常微分方程理论在数学建模中的简单应用
|
摘要
众所周知,自然界中一切物质都按照自身的规律在运动和演变,不同物质的运动规律总是在时间和空间中运动着的,虽然物质的运动形式千差万别,但我们总可以找到它们共性的一面,即具有共同的量的变化规律。为了能够定性和定量的研究一些特定的运动和演变过程,就必须将物质运动和演变过程中相关的因素进行数学化。这种数学化的过程就是数学建模的过程,即根据运动和演变规律找出不同变量之间互相制约、互相影响的关系式。由于大量的实际问题中,稍微复杂一些的运动过程往往不能直接写出他们的函数,却容易建立变量及其导数(或微分)间的关系式,即微分方程。微分方程描述的是物质运动的瞬时规律。将常微分方程应用于数学建模是因为常微分方程理论是用数学方法解决实际问题的强有力的工具,是一门有着重要背景应用的学科,具有悠久的历史,系统理论日臻完善,而且继续保持着进一步发展的活力,其主要原因是它的根源深扎在各种实际问题中。
作者通过查阅近几十年来的参考文献或书籍资料,对常微分方程在数学建模中的应用进行了详尽的综述。主要内容包括以下几个方面：
1.数学模型和常微分方程的发展情况。介绍了数学模型在不同学科与针对不同实际问题而产生的不同的定义,以及根据不同标准进行的详细分类；简要给出了数学建模的具体步骤以及每个步骤中需要解决的关键问题,其中采用的方法中最主要的就是使用常微分方程理论；最后通过讨论如何在雨中行走才能减少淋雨的程度这一实例给出了如何使用数学模型结合常微分方程理论解决实际问题的示范。
2.详细介绍了常微分方程建模的原理、主要方法和步骤。介绍了常微分方程建模的具体步骤,几种主要方法和注意事项；给出了一阶常微分方程建模的两个典型实例,打假问题模型和商业广告模型,通过这些实例详细演示了常微分方程建模的步骤与方法在实际问题中的应用；第二,介绍了一阶非线性常微分方程模型,研究了最速降线和单种群密度制约两个经典问题的数学模型的建立。将数学建模的几个主要步骤具体应用到实际问题中得到了基于常微分方程的数学模型。第三介绍了二阶常微分方程模型,包括追击问题和振动问题的数学模型建立,构建了二阶的追击运动轨迹模型和弹簧振子的二阶线性微分方程模型；并考虑共振问题中有阻力和无阻力的两种应用。
3.介绍了常微分力程模型中的稳定性问题。首先介绍了常微分方程中的稳定性基本理论,分别介绍了一维和二维动力系统中的平衡点与稳定性理论,以及非线性系统线性计时系统的稳定性；其次介绍了捕鱼持续收获模型和种群相互竞争模型的构建,重点进行了模型的稳定性分析与问题的求解。
It is well known that all of the substances in nature have, their own rules in the movement and evolution. Although the forms of material movement vary greatly, the different material always moves in time and space, one can always figure out their common properties, namely, the evolution rules of the common variables. In order to qualitatively and quantitatively research the specific movement and evolution, the relevant factors in the material movement and the process of evolution are needed to be represented using mathematical varibles and mathematical equations are employed to describe their relations. This kind of quantization process is the procedure of mathematical modeling, that is, identify the relations among different variables according to the movement.
Due to various of practical problems, the functions often can not be represented directly according to some slightly complicated processes of movement, but it is easy to set up the relationship between variable and their derivatives (or differentials); This relationship is called Differential Equations in mathematics. The application of Ordinary Differential Equations to mathematical modeling has its deep background in that the theory of Ordinary Differential Equations is a powerful tool to solve practical problems in mathematical modeling, which is a subject with vital background application Ordinary Differential Equation theory has a long history and it is a systematic theory which will be perfected gradually in continuously further development. The main reason for this is that its sources are deeply rooted in various practical problems.
The author has made detailed and integrated descriptions as to the applications of Ordinary Differential Equations to mathematical modeling by consulting reference literature, materials and books. The main results are given as follows:
1. A short introduction on the development of mathematical modeling and Ordinary Differential Equation. We introduce different definitions of mathematical models according to different practical problems and the classification of mathematical modeling based on the definitions. Detailed steps in mathematical modeling are concluded and the key point in every step is pointed out, where the main mathematical method to solve is Ordinary Differential Equation theory. Finally, an practical example, how to walk in the rain to maximally avoid the rain, is given to show how to do the mathematical modeling combined with Ordinary Differential Equation theory.
