数学建模的步骤

数学建模的步骤 伴随着当今社会的科学技术的飞速发展，数学已经渗透到各个领域，数学建模也显得尤为重要。数学建模在人们生活中扮演着重要的角色，而且随着计算机技术的发展，数学建模更是在人类的活动中起着重要作用，数学建模也更好的为人类服务。 一、数学模型 数学模型是对于现实世界的一个特定对象,一个特定目的,根据特有的内在规律,做出一些必要的假设,运用适当的数学工具,得到一个数学结构. 简单地说:就是系统的某种特征的本质的数学表达式(或是用数学术语对部分现实世界的描述),即用数学式子(如函数,图形,代数方程,微分方程,积分方程,差分方程等)来描述(表述,模拟)所研究的客观对象或系统在某一方面的存在规律. 随着社会的发展,生物,医学,社会,经济……,各学科,各行业都涌现现出大量的实际课题,急待人们去研究,去解决.但是,社会对数学的需求并不只是需要数学家和专门从事数学研究的人才,而更大量的是需要在各部门中从事实际工作的人善于运用数学知识及数学的思维 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 来解决他们每天面临的大量的实际问题,取得经济效益和社会效益.他们不是为了应用数学知识而寻找实际问题(就像在学校里做数学应用题),而是为了解决实际问题而需要用到数学.而且不止是要用到数学,很可能还要用到别的学科,领域的知识,要用到工作经验和常识.特别是在现代社会,要真正解决一个实际问题几乎都离不开计算机.可以这样说,在实际工作中遇到的问题,完全纯粹的只用现成的数学知识就能解决的问题几乎是没有的.你所能遇到的都是数学和其他东西混杂在一起的问题,不是"干净的"数学,而是"脏"的数学.其中的数学奥妙不是明摆在那里等着你去解决,而是暗藏在深处等着你去发现.也就是说,你要对复杂的实际问题进行分析,发现其中的可以用数学语言来描述的关系或规律,把这个实际问题化成一个数学问题,这就称为数学模型. 数学模型具有下列特征:数学模型的一个重要特征是高度的抽象性.通过数学模型能够将形象思维转化为抽象思维,从而可以突破实际系统的约束,运用已有的数学研究成果对研究对象进行深入的研究.数学模型的另一个特征是经济性.用数学模型研究不需要过多的专用设备和工具,可以节省大量的设备运行和维护费用,用数学模型可以大大加快研究工作的进度,缩短研究周期,特别是在电子计算机得到广泛应用的今天,这个优越性就更为突出.但是,数学模型具有局限性,在简化和抽象过程中必然造成某些失真.所谓"模型就是模型"(而不是原型),即是指该性质. 二、数学建模 数学建模是利用数学方法解决实际问题的一种实践.即通过抽象,简化,假设,引进变量等处理过程后,将实际问题用数学方式表达,建立起数学模型,然后运用先进的数学方法及计算机技术进行求解.简而言之,建立数学模型的这个过程就称为数学建模. 模型是客观实体有关属性的模拟.陈列在橱窗中的飞机模型外形应当象真正的飞机,至于它是否真的能飞则无关紧要;然而参加航模比赛的飞机模型则全然不同,如果飞行性能不佳,外形再象飞机,也不能算是一个好的模型.模型不一定是对实体的一种仿照,也可以是对实体的某些基本属性的抽象,例如,一张地质图并不需要用实物来模拟,它可以用抽象的符号,文字和数字来反映出该地区的地质结构.数学模型也是一种模拟,是用数学符号,数学式子,程序,图形等对实际课题本质属性的抽象而又简洁的刻划,它或能解释某些客观现象,或能预测未来的发展规律,或能为控制某一现象的发展提供某种意义下的最优策略或较好策略.数学模型一般并非现实问题的直接翻版,它的建立常常既需要人们对现实问题深入细微的观察和分析,又需要人们灵活巧妙地利用各种数学知识.这种应用知识从实际课题中抽象,提炼出数学模型的过程就称为数学建模.实际问题中有许多因素,在建立数学模型时你不可能,也没有必要把它们毫无遗漏地全部加以考虑,只能考虑其中的最主要的因素,舍弃其中的次要因素.数学模型建立起来了,实际问题化成了数学问题,就可以用数学工具,数学方法去解答这个实际问题.如果有现成的数学工具当然好.如果没有现成的数学工具,就促使数学家们寻找和发展出新的数学工具去解决它,这又推动了数学本身的发展.例如,开普勒由行星运行的观测数据总结出开普勒三定律,牛顿试图用自己发现的力学定律去解释它,但当时已有的数学工具是不够用的,这促使了微积分的发明.求解数学模型,除了用到数学推理以外,通常还要处理大量数据,进行大量计算,这在电子计算机发明之前是很难实现的.因此,很多数学模型,尽管从数学理论上解决了,但由于计算量太大而没法得到有用的结果,还是只有束之高阁.而电子计算机的出现和迅速发展,给用数学模型解决实际问题打开了广阔的道路.而在现在,要真正解决一个实际问题,离了计算机几乎是不行的.数学模型建立起来了,也用数学方法或数值方法求出了解答,是不是就万事大吉了呢 不是.既然数学模型只能近似地反映实际问题中的关系和规律,到底反映得好不好,还需要接受检验,如果数学模型建立得不好,没有正确地描述所给的实际问题,数学解答再正确也是没有用的.因此,在得出数学解答之后还要让所得的结论接受实际的检验,看它是否合理,是否可行,等等.如果不符合实际,还应设法找出原因,修改原来的模型,重新求解和检验,直到比较合理可行,才能算是得到了一个解答,可以先付诸times makes balance of table views "sense volume" into 0.1mg/grid, and 0.01mg/grid, and 0.001mg/grid. Therefore, also known as one out of 10,000 scales, one out of 10,000, one out of 10,000 balance scales. Due to the lower actual sense, so that it can soon reach a balance to achieve fast weight, but also improve the accuracy of reading, and in the 10mg (of one out of 10,000 balance 1mg, 0.1mg one out of 10,000 balance) weighing no less than plus weight readings directly from the screen, is a balance of people love to use. Third, electro-optical balance weighing method 1. direct weighing methods: the alleged material such as small crucibles, beakers, petri ... ... Directly, such as electro-optical balance to the left (or right) on the weighing pan, add weights or turn the index to balance projection screen, plus weight is the weight of the material. 