一类新复合型积分因子的存在充要条件

本文根据微分方程复合类型积分因子的定义,得到了复合类型积分因子存在的充要条件和计算公式,为解决某些非全微分方程求解问题提供了更加快捷的工具,避免了传统求解方法的繁琐及盲目。     

一类分数阶方程的数值解法

文章用有限差分法对Fisher分数阶微分方程进行近似和求解,对所建立的差分格式进行了合理的收敛性和稳定性分析,最后通过数值算例得到了方程的数值解表达式,并验证了数值解与精确解高度拟合,进而证明了该差分格式的可行性。     

对一道一阶直线型微分方程给出3种解法

对一道一阶直线型微分方程进行了探讨,并从多角度和不同思维方法进行分析求解,展示一题多解的技巧。     

Haar小波求解Fredholm-Volterra方程

本文运用Haar小波求解Fredholm-Volterra方程,建立了Haar小波的算子矩阵,利用Haar小波方法求解积分方程的基本思想是将求解积分方程的问题转化为求解一组代数方程组的问题。由于积分方程多出现在物理、工程等诸多应用性研究领域,且解析解难以求出,因此研究其数值解具有重要意义。     

常微分方程课程考试中的一道一阶隐式微分方程的“一题多解”

对常微分方程课程考试中的一道一阶隐式微分方程,通过参数法和变量分离法给出求解过程,显示初等解法的多样性与灵活性。     

一种新的高效的有限体积法

有限体积法是一种新的流体力学数值计算方法,其对于数值求解流体力学中常常遇到的偏微分方程十分有效。这篇文章采用有限体积法离散得到数值逼近格式,由原来的九点格式变成现在的五点格式,减少了节点的使用个数,且系数矩阵的条件数比较小,能保证离散的方程组有唯一解,且有较高的精度,并用提出的数值解方法求解二维的Laplace方程。数值计算结果与精确解作比较,吻合良好,且降低了计算机计算时所需要的时间。     

一阶常微分方程的一种新的数值求解方法

常微分方程作为微分方程的基本类型之一,在自然界与工程界有很广泛的应用.很多问题的数学表述都可以归结为常微分方程的定解问题,实际生活中很多问题的数学模型都是微分方程.但在许多情况下,首先找到问题的解析解,然后再进行相关的计算往往非常困难,有时甚至是行不通的,基于此理由,我们可以避免求解析解而直接求相应的数值解.本论文就是对目前已有的常微分方程的数值方法进行研究,并大胆地提出一种新的数值方法--欧拉-牛顿法.     

《数学实验》课程常微分方程求解的教学探索

常微分方程求解不仅是《高等数学》中的一个教学难点,同时也是《数学实验》课程中的教学重点及教学难点之一。本文首先介绍数值方法求解微分方程的思想,然后采用龙格库塔算法,通过matlab编程实现并画出图形,并用Simulink仿真模拟,最后实际应用到混沌系统的求解。完成了从应用数学知识建立数学模型、借助于matlab软件解决实际问题的一个过程。     

一阶常微分方程的一种新的数值求解方法

常微分方程作为微分方程的基本类型之一，在自然界与工程界有很广泛的应用。很多问题的数学表述都可以归结为常微分方程的定解问题，实际生活中很多问题的数学模型都是微分方程。但在许多情况下，首先找到问题的解析解，然后再进行相关的计算往往非常困难，有时甚至是行不通的，基于此理由，我们可以避免求解析解而直接求相应的数值解。本论文就是对目前已有的常微分方程的数值方法进行研究，并大胆地提出一种新的数值方法——欧拉-牛顿法。     

基于双指数跳扩散过程的亚式期权的解析定价

研究了双指数跳-扩散模型下亚式期权的定价,得到了这些期权定价得解析公式。在风险中性下,亚式期权的值在恰当的边际条件和终值条件下满足广义Black-Scholes方程;我们提出一种在跳扩散模型下亚式期权定价的新方法。该方法在于为亚式期权所满足的偏积分——微分方程指定恰当的边际条件和终值条件;然后,利用拉普拉斯变换求解该方程,得到了亚式期权的解析定价公式。     

基于Gauss求积公式的Newton迭代法

利用Gauss数值积分公式构造牛顿迭代法的变形格式,得到牛顿迭代方法的三个新格式,并证明了它们的收敛阶都为3。通过matlab编程进行数值试验,结果表明三个新格式具有较好的收敛速度。它们丰富了非线性方程求根的方法,在理论上和应用上都有一定的价值。     

Numerical solution of dynamic equilibrium models under Poisson uncertainty

We propose a simple and powerful numerical algorithm to compute the transition process in continuous-time dynamic equilibrium models with rare events. In this paper we transform the dynamic system of stochastic differential equations into a system of functional differential equations of the retarded type. We apply the Waveform Relaxation algorithm, i.e., we provide a guess of the policy function and solve the resulting system of (deterministic) ordinary differential equations by standard techniques. For parametric restrictions, analytical solutions to the stochastic growth model and a novel solution to Lucas' endogenous growth model under Poisson uncertainty are used to compute the exact numerical error. We show how (potential) catastrophic events such as rare natural disasters substantially affect the economic decisions of households.     

