本文選題：B樣條小波 + 分數(shù)階微積分�。� 參考：《寧夏大學》2017年碩士論文

【摘要】：分數(shù)階微積分被稱為現(xiàn)實世界和數(shù)學理論完美結(jié)合的一種嶄新的數(shù)學工具,被廣泛應(yīng)用到粘彈性力學、統(tǒng)計與隨機過程、信號分析處理等各個不同的領(lǐng)域.分數(shù)階微積分方程比整數(shù)階微積分方程更能真實客觀地刻畫許多復雜的物理過程,成為復雜力學與物理過程數(shù)學建模的重要工具之一.因此,有關(guān)分數(shù)階微積分方程的理論與計算方法的研究就顯得尤為迫切,在應(yīng)用領(lǐng)域起著至關(guān)重要的作用.遺憾的是,大部分分數(shù)階積分微分方程的解析解很復雜,計算困難,消耗大量時間;況且并非所有的分數(shù)階微積分方程都能得到其解析解.因此,發(fā)展新數(shù)值算法,建立分數(shù)階微積分方程的數(shù)值方法是非常必要的,有著重要的理論意義和實際應(yīng)用價值.本文主要求解了幾類Riemann-Liouville分數(shù)階微積分方程.第一章簡要介紹了研究背景、研究意義及國內(nèi)外研究現(xiàn)狀.第二章推導了半正交B樣條小波的分數(shù)階積分算子矩陣.第三章證明了第二類分數(shù)階Fredholm積分方程解的存在唯一性,利用半正交B樣條小波求解了第二類分數(shù)階Fredholm積分方程的數(shù)值解.算例針對精確解未知的方程給出其數(shù)值解.第四章證明了第二類分數(shù)階Fredholm積分方程組解的存在唯一性,利用半正交B樣條小波求解第二類分數(shù)階Fredholm積分方程組的數(shù)值解.并針對精確解未知的情況給出誤差分析.第五章研究了分數(shù)階非線性Fredholm積分微分方程,B樣條小波分數(shù)階積分算子矩陣將積分微分方程離散為代數(shù)方程組,數(shù)值算例驗證了此方法的可行性和有效性.第六章總結(jié)歸納了本文所做的工作并對將來的工作加以展望.
[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.
【學位授予單位】：寧夏大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.8

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