2. Specifies the weighing method: when weighing is not stable in water, in the air samples, such as metal specimens, ore ... ... And so on, this method can be used. The procedures of this method is: first known container weight (such as small beakers, Petri, such as paper, aluminium shovel, sulfuric acid), and then adds the index weights or disk to the specified weight, gently vibrating the sample slowly with a horn Spoon into containers, make balance consistent with said container. 3. decrement technique: when the sample is easy to absorb water and oxygen-sensitive or when reacting with carbon dioxide, then the sample is placed in a weighing bottle, weighed sample plus the total weight of the weighing bottle W1 g, and then pour a portion of the sample, and said the remaining sample plusComponents have the same statutory status as the SI units. Form 6, by a combination of the above units units of two, basic use of legal units of measurement 1. name of the legal units of measurement (1) name of the legal units of measurement, unless otherwise indicated, the name generally refers to the legal units of measurement, used in the descriptive text and the oral, is generally not used in formulas, data tables, drawings etc. Abbreviation of organization names are given in the table can be used as the unit of Chinese symbols (referred to as "Chinese symbol"). Chinese symbol only at primary and junior high school textbooks used when necessary and ordinary books. (2) name combination units, in principle, with the symbols of the order. Multiplication sign symbol does not correspond to the name, as long as the name read out in succession. For example: n • m name: NM abbreviation: NM divide slash (/), the corresponding name is "per", and there are several units in the denominator, "per" only shows up in front of the denominator. For example: j/(kg • K) name: joule per thousand kekaierwen abbreviations: kJ/kg open wrong: W/K/m W/K • m (3) if the unit with the power, the power of names shall be before the. Second power for the quadratic, cubic power three times, in turn, and so on. M 4 name: four meters (4) if the length of the second and third power representing area and volume, the corresponding index name are called the square and cubic, and before the unit of length such as: M2 name: sq m 2/s when used to represent the kinematic viscosity, and named "two meters per second"; When used to represent coverage rate, or "meters per second". (5) the powers of negative numbers meaning in addition to, either by using the power of names, you can also use "per". 2. legal units of measurement and the prefix symbols (1) unit of measure symbol, symbol is divided into units (international symbol) and unit symbols in 实施.但是,十全十美的 答案 八年级地理上册填图题岩土工程勘察试题省略号的作用及举例应急救援安全知识车间5s试题及答案 是没有的,已得到的解答仍有改进的余地,可以根据实际情况,或者继续研究和改进;或者暂时告一段落,待将来有新的情况和要求后再作改进. 应用数学知识去研究和和解决实际问题,遇到的第一项工作就是建立恰当的数学模型.从这一意义上讲,可以说数学建模是一切科学研究的基础.没有一个较好的数学模型就不可能得到较好的研究结果,所以,建立一个较好的数学模型乃是解决实际问题的关键之一.数学建模将各种知识综合应用于解决实际问题中,是培养和提高同学们应用所学知识分析问题,解决问题的能力的必备手段之一. 三、数学建模的一般方法 建立数学模型的方法并没有一定的模式,但一个理想的模型应能反映系统的全部重要特征:模型的可靠性和模型的使用性 建模的一般方法: 1.机理分析 机理分析就是根据对现实对象特性的认识,分析其因果关系,找出反映内部机理的规律,所建立的模型常有明确的物理或现实意义. (1) 比例分析法--建立变量之间函数关系的最基本最常用的方法. (2) 代数方法--求解离散问题(离散的数据,符号,图形)的主要方法. (3) 逻辑方法--是数学理论研究的重要方法,对社会学和经济学等领域的实际 问题,在决策,对策等学科中得到广泛应用. (4) 常微分方程--解决两个变量之间的变化规律,关键是建立"瞬时变化率" 的表达式. (5) 偏微分方程--解决因变量与两个以上自变量之间的变化规律. 2.