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本研究提出了一种基于3-尺度Legendre小波的高阶微分方程特征值问题数值解法。通过规范化Legendre多项式构造该小波基函数，证明其对未知函数展开逼近的一致收敛性；借助Laplace变换推导其Riemann-Liouville分数阶积分公式。以四阶、六阶微分方程特征值问题为算例，与Haar小波方法进行对比验证，结果表明：3-尺度Legendre小波采用更少的配置点可得到更高精度的特征值，且采用Gauss点作为配置点的数值结果优于Newton-cotes点的结果。
Abstract：A numerical method for solving eigenvalue problems of higher-order differential equations based on scale-3 Legendre wavelets is proposed. The wavelet basis functions are constructed by normalizing Legendre polynomials, and the uniform convergence of their expansion approximation for the unknown functions is proven.The Riemann-Liouville fractional integral formula is derived via the Laplace transform. Fourth-order and sixth-order differential equation eigenvalue problems are taken as numerical examples and compared with the Haar wavelet method for validation. The results show that the scale-3 Legendre wavelet method achieves higher accuracy in eigenvalues with fewer collocation points, and the numerical results using Gauss points as collocation points are superior to those using Newton-Cotes points.
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基本信息:
中图分类号:O241.8
引用信息:
[1]秦玺鹏,许小勇,汤思颖.基于3-尺度Legendre小波的微分方程特征值问题数值解[J].井冈山大学学报(自然科学版),2026,47(05):12-20.
基金信息:
江西省自然科学基金项目（2020BABL201006）
2026-09-10
2026-09-10