【摘要】：Haar小波由一系列分段常函數(shù)組成,是具有緊支集的最簡(jiǎn)單的正交小波,而斜坡Haar小波是對(duì)Haar小波通過(guò)變形得到的新小波,它由分段函數(shù)組成,仍具有類似Haar小波良好的性質(zhì),已應(yīng)用于圖像處理、信號(hào)處理、密碼學(xué)等工程方面.本文利用斜坡Haar小波求解發(fā)展方程,為微分方程的求解提供了新的工具。本文首先根據(jù)斜坡Haar小波的定義,研究了斜坡Haar小波積分公式,為數(shù)值求解發(fā)展方程做了準(zhǔn)備工作;其次研究了一維熱傳導(dǎo)方程和波動(dòng)方程的斜坡Haar小波數(shù)值解,其方法是將出現(xiàn)在方程中的最高階微分用斜坡Haar小波表示,其它的微分和函數(shù)通過(guò)多次積分獲得,把所有這些積分代入方程系統(tǒng),然后離散化,得線性代數(shù)方程組,使用Matlab進(jìn)行數(shù)值模擬;最后,使用斜坡Haar小波求解了二維拋物型方程和波動(dòng)方程,二維熱傳導(dǎo)方程作為拋物型方程的特例研究,使用張量積二維斜坡Haar函數(shù)把出現(xiàn)在方程中的最高階偏微分展開成二維斜坡Haar小波,然后多次積分其表達(dá)式,最后類似一維的方法得線性方程組,用Matlab數(shù)值求解.在一維和二維情形中,數(shù)值的結(jié)果與精確解進(jìn)行了比對(duì),說(shuō)明了本文所采用的方法的有效性和可行性。
[Abstract]:Haar wavelet is composed of a series of piecewise constant functions, which is the simplest orthogonal wavelet with a compact set, and the slope Haar wavelet is a new wavelet which is obtained by the deformation of Haar wavelet. It is composed of a segmentation function, which still has a good property like Haar wavelet and has been applied to image processing. Signal processing, cryptography, and other engineering aspects. This paper uses the slope Haar wavelet to solve the development equation, and provides a new tool for solving the differential equation. In this paper, based on the definition of the Haar wavelet of the slope, the Haar wavelet integral formula of the slope is studied, and the preparation for the numerical solution of the development equation is studied. Secondly, the numerical solution of the slope Haar wavelet of the one-dimensional heat conduction equation and the wave equation is studied. The method is that the most high order differential slope Haar wavelet which appears in the equation is represented by a slope Haar wavelet, and the other differential and function are obtained through multiple integral, and all the points are substituted into the equation system, then the linear algebraic equations are obtained, and the numerical simulation is carried out by using Matlab; and finally, The two-dimensional parabolic equation and the wave equation are solved by using the slope Haar wavelet, and the two-dimensional heat conduction equation is used as a special case study of the parabolic equation. The two-dimensional slope Haar function of the tensor product is used to expand the highest order partial differential appearing in the equation into two-dimensional slope Haar wavelet. The expressions are then integrated many times, and the system of linear equations is obtained by using the method of Matlab. In a two-dimensional and two-dimensional situation, the result of the numerical value is compared with the exact solution, and the effectiveness and feasibility of the method adopted in this paper are described.
【學(xué)位授予單位】：西安建筑科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.8

【參考文獻(xiàn)】

相關(guān)期刊論文 前7條

1 岳彥剛;石智;張卓;;基于Shearlet變換的圖像邊緣檢測(cè)[J];計(jì)算機(jī)應(yīng)用與軟件;2014年04期

2 王苗苗;趙鳳群;李娜;劉麗;;分?jǐn)?shù)階微分方程的Haar小波算法研究[J];計(jì)算力學(xué)學(xué)報(bào);2013年01期

3 石智;張卓;岳彥剛;;基于Shearlet變換的自適應(yīng)圖像融合算法[J];光子學(xué)報(bào);2013年01期

4 權(quán)豫西;石智;;小波伽遼金方法應(yīng)用于變系數(shù)波動(dòng)方程[J];蘭州大學(xué)學(xué)報(bào)(自然科學(xué)版);2008年05期

5 石智;鄧麗媛;;求解對(duì)流擴(kuò)散方程的Haar小波方法(英文)[J];應(yīng)用數(shù)學(xué);2008年01期

6 石智;葉中華;張馳;;長(zhǎng)方形區(qū)域上非分離非均勻Haar小波[J];純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué);2007年03期

7 呂桂霞;馬富明;;二維熱傳導(dǎo)方程有限差分區(qū)域分解算法[J];數(shù)值計(jì)算與計(jì)算機(jī)應(yīng)用;2006年02期

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本文編號(hào)：2491847