该微分性质中推导得到新颖有趣的恒等式。所采用的配点法假设问题的解可表示为移位Horadam多项式级数，为确定该级数解的展开系数，我们在移位Horadam多项式的零点处进行配点，将所研究的边值问题转化为一组非线性代数方程，随后采用牛顿迭代法求解这些代数方程，以得到移位Horadam多项式级数解的展开系数的数值。本文考虑了所提出的非线性边值问题的多个算例，以验证该方法的准确性、效率和可靠性，将获得的数值解和误差与现有解进行比较，通过表格和图形展示的结果对比清晰表明，移位Horadam配点法的性能优于现有方法。
英文摘要
In this paper, numerical solutions of a class of nonlinear ordinary differential equations are obtained using a collocation method based on the shifted Horadam polynomials. The proposed problem, which is of the fourth-order, satisfies a class of two-point boundary conditions. We first discuss definitions and basic properties of the Horadam polynomials before presenting new and useful differentiation formulae for them. Novel interesting identities are deduced from the differentiation properties. The collocation method under consideration assumes that the solution of the proposed problem can be expressed as a shifted Horadam polynomial series. To determine the expansion coefficients of the series solution, one collocates at the zeros of the shifted Horadam polynomials, and the proposed boundary value problem is subsequently reduced to a set of nonlinear algebraic equations. These algebraic equations are then solved using Newton's iterative method to obtain the numerical values of the expansion coefficients of the shifted Horadam polynomial series solution. Several examples of the proposed nonlinear boundary value problem are considered to demonstrate the accuracy, efficiency, and reliability of the proposed method. Numerical solutions and errors obtained are compared with existing solutions. Comparisons of results, which are shown in tables and graphs, clearly reveal that the shifted Horadam collocation method outperforms the existing methods.
