谱Lanczos法tau方法用于非线性积分微分方程组
Spectral Lanczos' tau method for systems of nonlinear integro-differential equations
AI总结：
本文提出将谱Lanczos法tau方法扩展到非线性积分微分方程组系统，通过线性化系数、递推关系和正交多项式评估提升数值稳定性与精度，验证了方法的有效性。
摘要：
本文提出将谱Lanczos法tau方法扩展到非线性积分微分方程组系统。该扩展包括（i）由非线性项导出的正交多项式乘积的线性化系数；（ii）在需要矩阵求逆时实现递推关系；（iii）直接在正交基上评估正交多项式。所有改进确保了近似解的数值稳定性和准确性。详细阐述的该新方法在数值近似方面显著优于其他方法及不同tau实现。数值结果集展示了所引入数学技术的影响。
英文摘要：
In this paper an extension of the spectral Lanczos' tau method to systems of nonlinear integro-differential equations is proposed. This extension includes (i) linearization coefficients of orthogonal polynomials products issued from nonlinear terms and (ii) recursive relations to implement matrix inversion whenever a polynomial change of basis is required and (iii) orthogonal polynomial evaluations directly on the orthogonal basis. All these improvements ensure numerical stability and accuracy in the approximate solution. Exposed in detail, this novel approach is able to significantly outperform numerical approximations with other methods as well as different tau implementations. Numerical results on a set of problems illustrate the impact of the mathematical techniques introduced.
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