Volterra型比例延迟积分微分方程的数值解：基于移位雅可比多项式的配置法
Numerical Solution of Pantograph Delay Integrodifferential Equation of Volterra Type: Collocation Method Based on Shifted Jacobi Polynomials
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中文总结 AI 辅助
针对Volterra型一阶比例延迟积分微分方程，提出基于移位雅可比多项式的配置法，通过牛顿法求解代数方程组，数值算例表明其逼近效果优于对比方法。
中文摘要 AI 辅助
比例延迟（Pantograph）问题出现在电力机车、材料建模以及量子点激光器建模中。比例延迟积分微分方程是一类含比例延迟的积分微分方程，广泛应用于电动力学、流行病学、控制理论、天体物理学、经济学及工程等领域。本研究针对一类带初值条件的一阶Volterra型比例延迟积分微分方程，提出一种基于移位雅可比多项式（shifted Jacobi polynomials）的高效配置法以获取其数值解。该方法将控制方程的解表示为待确定展开系数的移位雅可比多项式级数，通过在移位雅可比多项式的根处进行配置，将原问题转化为关于该级数解未知展开系数的代数方程组。随后采用牛顿法求解该代数方程组，得到展开系数的数值，再将这些系数代入假设的级数解中，即可得到所需的数值解。通过数值算例验证了移位雅可比配置法的适用性、可靠性、效率与精度，将该方法的结果与精确解及其他已发表结果进行比较，表格中呈现的误差对比表明，本方法对解的逼近效果优于所对比的方法。
英文摘要
Pantograph arises in electric trains, material modelling, and the modelling of quantum dot lasers. Pantograph integrodifferential equations are integrodifferential equations involving proportional delays; and they appear in fields such as electrodynamics, epidemiology, control theory, astrophysics, economics, and engineering. This research presents an efficient collocation method based on shifted Jacobi polynomials for obtaining numerical solutions of a class of first order pantograph delay integrodifferential equation of Volterra type with an initial condition. The proposed method expresses the solution of the governing equation as a shifted Jacobi polynomial series with expansion coefficients which are to be determined. Collocating at the roots of the shifted Jacobi polynomials, the underlying problem is reduced to a system of algebraic equations in the unknown expansion coefficients of the shifted Jacobi polynomial series solution. Newton's method is subsequently used to solve the resulting system of algebraic equations and numerical values of the expansion coefficients are obtained. The obtained coefficients are substituted into the assumed series solution to obtain the required numerical solutions. The applicability, reliability, efficiency, and accuracy of the shifted Jacobi collocation method are demonstrated through illustrative examples. Results obtained using the proposed method are compared with exact solutions and other published results. Comparisons of errors which are presented in tables reveal that our method approximates the solution better than the methods under comparison.
