第二类Fredholm积分方程和Fisher方程的小波数值解

发布时间：2018-03-17 15:40

本文选题：第二类Fredholm积分方程　切入点：Fisher方程　出处：《海南师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：随着信息科技与经济的快速发展,现实生活中的许多物理现象所涉及到的数学问题大多都化为对积分方程和微分方程的求解问题,但是这些方程的解析解我们很难得到,于是需要利用计算机技术来求其数值解.常用的数值求解方法有差分变换法、Runge-Kutta法、有限元法等.这些常用的数值方法在求具有规则的、光滑性的解的方程时有很好的效果,但是这些数值解法尤其独特的优点的同时,也存在着不足,如在求解具有奇异性解的方程时,这些方法就显得力不从心.随着小波分析的发展,人们发现小波所具有的良好的时频局部化特征,以及其具有正交性、消失矩、紧支撑性等良好的性质,使得以小波函数作为基函数来求解积分方程和微分方程的数值解是可行的和有效的,给数值计算带来了方便.本文主要研究小波在积分方程和微分方程的数值解方面的应用.主要完成了以下工作:(1)介绍了小波分析的发展和应用,以及积分方程和微分方程的小波数值解法的研究进展.(2)简单阐述了小波分析的基本理论,包括小波的定义,连续小波变换、离散小波变换及其重构公式、小波多分辨分析以及与之相关的重要定理.(3)重点介绍分片多项式小波的构造方法.首先利用压缩映射以及小波的多分辨分析的性质和小波的正交性,在有界区域上构造分片多项式小波,然后利用Legendre多项式和小波的正交性得到分片线性多项式小波和分片二次多项式小波的显式表达式.显式表达式的给出为我们的数值计算带来了极大的方便.(4)利用分片多项式小波求第二类Fredholm积分方程组和Fisher方程的小波数值解.首先将分片线性多项式小波作为基函数,采用函数逼近的方法将第二类Fredholm积分方程组中的未知函数进行函数逼近,通过函数逼近可将第二类Fredholm积分方程组离散为关于分片线性多项式小波系数的代数方组,最后通过求解离散后的代数方程组从而求得第二类Fredholm积分方程组的小波数值解.同时,利用分片二次多项式小波作为基函数采用函数逼近将Fish-er方程中的微分算子进行函数逼近,然后采用配置法将Fisher方程转化为关于分片二次多项式小波系数的代数方程组,最后利用计算机编程可求得Fisher方程的小波数值解.最后的数值例子表明,本文的小波方法是可行的和有效的.
[Abstract]:With the rapid development of information technology and economy, the mathematical problems involved in many physical phenomena in real life are mostly reduced to solving integral equations and differential equations, but the analytical solutions of these equations are difficult to obtain. So we need to use computer technology to find its numerical solution. The commonly used numerical methods are the difference transformation method Runge-Kutta method, the finite element method and so on. These commonly used numerical methods have very good effect in solving the equation with regular and smooth solution. However, while these numerical methods have special advantages, they also have some disadvantages. For example, in solving equations with singular solutions, these methods seem to be unable to do the same. With the development of wavelet analysis, It is found that wavelet has good time-frequency localization characteristics, and it has good properties such as orthogonality, vanishing moment, compact support, etc. So that it is feasible and effective to use wavelet function as the basis function to solve the numerical solutions of integral equations and differential equations. This paper mainly studies the application of wavelet in the numerical solution of integral equation and differential equation. The main work is as follows: 1) the development and application of wavelet analysis are introduced. The basic theory of wavelet analysis, including the definition of wavelet, continuous wavelet transform, discrete wavelet transform and its reconstruction formula, is briefly described. The construction of piecewise polynomial wavelets is introduced in detail. Firstly, by using the properties of compression mapping and wavelet multi-resolution analysis and the orthogonality of wavelets, we introduce the method of constructing piecewise polynomial wavelets. The piecewise polynomial wavelets are constructed on the bounded domain. Then the explicit expressions of piecewise linear polynomial wavelets and piecewise quadratic polynomial wavelets are obtained by using the orthogonality of Legendre polynomials and wavelets. The piecewise polynomial wavelets are used to solve the Fredholm integral equations of the second kind and the wavelet numerical solutions of the Fisher equations. Firstly, the piecewise linear polynomial wavelets are taken as the basis functions. The unknown functions of the second kind of Fredholm integral equations are approximated by the method of function approximation. The second kind of Fredholm integral equations can be discretized into algebraic groups of wavelet coefficients of piecewise linear polynomials by function approximation. Finally, by solving the discrete algebraic equations, the wavelet numerical solution of the second kind of Fredholm integral equations is obtained. At the same time, the differential operators in the Fish-er equation are approximated by the piecewise quadratic polynomial wavelet as the basis function. Then the Fisher equation is transformed into algebraic equations about piecewise quadratic polynomial wavelet coefficients by collocation method. Finally, the wavelet numerical solution of Fisher equation can be obtained by computer programming. The wavelet method in this paper is feasible and effective.
【学位授予单位】：海南师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.8

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