2. A detailed description on the principle,method and process of mathematical modeling using Ordinary Differential Equation theory is given. First, the detailed steps to mathematical modeling by Ordinary Differential Equation theory are first introduced. We also present some methods to use and the key points which should be mentioned. By two typical examples, counterfeit and advertising, the process of mathematical modeling are explained in detail which implies that the practical application of mathematical modeling. Second, mathematical modeling based on first-order nonlinear Ordinary Differential Equation is introduced to consider the mathematical modeling of two classical problems, Brachistochrone and biotic population dynamics. Main procedure of mathematical modeling is applied to investigate practical problems. Third, second-order nonlinear Ordinary Differential Equations are introduced to construct the mathematical modeling in chasing problem and vibration problem. We establish the second-order chasing dynamics model as well as vibration equation.
3. The stability of Ordinary Differential Equations models are introduced. The fundamental theory on stability of Ordinary Differential Equations are discussed, equilibriums and stability of one-dimensional and two-dimensional of Ordinary Differential Equations are presented, respectively. The mathematical model to maintain continual harvest in the fishing and the competition among biotic population model are presented. The stability of both models are analyzed and two problems are solved based on the stability analysis
作者通过查阅近几十年来的参考文献或书籍资料,对常微分方程在数学建模中的应用进行了详尽的综述。主要内容包括以下几个方面：
1.数学模型和常微分方程的发展情况。介绍了数学模型在不同学科与针对不同实际问题而产生的不同的定义,以及根据不同标准进行的详细分类；简要给出了数学建模的具体步骤以及每个步骤中需要解决的关键问题,其中采用的方法中最主要的就是使用常微分方程理论；最后通过讨论如何在雨中行走才能减少淋雨的程度这一实例给出了如何使用数学模型结合常微分方程理论解决实际问题的示范。
2.详细介绍了常微分方程建模的原理、主要方法和步骤。介绍了常微分方程建模的具体步骤,几种主要方法和注意事项；给出了一阶常微分方程建模的两个典型实例,打假问题模型和商业广告模型,通过这些实例详细演示了常微分方程建模的步骤与方法在实际问题中的应用；第二,介绍了一阶非线性常微分方程模型,研究了最速降线和单种群密度制约两个经典问题的数学模型的建立。将数学建模的几个主要步骤具体应用到实际问题中得到了基于常微分方程的数学模型。第三介绍了二阶常微分方程模型,包括追击问题和振动问题的数学模型建立,构建了二阶的追击运动轨迹模型和弹簧振子的二阶线性微分方程模型；并考虑共振问题中有阻力和无阻力的两种应用。
3.介绍了常微分力程模型中的稳定性问题。首先介绍了常微分方程中的稳定性基本理论,分别介绍了一维和二维动力系统中的平衡点与稳定性理论,以及非线性系统线性计时系统的稳定性；其次介绍了捕鱼持续收获模型和种群相互竞争模型的构建,重点进行了模型的稳定性分析与问题的求解。
It is well known that all of the substances in nature have, their own rules in the movement and evolution. Although the forms of material movement vary greatly, the different material always moves in time and space, one can always figure out their common properties, namely, the evolution rules of the common variables. In order to qualitatively and quantitatively research the specific movement and evolution, the relevant factors in the material movement and the process of evolution are needed to be represented using mathematical varibles and mathematical equations are employed to describe their relations. This kind of quantization process is the procedure of mathematical modeling, that is, identify the relations among different variables according to the movement.
Due to various of practical problems, the functions often can not be represented directly according to some slightly complicated processes of movement, but it is easy to set up the relationship between variable and their derivatives (or differentials); This relationship is called Differential Equations in mathematics. The application of Ordinary Differential Equations to mathematical modeling has its deep background in that the theory of Ordinary Differential Equations is a powerful tool to solve practical problems in mathematical modeling, which is a subject with vital background application Ordinary Differential Equation theory has a long history and it is a systematic theory which will be perfected gradually in continuously further development. The main reason for this is that its sources are deeply rooted in various practical problems.
