A Study Of Wavelet Method For Solving Three Kinds Of Problems Of Variable-order Fractional Calculus

With rapid development of fractional calculus in the theory and practical applications, researches on fractional calculus gradually emerge. Variable-order fractional calculus not only provides people with a great research space in theory sense, but also has a great potential in practical modeling and applications. Therefore, the research value of this field is very significant.Due to the existence of the variable-order fractional calculus operators, the deduction and calculation of these kinds of problems is more difficult compared with the traditional fractional calculus. Along with the development of the variable-order fractional calculus, the researches of the relevant efficient numerical methods are especially important. The numerical calculation of the variable-order fractional calculus operators and the numerical solutions of the fractional calculus equations are two vital topics in this paper.With decades of development, the theoretical basis of wavelet analysis has been very profound. Wavelet transform has been widely applied to solve various kinds of problems in the processing of signals and images, such as decomposition, reconstruction, enhancement, denoising etc. Additionally, the good properties of the orthogonal wavelet bases combined with the theory of function approximation can be efficiently applied to numerical calculation. In this paper, the above two types of applications of wavelet analysis are considered under the background of variable-order fractional calculus.Firstly, this paper introduces the development history and research status of variable-order fractional calculus. Then, the research background and research status of wavelet analysis are also provided. Moreover, the paper gives a brief introduction of the basic knowledge of variable-order fractional calculus and wavelet analysis. Above all, this part introduces the basic definitions and properties of Legendre wavelets functions.Secondly, the paper adopts the Legendre wavelets functions to derive the operational matrices of variable-order fractional differential operators. Accordingly, the numerical solutions of the variable-order fractional ordinary differential equation and the variable-order fractional diffusion equation are found in Chapter 3 and Chapter 4 respectively. In Chapter 3, a comparison between the finite difference method and the proposed method is also given. In Chapter 4, not only the original equations are rewritten to algebraic equations by using the provided operational matrices, but also the corresponding boundary conditions are transformed in similar forms based on the piecewise property of Legendre wavelets. The numerical examples verify that the given algorithms are feasible and valid in the two chapters.Finally, in Chapter 5, the paper combines the wavelet denoising and the polynomial curve fitting method to calculate the variable-order fractional numerical differentiation of the noisy signal. The numerical results show that the algorithm is simple and effective and the comparison of the results with the existing literature is also given.