测试分析方法 测试分析方法就是将研究对象视为一个"黑箱"系统,内部机理无法直接寻求,通过测量系统的输入输出数据,并以此为基础运用统计分析方法,按照事先确定的准则在某一类模型中选出一个数据拟合得最好的模型. (1) 回归分析法--用于对函数f(x)的一组观测值(xi,fi)i=1,2,…,n, 确定函数的表达式,由于处理的是静态的独立数据,故称为数理统计方法. (2) 时序分析法--处理的是动态的相关数据,又称为过程统计方法. (3) 回归分析法--用于对函数f(x)的一组观测值(xi,fi)i=1,2,…,n, 确定函数的表达式,由于处理的是静态的独立数据,故称为数理统计方法. (4) 时序分析法--处理的是动态的相关数据,又称为过程统计方法. 将这两种方法结合起来使用,即用机理分析方法建立模型的结构,用系统测试方法来确定模型的参数,也是常用的建模方法, 在实际过程中用那一种方法建模主要是根据我们对研究对象的了解程度和建模目的来决定.机理分析法建模的具体步骤大致可见左图. 3.仿真和其他方法 (1) 计算机仿真(模拟)--实质上是统计估计方法,等效于抽样试验. ? 离散系统仿真--有一组状态变量. ? 连续系统仿真--有解析表达式或系统结构图. (2) 因子试验法--在系统上作局部试验,再根据试验结果进行不断分析修改,求得所需的模型结构. (3) 人工现实法--基于对系统过去行为的了解和对未来希望达到的目标,并考虑到系统有关因素的可能变化,人为地组成一个系统.(参见:齐欢《数学模型方法》,华中理工大学出版社,1996) 四、数学模型的分类 数学模型可以按照不同的方式分类,下面介绍常用的几种. 1.按照模型的应用领域(或所属学科)分:如人口模型,交通模型,环境模型,生态模型,城镇规划模型,水资源模型,再生资源利用模型,污染模型等.范畴更大一些则形成许多边缘学科如生物数学,医学数学,地质数学,数量经济学,数学社会学等. 2.按照建立模型的数学方法(或所属数学分支)分:如初等数学模型,几何模型,微分方程模型,图论模型,马氏链模型,规划论模型等. 按第一种方法分类的数学模型教科书中,着重于某一专门领域中用不同方法建立模型,而按第二种方法分类的书里,是用属于不同领域的现成的数学模型来解释某种数学技巧的应用.在本书中我们重点放在如何应用读者已具备的基本数学知识在各个不同领域中建模. 3.按照模型的表现特性又有几种分法: 确定性模型和随机性模型 取决于是否考虑随机因素的影响.近年来随着数学的发展,又有所谓突变性模型和模糊性模型. 静态模型和动态模型 取决于是否考虑时间因素引起的变化. 线性模型和非线性模型 取决于模型的基本关系,如微分方程是否是线性的. 离散模型和连续模型 指模型中的变量(主要是时间变量)取为离散还是连续的. times makes balance of table views "sense volume" into 0.1mg/grid, and 0.01mg/grid, and 0.001mg/grid. Therefore, also known as one out of 10,000 scales, one out of 10,000, one out of 10,000 balance scales. Due to the lower actual sense, so that it can soon reach a balance to achieve fast weight, but also improve the accuracy of reading, and in the 10mg (of one out of 10,000 balance 1mg, 0.1mg one out of 10,000 balance) weighing no less than plus weight readings directly from the screen, is a balance of people love to use. Third, electro-optical balance weighing method 1. direct weighing methods: the alleged material such as small crucibles, beakers, petri ... ... Directly, such as electro-optical balance to the left (or right) on the weighing pan, add weights or turn the index to balance projection screen, plus weight is the weight of the material. 2. Specifies the weighing method: when weighing is not stable in water, in the air samples, such as metal specimens, ore ... ... And so on, this method can be used. The procedures of this method is: first known container weight (such as small beakers, Petri, such as paper, aluminium shovel, sulfuric acid), and then adds the index weights or disk to the specified weight, gently vibrating the sample slowly with a horn Spoon into containers, make balance consistent with said container. 3. decrement technique: when the sample is easy to absorb water and oxygen-sensitive or when reacting with carbon dioxide, then the sample is placed in a weighing bottle, weighed sample plus the total weight of the weighing bottle W1 g, and then pour a portion of the sample, and said the remaining sample plusComponents have the same statutory status as the SI units. Form 6, by a combination of the above units units of two, basic use of legal units of measurement 1. name of the legal units of measurement (1) name of the legal units of measurement, unless otherwise indicated, the name generally refers to the legal units of measurement, used in the descriptive text and the oral, is generally not used in formulas, data tables, drawings etc. Abbreviation of organization names are given in the table can be used as the