The author has made detailed and integrated descriptions as to the applications of Ordinary Differential Equations to mathematical modeling by consulting reference literature, materials and books. The main results are given as follows:
1. A short introduction on the development of mathematical modeling and Ordinary Differential Equation. We introduce different definitions of mathematical models according to different practical problems and the classification of mathematical modeling based on the definitions. Detailed steps in mathematical modeling are concluded and the key point in every step is pointed out, where the main mathematical method to solve is Ordinary Differential Equation theory. Finally, an practical example, how to walk in the rain to maximally avoid the rain, is given to show how to do the mathematical modeling combined with Ordinary Differential Equation theory.
2. A detailed description on the principle,method and process of mathematical modeling using Ordinary Differential Equation theory is given. First, the detailed steps to mathematical modeling by Ordinary Differential Equation theory are first introduced. We also present some methods to use and the key points which should be mentioned. By two typical examples, counterfeit and advertising, the process of mathematical modeling are explained in detail which implies that the practical application of mathematical modeling. Second, mathematical modeling based on first-order nonlinear Ordinary Differential Equation is introduced to consider the mathematical modeling of two classical problems, Brachistochrone and biotic population dynamics. Main procedure of mathematical modeling is applied to investigate practical problems. Third, second-order nonlinear Ordinary Differential Equations are introduced to construct the mathematical modeling in chasing problem and vibration problem. We establish the second-order chasing dynamics model as well as vibration equation.
3. The stability of Ordinary Differential Equations models are introduced. The fundamental theory on stability of Ordinary Differential Equations are discussed, equilibriums and stability of one-dimensional and two-dimensional of Ordinary Differential Equations are presented, respectively. The mathematical model to maintain continual harvest in the fishing and the competition among biotic population model are presented. The stability of both models are analyzed and two problems are solved based on the stability analysis
