爆破型Volterra积分微分方程的高精度计算  

High Precision Computation for Volterra Integro-differential Equations with Blow-up

作　　者：贾成旺 王雨轩 王同科[1] JIA Chengwang;WANG Yuxuan;WANG Tongke(School of Mathematical Sciences,Tianjin Normal University,Tianjin 300387,China)

机构地区：[1]天津师范大学数学科学学院,天津300387

出　　处：《应用数学》2024年第3期792-804,共13页Mathematica Applicata

基　　金：天津师范大学研究生科研创新资助项目(2023KYCX055Y);国家自然科学基金资助项目(11971241)。

摘　　要：本文求解爆破型非线性Volterra积分微分方程,使用Picard迭代求出解在初始点的有限项分数阶级数展开式.使用级数解分离出方程的初始奇点,提出一种高效的Chebyshev配置法.对于爆破问题,使用待定系数法得到方程解在爆破时刻的渐近展开式主项,并把级数解Padé逼近分母的最小正根作为爆破时间的初步估计,进一步通过积分或微分变换提高爆破时间的预测精度.最后,数值算例验证级数解、Chebyshev配置解和爆破时间预测方法的正确性和有效性.In this paper,we solve the nonlinear Volterra integro-differential equation with blow-up.The finite-term fractional series expansion for the solution around the initial point is obtained by using the Picard iteration.By separating the initial singularity of the equation from the series solution,we propose an efficient Chebyshev collocation method.For the blow-up problem,the leading term of the asymptotic expansion for the equation around the blow-up time is obtained by the method of undetermined coefficients.The smallest positive root of the denominator of the Pade approximant generated by the series solution is taken as the preliminary estimate of the blow-up time,and the predicted accuracy of the blow-up time is further improved by integral or differential transformation.Finally,numerical examples are given to confirm the correctness and effectiveness of the series solution,Chebyshev collocation solution and blow-up time prediction method.

关 键 词：积分微分方程 级数解 PADÉ逼近 Chebyshev配置法 爆破时间估计 

分 类 号：O241.83[理学—计算数学]