unit of Chinese symbols (referred to as "Chinese symbol"). Chinese symbol only at primary and junior high school textbooks used when necessary and ordinary books. (2) name combination units, in principle, with the symbols of the order. Multiplication sign symbol does not correspond to the name, as long as the name read out in succession. For example: n • m name: NM abbreviation: NM divide slash (/), the corresponding name is "per", and there are several units in the denominator, "per" only shows up in front of the denominator. For example: j/(kg • K) name: joule per thousand kekaierwen abbreviations: kJ/kg open wrong: W/K/m W/K • m (3) if the unit with the power, the power of names shall be before the. Second power for the quadratic, cubic power three times, in turn, and so on. M 4 name: four meters (4) if the length of the second and third power representing area and volume, the corresponding index name are called the square and cubic, and before the unit of length such as: M2 name: sq m 2/s when used to represent the kinematic viscosity, and named "two meters per second"; When used to represent coverage rate, or "meters per second". (5) the powers of negative numbers meaning in addition to, either by using the power of names, you can also use "per". 2. legal units of measurement and the prefix symbols (1) unit of measure symbol, symbol is divided into units (international symbol) and unit symbols in 虽然从本质上讲大多数实际问题是随机性的,动态的,非线性的,但是由于确定性,静态,线性模型容易处理,并且往往可以作为初步的近似来解决问题,所以建模时常先考虑确定性,静态,线性模型.连续模型便于利用微积分方法求解,作理论分析,而离散模型便于在计算机上作数值计算,所以用哪种模型要看具体问题而定.在具体的建模过程中将连续模型离散化,或将离散变量视作连续,也是常采用的方法. 4.按照建模目的分:有描述模型,分析模型,预报模型,优化模型,决策模型,控制模型等. 5.按照对模型结构的了解程度分:有所谓白箱模型,灰箱模型,黑箱模型.这是把研究对象比喻成一只箱子里的机关,要通过建模来揭示它的奥妙.白箱主要包括用力学,热学,电学等一些机理相当清楚的学科描述的现象以及相应的工程技术问题,这方面的模型大多已经基本确定,还需深入研究的主要是优化设计和控制等问题了.灰箱主要指生态,气象,经济,交通等领域中机理尚不十分清楚的现象,在建立和改善模型方面都还不同程度地有许多工作要做.至于黑箱则主要指生命科学和社会科学等领域中一些机理(数量关系方面)很不清楚的现象.有些工程技术问题虽然主要基于物理,化学原理,但由于因素众多,关系复杂和观测困难等原因也常作为灰箱或黑箱模型处理.当然,白,灰,黑之间并没有明显的界限,而且随着科学技术的发展,箱子的"颜色"必然是逐渐由暗变亮的. 五、数学建模的一般步骤 建模的步骤一般分为下列几步: 1.模型准备.首先要了解问题的实际背景,明确题目的要求,搜集各种必要的信息. 2.模型假设.在明确建模目的,掌握必要资料的基础上,通过对资料的分析计算,找出起主要作用的因素,经必要的精炼,简化,提出若干符合客观实际的假设,使问题的主要特征凸现出来,忽略问题的次要方面.一般地说,一个实际问题不经过简化假设就很难翻译成数学问题,即使可能,也很难求解.不同的简化假设会得到不同的模型.假设作得不合理或过份简单,会导致模型失败或部分失败,于是应该修改和补充假设;假设作得过分详细,试图把复杂对象的各方面因素都考虑进去,可能使你很难甚至无法继续下一步的工作.通常,作假设的依据,一是出于对问题内在规律的认识,二是来自对数据或现象的分析,也可以是二者的综合.作假设时既要运用与问题相关的物理,化学,生物,经济等方面的知识,又要充分发挥想象力,洞察力和判断力,善于辨别问题的主次,果断地抓住主要因素,舍弃次要因素,尽量将问题线性化,均匀化.经验在这里也常起重要作用.写出假设时,语言要精确,就象做习题时写出已知条件那样. 3.模型构成.根据所作的假设以及事物之间的联系, 利用适当的数学工具去刻划各变量之间的关系,建立相应的数学结构――即建立数学模型.把问题化为数学问题.要注意尽量采取简单的数学工具,因为简单的数学模型往往更能反映事物的本质,而且也容易使更多的人掌握和使用. 4.模型求解.利用已知的数学方法来求解上一步所得到的数学问题,这时往往还要作出进一步的简化或假设.在难以得出解析解时,也应当借助计算机求出数值解. 5.模型分析.对模型解答进行数学上的分析,有时要根据问题的性质分析变量间的依赖关系或稳定状况,有时是根据所得结果给出数学上的预报,有时则可能要给出数学上的最优决策或控制,不论哪种情况还常常需要进行误差分析,模型对数据的稳定性或灵敏性分析等. 6.模型检验.分析所得结果的实际意义,与实际情况进行比较,看是否符合实际,如果结果不够理想,应该修改,补充假设或重新建模,有些模型需要经过几次反复,不断完善. 7.模型应用.所建立的模型必须在实际中应用才能产生效益,在应用中不断改进和完善.