引文
[1]汪娜.微分方程在数学建模中的应用[J].安庆师范学院学报(自然科学版),2009,(01)
[2]廖军.浅析数学建模在数学教学中的重要性[J].文山师范高等专科学校学报,2001,(01)
[3]边学军.增强高等数学课教学效果初探[J].边疆经济与文化,2006,(06)
[4]高国继.对高职院校数学教学的一些思考[J].甘肃科技纵横,2007,(04)
[5]张云霞.数学建模与高等数学教学[J].山西财政税务专科学校学报2001,(04)
[6]苏有慧,姜英姿.用数学建模教育活动推动高校数学教学改革[J].徐州教育学院学报,2008,(03)
[7]郝美艳.人文主义教育思想对高中数学教学的启示[J].科技信息2009,(01)
[8]宋丹萍.在数学教学中渗透建模思想[J].科技资讯,2008,(36)
[9]蒋继宏,严尚安,蒋银华,赵静.以数学建模为突破口促进数学教学改革[J].工科数学,2002,(01)
[10]宋益荣,张瑛.在数学教学中培养学生的创造思维能力例谈[J].科技信息(学术研究),2008,(27)
[11]王树禾,数学模型选讲,北京科学出版社,2008
[12]任善强,雷鸣编著,数学模型,重庆大学出版社出版,1998年4月第2版
[13]Frank R.Giordano Maurice D.weir William P.Fox著A First Course in Marhcmatical Modeling机械工业出版社
[14]王兵团主编,数学建模基础,清华大学出版社,北京交通大学出版社,
[15]王庚,王敏生著,现代数学建模方法,科学出版社,
[16]湖北省大学生数学建模竞赛专家组编,数学建模,华中科技大学出版社,
[17]冯杰,黄力伟,王勤,尹成义编著,数学建模原理与案例,科学出版社,
[18]蔡锁章主编,数学建模原理与方法,海洋出版社。
[19]王贺元,徐美进.搞好数学建模参赛工作促进工科数学教学改革[J].辽宁工学院学报(社会科学版),2007,(04)
[20]蒋利平,董玉成.大学生数学建模竞赛的独特魅力[J].数学的实践与认识,2002,(02)
[21]高秀娟,李占培.数学教学改革与大学生创新能力培养[J].白城师范学院学报,2005,(05)
[22]刘宝炜,刘凤华.关注大学生数学建模竞赛推动数学教学改革[J].沧州师范专科学校学报,2004,(01)
[23]凌巍炜.高职院校数学建模活动的探索与实践[J].职业教育研究2007,(12)
[24]李继成,朱旭,王绵森,武忠祥.《数学实验》课程建设及分层次教学与实践[J].大学数学,2005,(06)
[25]严培胜.数学建模与经济数学教学改革[J].科协论坛(下半月)2007,(05)
[26]唐小峰,张嘎.浅谈大学生数学建模竞赛和大学数学教学改革[J].科教文汇(上半月),2007,(03)
[27]罗卫民,李昌兴,史克刚.“数学实验”与“数学建模”课程教学改革[J].高等工程教育研究,2005,(06)
[28]杨红伟.数学建模与创新能力的培养[J].兵团教育学院学报,2003,(01)
[29]赵建昕.提高数学建模能力的策略研究[J].数学教育学报,2004,(03)
[30]周义仓,赫孝良.数学建模与创新人才培养[J].西安交通大学学报(社会科学版),2000,(S1)
[31]崔建斌.以数学建模为切入点全面推进大学数学教学改革[J].宜春学院学报,2004,(04)
[32]于海涛.微分方程的应用[J].齐齐哈尔师范高等专科学校学报,2008,(06)
[33]刘琴.微分方程在经济领域的应用[J].考试周刊,2008,(18)
[34]熊佐亮,蒋鹏,朱向洪,黄先玖.微分方程应用若干举例[J].江西教育学院学报,2006,(06)
[35]许敏伟,吴炳华.变量代换法在求解微分方程问题中的应用[J].徐州教育学院学报,2008,(03)
[36]卢志中,王长义.高阶微分方程在工程中的应用[J].兰州工业高等专科学校学报,1997,(02)
[37]袁建清.积分不等式的应用[J].太原城市职业技术学院学报,2008,(05)
[38]王春草.常数变易法在求解微分方程中的应用[J].杨凌职业技术学院学报,2007,(04)
[39]林振聲.近乎線性微分方程系的迥期解[J].厦门大学学报(哲学社会科学版),1954,(04)
[40]敖恩.一个不等式及其应用[J].昭乌达蒙族师专学报,2003,(02)
[41]马玉琼.计算机辅助设计在教学中的应用[J].沧州师范专科学校学报2003,(03)
[42]欧阳瑞,孙要伟.常微分方程在数学建模中的应用[J].宿州教育学院学报,2008,(02)
[43]郭爽,李秀丽,高云伟.如何应用微分方程理论进行数学建模[J].大庆师范学院学报,2007,(02)
[44]刘慧,宋广华.微分方程在数学建模中的应用探究[J].科技信息(科学教研),2008,(13)
[45]贺玉兰.数学建模初探[J].当代教育论坛,2005,(08)
[46]王慧群.浅谈数学建模及其特点[J].晋东南师范专科学校学报,2001,(03)