一种改进的遗传算法及其在函数优化中的应用

使用简单遗传算法(SGA)求解线性方程组时,由于易发生"早熟"现象,简单遗传算法求出的数值解误差很大甚至会失真。针对此问题,本文提出了一种改进的遗传算法(IGA),并设计了选择算子,交叉算子和变异算子。为了提高简单遗传算法抗"早熟"的能力,采用遗传算子结合惩罚函数,最佳个体保留以及种群迁移等措施。最后以核磁共振测井数学模型线性化后的大型病态线性方程组为例,对算法进行了测试。实验结果表明:同简单遗传算法相比较,IGA在一定程度上提高了数值解的精度。     

Evaluation of American option prices in a path integral framework using Fourier–Hermite series expansions

In this paper we review the path integral technique which has wide applications in statistical physics and relate it to the backward recursion technique which is widely used for the evaluation of derivative securities. We formulate the pricing of equity options, both European and American, using the path integral framework. Discretising in the time variable and using expansions in Fourier–Hermite series for the continuous representation of the underlying asset price, we show how these options can be evaluated in the path integral framework. For American options, the solution technique facilitates the accurate determination of the early exercise boundary as part of the solution. Additionally, the continuous representation of the state variable allows the relatively accurate and efficient evaluation of the option prices and the delta hedge ratio.     

The stochastic lake game: A numerical solution

In this paper, we numerically solve a stochastic dynamic programming problem for the solution of a stochastic dynamic game for which there is a potential function. The players select a mean level of control. The state transition dynamics is a function of the current state of the system and a multiplicative noise factor on the control variables of the players. The particular application is for lake water usage. The control variables are the levels of phosphorus discharged (typically by farmers) into the watershed of the lake, and the random shock is the rainfall that washes the phosphorus into the lake. The state of the system is the accumulated level of phosphorus in the lake. The system dynamics are sufficiently nonlinear so that there can be two Nash equilibria. A Skiba-like point can be present in the optimal control solution.We analyze (numerically) how the dynamics and the Skiba-like point change as the variance of the noise (the rain) increases. The numerical analysis uses a result of Dechert (1978. Optimal control problems from second order difference equations. Journal of Economic Theory 19, 50–63) to construct a potential function for the dynamic game. This greatly reduces the computational burden in finding Nash equilibria solutions for the dynamic game.     

基于良好率的航材保障经费预测

采用了航材保障良好率作为航材保障的效能指标,建立了航材保障良好率的数学模型。以航材保障经费为资源,平均航材保障良好率达到最大为目标函数,建立了航材保障经费的优化配置模型。用边际分析法求解,并建立了最优递推方程。通过实例演示了如何利用配置模型进行预测,计算工具是Matlab,进行了数值实验和理论分析,结果表明该模型具有良好的优化效果和实用性。     

How should a convertible bond be decomposed?

In this paper, a new decomposition approach for valuing convertible bonds (CBs) is presented. Through developing an appropriate integral representation for the value of convertible bonds, we show that an extra premium associated with the holder??s early conversion right exists in addition to the traditional split of a convertible bond into a straight bond plus an option. Three distinct cases, CBs with zero-coupon, continuous coupon and discrete coupon payments, are discussed in this article. Also, to validate our integral formulation, a numerical implementation for the solution of the CB value is conducted and some preliminary results are presented.     

Extend (r, Q) Inventory Model Under Lead Time and Ordering Cost Reductions When the Receiving Quantity is Different from the Ordered Quantity

This paper investigates the continuous review inventory model involving variable lead time with partial backorders, where the amount received is uncertain. The options of investing in ordering cost reduction is included, and lead time can be shortened at an extra crashing cost. The objective of this article is to simultaneously optimize the order quantity, reorder point, ordering cost and lead time. We first assume that the lead time demand follows a normal distribution and develop an algorithm to find the optimal solution. Then, we relax the assumption of normality to consider a distribution free case where only the mean and standard deviation of lead time demand are known. We apply the minimax distribution free procedure to solve this problem. For both cases, we also show that the objective cost function to be minimized is jointly convex in the decision variables. Furthermore, two numerical examples are given to illustrate the results.     

Semi-nonparametric estimation of independently and identically repeated first-price auctions via an integrated simulated moments method

In this paper we propose to estimate the value distribution of independently and identically repeated first-price auctions directly via a semi-nonparametric integrated simulated moments sieve approach. Given a candidate value distribution function in a sieve space, we simulate bids according to the equilibrium bid function involved. We take the difference of the empirical characteristic functions of the actual and simulated bids as the moment function. The objective function is then the integral of the squared moment function over an interval. Minimizing this integral to the distribution functions in the sieve space involved and letting the sieve order increase to infinity with the sample size then yields a uniformly consistent semi-nonparametric estimator of the actual value distribution. Also, we propose an integrated moment test for the validity of the first-price auction model, and an data-driven method for the choice of the sieve order. Finally, we conduct a few numerical experiments to check the performance of our approach.