第二类分数阶Fredholm积分方程的配置法研究

发布时间：2018-12-14 07:12

【摘要】：积分方程这一重要的数学工具在力学、气象预报、振动理论、博弈论和粒子物理等学科中得到了广泛地应用。近年来,对于分数阶微积分的理论研究也得到了科学工作者的日益重视,并且在流体力学、模型设计、地质勘探、色散理论等方面得到了较好的效果。在有关分数阶积分的问题中,常常会涉及到对一些分数阶形式的积分方程进行计算。但是目前对于分数阶积分方程来说,求得其理论解比较困难,通常采用的办法是通过数值方法来得到它的数值结果。所以进一步完善它的数值解理论很有必要。本篇论文首先研究了第一类弱奇异Volterra积分方程的数值解法,通过引入广义Jacobi函数,将方程中未知函数进行了替换,并将方程转化为线性方程组的形式,从而得到其数值求解的格式,并对方程解的唯一性进行了证明。同时还对一类积分微分方程,通过采用广义Jacobi函数方法,给出了其配置法求解格式。其次对第二类分数阶Fredholm积分方程的数值解法进行了研究,通过引入广义Jacobi函数,将方程中未知函数进行了替换,通过配置法进行求解,将方程转化为线性方程组的形式,从而得到其数值求解的格式,并给出了方程解存在唯一的一个充分条件,同时,对方法的误差进行了分析,最后通过几个数值算例验证了方法是有效的。
[Abstract]:Integral equation, an important mathematical tool, has been widely used in mechanics, weather forecast, vibration theory, game theory and particle physics. In recent years, more and more attention has been paid to the theory of fractional calculus, and good results have been obtained in fluid mechanics, model design, geological exploration, dispersion theory and so on. In the problem of fractional integral, it is often involved in the calculation of some fractional integral equations. However, for fractional integral equation, it is difficult to obtain its theoretical solution, usually the method is to obtain its numerical results by numerical method. Therefore, it is necessary to further improve its numerical solution theory. In this paper, we first study the numerical solution of the first kind of weakly singular Volterra integral equation. By introducing the generalized Jacobi function, we replace the unknown function in the equation and transform the equation into the form of linear equation system. The scheme of numerical solution is obtained, and the uniqueness of the solution of the equation is proved. At the same time, for a class of integro-differential equations, the scheme of collocation method is given by using the generalized Jacobi function method. Secondly, the numerical solution of the second kind of fractional Fredholm integral equation is studied. By introducing the generalized Jacobi function, the unknown functions in the equation are replaced and solved by collocation method, and the equation is transformed into the form of linear equations. A sufficient condition for the existence and uniqueness of the solution of the equation is given. At the same time, the error of the method is analyzed. Finally, several numerical examples show that the method is effective.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.83

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