[47]罗威.数学建模探析[J].辽宁经济职业技术学院.辽宁经济管理干部学院学报,2004,(02)
[48]王庚,张珠宝.数学建模融入微积分教学单元[J].大学数学,2006,(04)
[2]廖军.浅析数学建模在数学教学中的重要性[J].文山师范高等专科学校学报,2001,(01)
[3]边学军.增强高等数学课教学效果初探[J].边疆经济与文化,2006,(06)
[4]高国继.对高职院校数学教学的一些思考[J].甘肃科技纵横,2007,(04)
[5]张云霞.数学建模与高等数学教学[J].山西财政税务专科学校学报2001,(04)
[6]苏有慧,姜英姿.用数学建模教育活动推动高校数学教学改革[J].徐州教育学院学报,2008,(03)
[7]郝美艳.人文主义教育思想对高中数学教学的启示[J].科技信息2009,(01)
[8]宋丹萍.在数学教学中渗透建模思想[J].科技资讯,2008,(36)
[9]蒋继宏,严尚安,蒋银华,赵静.以数学建模为突破口促进数学教学改革[J].工科数学,2002,(01)
[10]宋益荣,张瑛.在数学教学中培养学生的创造思维能力例谈[J].科技信息(学术研究),2008,(27)
[11]王树禾,数学模型选讲,北京科学出版社,2008
[12]任善强,雷鸣编著,数学模型,重庆大学出版社出版,1998年4月第2版
[13]Frank R.Giordano Maurice D.weir William P.Fox著A First Course in Marhcmatical Modeling机械工业出版社
[14]王兵团主编,数学建模基础,清华大学出版社,北京交通大学出版社,
[15]王庚,王敏生著,现代数学建模方法,科学出版社,
[16]湖北省大学生数学建模竞赛专家组编,数学建模,华中科技大学出版社,
[17]冯杰,黄力伟,王勤,尹成义编著,数学建模原理与案例,科学出版社,
[18]蔡锁章主编,数学建模原理与方法,海洋出版社。
[19]王贺元,徐美进.搞好数学建模参赛工作促进工科数学教学改革[J].辽宁工学院学报(社会科学版),2007,(04)
[20]蒋利平,董玉成.大学生数学建模竞赛的独特魅力[J].数学的实践与认识,2002,(02)
[21]高秀娟,李占培.数学教学改革与大学生创新能力培养[J].白城师范学院学报,2005,(05)
[22]刘宝炜,刘凤华.关注大学生数学建模竞赛推动数学教学改革[J].沧州师范专科学校学报,2004,(01)
[23]凌巍炜.高职院校数学建模活动的探索与实践[J].职业教育研究2007,(12)
[24]李继成,朱旭,王绵森,武忠祥.《数学实验》课程建设及分层次教学与实践[J].大学数学,2005,(06)
[25]严培胜.数学建模与经济数学教学改革[J].科协论坛(下半月)2007,(05)
[26]唐小峰,张嘎.浅谈大学生数学建模竞赛和大学数学教学改革[J].科教文汇(上半月),2007,(03)
[27]罗卫民,李昌兴,史克刚.“数学实验”与“数学建模”课程教学改革[J].高等工程教育研究,2005,(06)
[28]杨红伟.数学建模与创新能力的培养[J].兵团教育学院学报,2003,(01)
[29]赵建昕.提高数学建模能力的策略研究[J].数学教育学报,2004,(03)
[30]周义仓,赫孝良.数学建模与创新人才培养[J].西安交通大学学报(社会科学版),2000,(S1)
[31]崔建斌.以数学建模为切入点全面推进大学数学教学改革[J].宜春学院学报,2004,(04)
[32]于海涛.微分方程的应用[J].齐齐哈尔师范高等专科学校学报,2008,(06)
[33]刘琴.微分方程在经济领域的应用[J].考试周刊,2008,(18)
[34]熊佐亮,蒋鹏,朱向洪,黄先玖.微分方程应用若干举例[J].江西教育学院学报,2006,(06)
[35]许敏伟,吴炳华.变量代换法在求解微分方程问题中的应用[J].徐州教育学院学报,2008,(03)
[36]卢志中,王长义.高阶微分方程在工程中的应用[J].兰州工业高等专科学校学报,1997,(02)
[37]袁建清.积分不等式的应用[J].太原城市职业技术学院学报,2008,(05)
[38]王春草.常数变易法在求解微分方程中的应用[J].杨凌职业技术学院学报,2007,(04)
[39]林振聲.近乎線性微分方程系的迥期解[J].厦门大学学报(哲学社会科学版),1954,(04)
[40]敖恩.一个不等式及其应用[J].昭乌达蒙族师专学报,2003,(02)
[41]马玉琼.计算机辅助设计在教学中的应用[J].沧州师范专科学校学报2003,(03)
[42]欧阳瑞,孙要伟.常微分方程在数学建模中的应用[J].宿州教育学院学报,2008,(02)
[43]郭爽,李秀丽,高云伟.如何应用微分方程理论进行数学建模[J].大庆师范学院学报,2007,(02)
[44]刘慧,宋广华.微分方程在数学建模中的应用探究[J].科技信息(科学教研),2008,(13)
[45]贺玉兰.数学建模初探[J].当代教育论坛,2005,(08)
[46]王慧群.浅谈数学建模及其特点[J].晋东南师范专科学校学报,2001,(03)
[47]罗威.数学建模探析[J].辽宁经济职业技术学院.辽宁经济管理干部学院学报,2004,(02)
[48]王庚,张珠宝.数学建模融入微积分教学单元[J].大学数学,2006,(04)