【摘要】：信號(hào)處理、流體力學(xué)、控制理論等很多領(lǐng)域中的現(xiàn)象都能用分?jǐn)?shù)階積分微分方程描述,但此類(lèi)方程解析解的求解非常困難,因此相關(guān)領(lǐng)域研究者們將目光投向了對(duì)其數(shù)值解的研究上.目前求解分?jǐn)?shù)階積分微分方程的數(shù)值解法有很多,如有限元法、同倫攝動(dòng)法、Adomain分解法等,而將小波方法應(yīng)用于求解分?jǐn)?shù)階積分微分方程的文獻(xiàn)則相對(duì)較少.本文考慮用Bernoulli小波方法求解幾類(lèi)分?jǐn)?shù)階積分微分方程(組)的數(shù)值解.本文共分為六章.第一章對(duì)分?jǐn)?shù)階微積分的研究意義及分?jǐn)?shù)階積分微分方程數(shù)值解法的國(guó)內(nèi)外研究現(xiàn)狀進(jìn)行了概述.第二章簡(jiǎn)要介紹了分?jǐn)?shù)階微積分和Bernoulli小波的基本理論,推導(dǎo)了Bernoulli小波的乘積算子矩陣和分?jǐn)?shù)階積分算子矩陣.第三章利用Bernoulli小波的分?jǐn)?shù)階積分算子矩陣分別求解了非線性分?jǐn)?shù)階Fredholm積分微分方程、線性和非線性分?jǐn)?shù)階Fredholm積分微分方程組的數(shù)值解并證明了其解的存在唯一性.另外,從理論上證明了 Bernoulli小波方法求解此類(lèi)方程的收斂性.第四章利用Bernoulli小波的分?jǐn)?shù)階積分算子矩陣求解了線性和非線性分?jǐn)?shù)階Fredholm-Volterra積分微分方程以及弱奇異分?jǐn)?shù)階積分微分方程,數(shù)值算例說(shuō)明了此方法求解這幾類(lèi)方程的可行性.第五章利用Bernoulli小波的分?jǐn)?shù)階積分算子矩陣求解了微分項(xiàng)中階數(shù)不固定且滿(mǎn)足一定初始條件的非線性分?jǐn)?shù)階Volterra積分微分方程、分?jǐn)?shù)階Volterra積分微分方程組并對(duì)其收斂性給出了證明,數(shù)值算例說(shuō)明了該方法的有效性和準(zhǔn)確性.第六章對(duì)全文所做的工作進(jìn)行了總結(jié)并對(duì)今后進(jìn)一步的研究提出展望.
[Abstract]:The phenomena in signal processing, fluid mechanics, control theory and many other fields can be described by fractional integro-differential equations, but it is very difficult to solve the analytical solutions of such equations. Therefore, researchers in related fields have focused on the study of its numerical solution. At present, there are many numerical methods for solving fractional integrodifferential equations, such as finite element method, homotopy perturbation method, Adomain decomposition method, etc. In this paper, Bernoulli wavelet method is used to solve several kinds of fractional integrodifferential equations (systems). This paper is divided into six chapters. In the first chapter, the significance of fractional calculus and the present situation of numerical solution of fractional integrodifferential equation are summarized. In chapter 2, the basic theories of fractional calculus and Bernoulli wavelet are briefly introduced, and the product operator matrix and fractional integral operator matrix of Bernoulli wavelet are derived. In chapter 3, by using the fractional integral operator matrix of Bernoulli wavelet, the numerical solutions of nonlinear fractional Fredholm integrodifferential equations, linear and nonlinear fractional Fredholm integrodifferential equations are solved, and the existence and uniqueness of the solutions are proved. In addition, the convergence of Bernoulli wavelet method for solving this kind of equation is proved theoretically. In chapter 4, the linear and nonlinear fractional Fredholm-Volterra integrodifferential equations and weakly singular fractional integral differential equations are solved by using the fractional integral operator matrix of Bernoulli wavelet. Numerical examples show that this method is feasible to solve these kinds of equations. In chapter 5, we use the fractional integral operator matrix of Bernoulli wavelet to solve the nonlinear fractional Volterra integrodifferential equations with uncertain order and satisfying certain initial conditions. The convergence of fractional Volterra integro-differential equations is proved. Numerical examples show the effectiveness and accuracy of the method. The sixth chapter summarizes the work done in this paper and puts forward the prospect of further research in the future.
【學(xué)位授予單位】：寧夏大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.8

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