应用的方式自然取决于问题的性质和建模的目的. times makes balance of table views "sense volume" into 0.1mg/grid, and 0.01mg/grid, and 0.001mg/grid. Therefore, also known as one out of 10,000 scales, one out of 10,000, one out of 10,000 balance scales. Due to the lower actual sense, so that it can soon reach a balance to achieve fast weight, but also improve the accuracy of reading, and in the 10mg (of one out of 10,000 balance 1mg, 0.1mg one out of 10,000 balance) weighing no less than plus weight readings directly from the screen, is a balance of people love to use. Third, electro-optical balance weighing method 1. direct weighing methods: the alleged material such as small crucibles, beakers, petri ... ... Directly, such as electro-optical balance to the left (or right) on the weighing pan, add weights or turn the index to balance projection screen, plus weight is the weight of the material. 2. Specifies the weighing method: when weighing is not stable in water, in the air samples, such as metal specimens, ore ... ... And so on, this method can be used. The procedures of this method is: first known container weight (such as small beakers, Petri, such as paper, aluminium shovel, sulfuric acid), and then adds the index weights or disk to the specified weight, gently vibrating the sample slowly with a horn Spoon into containers, make balance consistent with said container. 3. decrement technique: when the sample is easy to absorb water and oxygen-sensitive or when reacting with carbon dioxide, then the sample is placed in a weighing bottle, weighed sample plus the total weight of the weighing bottle W1 g, and then pour a portion of the sample, and said the remaining sample plusComponents have the same statutory status as the SI units. Form 6, by a combination of the above units units of two, basic use of legal units of measurement 1. name of the legal units of measurement (1) name of the legal units of measurement, unless otherwise indicated, the name generally refers to the legal units of measurement, used in the descriptive text and the oral, is generally not used in formulas, data tables, drawings etc. Abbreviation of organization names are given in the table can be used as the unit of Chinese symbols (referred to as "Chinese symbol"). Chinese symbol only at primary and junior high school textbooks used when necessary and ordinary books. (2) name combination units, in principle, with the symbols of the order. Multiplication sign symbol does not correspond to the name, as long as the name read out in succession. For example: n • m name: NM abbreviation: NM divide slash (/), the corresponding name is "per", and there are several units in the denominator, "per" only shows up in front of the denominator. For example: j/(kg • K) name: joule per thousand kekaierwen abbreviations: kJ/kg open wrong: W/K/m W/K • m (3) if the unit with the power, the power of names shall be before the. Second power for the quadratic, cubic power three times, in turn, and so on. M 4 name: four meters (4) if the length of the second and third power representing area and volume, the corresponding index name are called the square and cubic, and before the unit of length such as: M2 name: sq m 2/s when used to represent the kinematic viscosity, and named "two meters per second"; When used to represent coverage rate, or "meters per second". (5) the powers of negative numbers meaning in addition to, either by using the power of names, you can also use "per". 2. legal units of measurement and the prefix symbols (1) unit of measure symbol, symbol is divided into units (international symbol) and unit symbols in