分数阶积分微分方程的B样条小波解法

发布时间：2018-05-05 17:37

本文选题：B样条小波 + 分数阶微积分　；参考：《宁夏大学》2017年硕士论文

【摘要】：分数阶微积分被称为现实世界和数学理论完美结合的一种崭新的数学工具,被广泛应用到粘弹性力学、统计与随机过程、信号分析处理等各个不同的领域.分数阶微积分方程比整数阶微积分方程更能真实客观地刻画许多复杂的物理过程,成为复杂力学与物理过程数学建模的重要工具之一.因此,有关分数阶微积分方程的理论与计算方法的研究就显得尤为迫切,在应用领域起着至关重要的作用.遗憾的是,大部分分数阶积分微分方程的解析解很复杂,计算困难,消耗大量时间;况且并非所有的分数阶微积分方程都能得到其解析解.因此,发展新数值算法,建立分数阶微积分方程的数值方法是非常必要的,有着重要的理论意义和实际应用价值.本文主要求解了几类Riemann-Liouville分数阶微积分方程.第一章简要介绍了研究背景、研究意义及国内外研究现状.第二章推导了半正交B样条小波的分数阶积分算子矩阵.第三章证明了第二类分数阶Fredholm积分方程解的存在唯一性,利用半正交B样条小波求解了第二类分数阶Fredholm积分方程的数值解.算例针对精确解未知的方程给出其数值解.第四章证明了第二类分数阶Fredholm积分方程组解的存在唯一性,利用半正交B样条小波求解第二类分数阶Fredholm积分方程组的数值解.并针对精确解未知的情况给出误差分析.第五章研究了分数阶非线性Fredholm积分微分方程,B样条小波分数阶积分算子矩阵将积分微分方程离散为代数方程组,数值算例验证了此方法的可行性和有效性.第六章总结归纳了本文所做的工作并对将来的工作加以展望.
[Abstract]:Fractional calculus is called a new mathematical tool which combines the real world and mathematical theory perfectly. It is widely used in various fields such as viscoelastic mechanics, statistics and stochastic processes, signal analysis and processing, etc. Fractional calculus equations can describe many complex physical processes more realistically and objectively than integer order calculus equations and become one of the important tools for mathematical modeling of complex mechanics and physical processes. Therefore, the research on the theory and calculation method of fractional calculus equation is particularly urgent and plays an important role in the application field. Unfortunately, the analytical solutions of most fractional integro-differential equations are very complex, difficult to calculate and consume a lot of time. Moreover, not all fractional calculus equations can obtain their analytical solutions. Therefore, it is very necessary to develop a new numerical algorithm and establish the numerical method of fractional calculus equation, which has important theoretical significance and practical application value. In this paper, several kinds of Riemann-Liouville fractional calculus equations are solved. The first chapter briefly introduces the research background, research significance and research status at home and abroad. In chapter 2, the fractional integral operator matrix of semi-orthogonal B-spline wavelet is derived. In chapter 3, we prove the existence and uniqueness of the solution of the second kind of fractional Fredholm integral equation, and use semi-orthogonal B-spline wavelet to solve the numerical solution of the second kind of fractional Fredholm integral equation. An example is given to give the numerical solution for the equation with unknown exact solution. In chapter 4, we prove the existence and uniqueness of the solution of the second kind of fractional Fredholm integral equations, and use semi-orthogonal B-spline wavelet to solve the numerical solution of the second kind of fractional Fredholm integral equations. The error analysis is given for the unknown exact solution. In chapter 5, the integral differential equations are discretized into algebraic equations by B-spline wavelet fractional integral operator matrix of fractional nonlinear Fredholm integro-differential equations. Numerical examples demonstrate the feasibility and validity of this method. The sixth chapter summarizes the work done in this paper and looks forward to the future work.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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