小波方法求解三类变分数阶微积分问题研究

发布时间：2018-02-10 19:26

本文关键词： 变分数阶微分方程 Legendre小波数值解小波去噪多项式拟合数值微分　出处：《燕山大学》2015年硕士论文　论文类型：学位论文

【摘要】：随着分数阶微积分在理论和实际应用上的迅速发展,变分数阶微积分的相关研究逐渐崭露头角。变分数阶微积分不仅在理论上给人们以巨大的研究空间,而且在实际建模和应用上也蕴藏着相当大的潜力。因此,这一领域相关问题的研究价值是十分显著的。由于变分数阶微积分算子中的变阶指数的存在,此类问题的推导和计算较之于传统的分数阶微积分就更为困难。随着变分数阶微积分的发展,开展相关高效数值计算方法的研究就尤为重要。变分数阶微积分算子的数值计算以及变分数阶微积分方程的数值求解就是论文要研究的其中两个重要课题。小波分析经过数十年的发展,其理论基础已经相当深厚,而且在实际应用中也发挥着强大的威力。小波变换已被广泛应用于解决信号和图像的分解、重构、增强、去噪等问题。此外,正交小波基函数的良好性质结合函数逼近理论也可以很好地解决一些数值计算问题。论文结合以上两类小波分析的应用对文中课题开展研究工作。首先,论文对变分数阶微积分的发展历程和研究现状作了介绍,接着介绍了小波分析的历史背景与研究现状。然后论文简要给出了变分数阶微积分和小波分析的相关基础知识。另外,这一部分重点介绍了Legendre小波函数的基本定义和性质。其次,在3、4章中,论文利用Legendre小波函数导出相关的变分数阶微分算子矩阵,分别求解了变分数阶常微分方程和变分数阶扩散方程的数值解。在第3章中,除给出了问题求解的新算法外,还给出了与有限差分算法的计算比较。在第4章中,在应用所给算子矩阵改写原方程外,还针对变分数阶扩散方程的特征结合Legendre小波函数的分段特征,对原方程的初始条件进行了特别处理。通过数值算例,说明两章所提的数值算法是有效可行的。最后,在第5章中,论文结合小波去噪方法与多项式曲线拟合方法求解了含噪信号的变分数阶数值微分,数值算例表明了所给算法是简洁有效的,同时也给出了与已有相关文献结果的比较。
[Abstract]:With the rapid development of fractional calculus in theory and practical application, the related research of variable fractional calculus is gradually emerging. Moreover, it has great potential in practical modeling and application. Therefore, the research value of the related problems in this field is very significant, because of the existence of variable-order exponents in variable-fractional calculus operators, The derivation and calculation of this kind of problem is more difficult than the traditional fractional calculus. It is very important to study the related and efficient numerical calculation methods. The numerical calculation of variational fractional calculus operator and the numerical solution of variational fractional calculus equation are two important topics to be studied in this paper. After decades of development, Wavelet transform has been widely used to solve the problems of signal and image decomposition, reconstruction, enhancement, denoising and so on. The good properties of orthogonal wavelet basis function and the approximation theory of function can also solve some problems of numerical calculation. This paper introduces the history and research status of variable fractional calculus, and then introduces the historical background and research status of wavelet analysis. Then the basic knowledge of variable fractional calculus and wavelet analysis is briefly given. In this part, the basic definition and properties of Legendre wavelet function are introduced. Secondly, in chapter 3, we use Legendre wavelet function to derive the related fractional differential operator matrix. The numerical solutions of variable fractional ordinary differential equations and variable fractional diffusion equations are solved, respectively. In chapter 3, in addition to the new algorithm for solving the problem, a comparison with the finite difference algorithm is given. In addition to rewriting the original equation by using the given operator matrix, the initial conditions of the original equation are specially treated according to the characteristics of the variational fractional diffusion equation and the piecewise characteristic of the Legendre wavelet function. Finally, in chapter 5, combining wavelet denoising method and polynomial curve fitting method, the variational fractional numerical differential of noisy signal is solved. Numerical examples show that the proposed algorithm is simple and effective.
【学位授予单位】：燕山大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O172

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