特殊的两点边值问题——微分方程数值解课程改革案例
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摘要
考虑一个二阶非线性常微分方程两点边值问题,其解的唯一性依赖于给定的边值条件.通过该问题的特殊性,指出传统微分方程数值解课程中介绍的打靶法和有限差分方法的不足,引导学生构造有针对性的算法求解该特殊问题.再进一步启发学生根据该问题的特点,构造具有类似特性的两点边值问题,并用所构造的算法对猜想进行检验.
We study a second order nonlinear two-point boundary value problem of ordinary differential equations. The uniqueness of the solution of this problem depends on the given boundary value conditions. Through this particular problem we point out the shortage of the traditional shooting method and finite difference schemes that are taught in the course of numerical solutions of differential equations, and then guide students to construct a corresponding algorithm to solve the special problem. We further inspire the students to construct some two-point boundary value problems that may have similar properties, and to use the constructed algorithm to verify the guess.
We study a second order nonlinear two-point boundary value problem of ordinary differential equations. The uniqueness of the solution of this problem depends on the given boundary value conditions. Through this particular problem we point out the shortage of the traditional shooting method and finite difference schemes that are taught in the course of numerical solutions of differential equations, and then guide students to construct a corresponding algorithm to solve the special problem. We further inspire the students to construct some two-point boundary value problems that may have similar properties, and to use the constructed algorithm to verify the guess.
引文
[1]唐玲艳,屈田兴.微分方程数值解课程教学的实践与探索[J].湖南工业大学学报,2010,24(2):96~98
[2]杨韧,张志让.《微分方程数值解》课程教学改革与实践[J].大学数学,2011,27(4):19~22
[3]邓斌,朱晓临,张瑞丰,等.《微分方程数值解》课程教学改革的探索和实践[J].大学数学,2014,12(增刊1):56~58
[4]黄鹏展.教学与科研相结合原则在偏微分方程数值解教学中的实践[J].数学教育学报,2015,24(4):48~50
[5]Chen Y,Baule A,Touchette H,Just W.Weak-noise limit of a piecewise-smooth stochastic differential equation[J].Physical Review E,2013,88:052103
[6]姜礼尚,陈亚浙,刘西垣,等.数学物理方程讲义[M].第2版.北京:高等教育出版社,2005
[7]丁同仁,李承治.常微分方程教程[M].北京:高等教育出版社,2006
[8]胡健伟,汤怀民.微分方程数值方法[M].北京:科学出版社,2007
[2]杨韧,张志让.《微分方程数值解》课程教学改革与实践[J].大学数学,2011,27(4):19~22
[3]邓斌,朱晓临,张瑞丰,等.《微分方程数值解》课程教学改革的探索和实践[J].大学数学,2014,12(增刊1):56~58
[4]黄鹏展.教学与科研相结合原则在偏微分方程数值解教学中的实践[J].数学教育学报,2015,24(4):48~50
[5]Chen Y,Baule A,Touchette H,Just W.Weak-noise limit of a piecewise-smooth stochastic differential equation[J].Physical Review E,2013,88:052103
[6]姜礼尚,陈亚浙,刘西垣,等.数学物理方程讲义[M].第2版.北京:高等教育出版社,2005
[7]丁同仁,李承治.常微分方程教程[M].北京:高等教育出版社,2006
[8]胡健伟,汤怀民.微分方程数值方法[M].北京:科学出版社,2007