发展方程的斜坡Haar小波函数数值解
发布时间：2019-06-03 09:35
【摘要】：Haar小波由一系列分段常函数组成,是具有紧支集的最简单的正交小波,而斜坡Haar小波是对Haar小波通过变形得到的新小波,它由分段函数组成,仍具有类似Haar小波良好的性质,已应用于图像处理、信号处理、密码学等工程方面.本文利用斜坡Haar小波求解发展方程,为微分方程的求解提供了新的工具。本文首先根据斜坡Haar小波的定义,研究了斜坡Haar小波积分公式,为数值求解发展方程做了准备工作;其次研究了一维热传导方程和波动方程的斜坡Haar小波数值解,其方法是将出现在方程中的最高阶微分用斜坡Haar小波表示,其它的微分和函数通过多次积分获得,把所有这些积分代入方程系统,然后离散化,得线性代数方程组,使用Matlab进行数值模拟;最后,使用斜坡Haar小波求解了二维抛物型方程和波动方程,二维热传导方程作为抛物型方程的特例研究,使用张量积二维斜坡Haar函数把出现在方程中的最高阶偏微分展开成二维斜坡Haar小波,然后多次积分其表达式,最后类似一维的方法得线性方程组,用Matlab数值求解.在一维和二维情形中,数值的结果与精确解进行了比对,说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.
【学位授予单位】：西安建筑科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2491847
[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.
【学位授予单位】：西安建筑科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
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