本文選題：時間分?jǐn)?shù)階擴(kuò)散方程 + 未知源識別��； 參考：《蘭州理工大學(xué)》2017年碩士論文

【摘要】：本文考慮兩類時間分?jǐn)?shù)階擴(kuò)散方程反問題,分別是未知源識別問題和反演初值問題.這兩類問題都是不適定問題,它們的解(如果存在)不連續(xù)依賴于測量數(shù)據(jù).本文第二章考慮一般有界區(qū)域上變系數(shù)時間分?jǐn)?shù)階擴(kuò)散方程未知源識別問題,這是一個不適定問題.本文采用Landweber迭代法恢復(fù)問題的不適定性,并且給出在先驗(yàn)和后驗(yàn)兩種正則化參數(shù)選取規(guī)則下的收斂性估計(jì).最后,數(shù)值實(shí)驗(yàn)表明本文所用的Landweber迭代法對解決此類問題是有效的.第三章討論一般有界區(qū)域上變系數(shù)齊次時間分?jǐn)?shù)階擴(kuò)散方程反演初值問題,這是一個不適定問題.本文應(yīng)用Landweber迭代法求解該問題,并且證明在先驗(yàn)和后驗(yàn)兩種正則化參數(shù)選取規(guī)則下的收斂性估計(jì).最后,數(shù)值結(jié)果表明Landweber迭代法對解決此類問題是有效和穩(wěn)定的.第四章考慮高維變系數(shù)非齊次時間分?jǐn)?shù)階擴(kuò)散方程反演初值問題,這個問題是不適定的.本文利用擬逆方法得到問題的正則解,并且給出先驗(yàn)和后驗(yàn)兩種正則化參數(shù)選取規(guī)則下精確解與正則解之間的收斂性估計(jì).最后,在一維和二維兩種情形下的數(shù)值例子表明擬逆方法對解決此類問題是有效的.其中,本文所解決的變系數(shù)非齊次時間分?jǐn)?shù)階擴(kuò)散方程反演初值問題,是一個比較新穎的不適定問題.本文中的理論結(jié)果和數(shù)值結(jié)果都能充分有效的說明所采用的正則化方法能夠很好地求解給定的不適定問題.
[Abstract]:In this paper, we consider two kinds of inverse problems of fractional diffusion equations in time, which are unknown source identification problem and inverse initial value problem, respectively. Both of these problems are ill-posed problems whose solutions (if any) are discontinuous dependent on measured data. In chapter 2, we consider the problem of identifying unknown sources of time fractional diffusion equations with variable coefficients in a general bounded domain, which is an ill-posed problem. In this paper, the Landweber iterative method is used to restore the ill-posed property of the problem, and the convergence estimates are given under the rules of the selection of priori and posteriori regularization parameters. Finally, numerical experiments show that the Landweber iterative method used in this paper is effective in solving such problems. In chapter 3, we discuss the inverse initial value problem of homogeneous time fractional diffusion equations with variable coefficients in a general bounded domain, which is an ill-posed problem. In this paper, the Landweber iterative method is used to solve the problem, and the convergence estimates are proved under the rules of the priori and posteriori regularization parameter selection. Finally, the numerical results show that the Landweber iterative method is effective and stable for solving such problems. In chapter 4, we consider the inverse initial value problem of nonhomogeneous time fractional diffusion equation with high dimensional variable coefficient, which is ill-posed. In this paper, we obtain the regular solution of the problem by using the quasi-inverse method, and give the convergence estimates between the exact solution and the regular solution under the rule of selecting two regularization parameters a priori and a posteriori. Finally, numerical examples in one and two dimensions show that the quasi-inverse method is effective for solving this kind of problem. The inhomogeneous time fractional diffusion equation with variable coefficients is a novel ill-posed problem. The theoretical and numerical results in this paper can fully and effectively illustrate that the regularization method used in this paper can solve the given ill-posed problem well.
【學(xué)位授予單位】：蘭州理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.8

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本文編號：1875774